From martin at eipye.com  Sun Mar  1 11:39:34 2015
From: martin at eipye.com (Martin Davis)
Date: Sun, 1 Mar 2015 08:39:34 -0800
Subject: [FOM] Fwd: Kreisel
In-Reply-To: <54F30443.4060206@sbcglobal.net>
References: <54F30443.4060206@sbcglobal.net>
Message-ID: <CA+cpueKdM=5Yozjx50rPSf2my-RGLhSWWEMxB9N9DeSxUpqOcw@mail.gmail.com>

---------- Forwarded message ----------
From: Kenneth Derus <khderus at sbcglobal.net>
Date: Sun, Mar 1, 2015 at 4:21 AM
Subject: Kreisel
To: davism at cs.nyu.edu


Georg Kreisel died in Salzburg at 2:30a local time.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150301/734b7925/attachment.html>

From dmytro at mit.edu  Sun Mar  1 12:45:59 2015
From: dmytro at mit.edu (Dmytro Taranovsky)
Date: Sun, 01 Mar 2015 12:45:59 -0500
Subject: [FOM] Set Theory with Reflective Sequences
Message-ID: <54F35057.8020807@mit.edu>

Within the language of set theory, one reaches higher and higher 
expressive power by climbing higher in the cumulative hierarchy of V.  
But how can we go further once the language allows quantification over 
the whole V?  Intuitively, we would want to continue the hierarchy above 
V, except that all sets are already in V.  The solution is to label a 
cardinal kappa such that V_kappa is sufficiently close to V, and 
continue the hierarchy above V_kappa. V_kappa represents V, and with 
kappa labeled in an extended language, hierarchy above V_kappa 
corresponds to higher order set theory.

To go further, we can iterate higher order set theory by picking 
lambda>kappa with V_lambda representing V, and then mu>lambda, and 
continuing to longer sequences of ordinals.  Thus, continuing the 
cumulative hierarchy above V corresponds to labeling certain ordinals 
that are sufficiently similar to Ord -- in other words, certain ordinals 
with sufficiently strong reflection properties -- while staying within V.

How do we choose the right kappa?  The answer is that we postulate a 
certain degree of symmetry and reflection in V.
Convergence Hypothesis (general form): For an appropriate type of 
objects, all objects of that type with sufficient reflection properties 
are, in a certain sense, indistinguishable from each other.
Convergence Hypothesis (ordinals): If alpha and beta are ordinals with 
sufficiently strong reflection properties, then phi(S, alpha) <==> 
phi(S, beta) whenever rank(S) < min(alpha, beta) and phi is a first 
order formula of set theory with two free variables.
Definition: kappa is a reflective cardinal, denoted by R(kappa), iff 
(V,in,kappa) has the same theory with parameters in V_kappa as 
(V,in,lambda) for every cardinal lambda>kappa with sufficiently strong 
reflection properties.
Example:  In L (assuming zero sharp), every Silver indiscernible has 
sufficient reflection properties, which allows us to define R^L in V.
Note:  A formalist can treat the Convergence Hypothesis as a guiding 
principle to get good systems of axioms.

To the extent that it holds, the Convergence Hypothesis allows us to 
define new reflective notions without ambiguity.  We specify a notion by 
stating its type (example: a pair of ordinals) along with the class of 
predicates for which it agrees with all objects of sufficient reflection 
properties.  See "Reflective Cardinals" (2012, 
http://arxiv.org/abs/1203.2270 ) for analysis, axiomatization, and 
theory of reflective cardinals.  Here, I will introduce the theory of 
infinite reflective sequences.

Definition: Given a length condition P (such as having length omega), an 
increasing sequence of ordinals S satisfying P is reflective iff
   (1) the theory of (V, in, S) with parameters in V_min(S) is correct, 
that is it agrees with (V, in, T) for every T satisfying P and having 
sufficient reflection properties and min(T)>min(S), and
   (2) for every alpha<sup(S) S\alpha is reflective under the above 
criterion (where the length condition is modified to correspond to 
S\alpha if S\alpha is "shorter" than S).
A variation on the notion is to omit the second condition.

To understand the hierarchy of reflective sequences, imagine a process 
of picking ordinals with sufficient reflection properties, and with the 
process itself having sufficient reflection properties.  Let S be a 
resulting sequence.  Here are some stages in the process:

1. n ordinals for finite n.  This is well-understood (see "Reflective 
Cardinals" for theory).
2. omega ordinals.  See "Reflective Cardinals" for some results. This 
notion contradicts V=HOD (assuming that removing the first element does 
not change its theory), which makes analysis and finding of canonical 
models difficult.  It may be related to Prikry forcing.
3. alpha ordinals for countable alpha.  We expect reflectiveness to be 
closed under subsequences that preserve order type.
4. omega_1 ordinals.  Assuming the Continuum Hypothesis (CH), we cannot 
require closure of reflectiveness under all subsequences of the same 
length. Even without CH, this limitation holds at omega_2.
5. alpha ordinals for a fixed ordinal alpha > omega_1 (min(S)>alpha).
6. More complicated conditions about the length of S, such as 
|S|=min(S). S excludes its limit points.
7. Length of S has sufficiently strong reflection properties relative to 
S. S excludes its limit points.
8. To go beyond (7), we assume that S includes its limit points at 
places with sufficient (relative to S) degree of reflectiveness. One may 
then state conditions on the length of S such as "S includes exactly 
omega limit points, and they are cofinal in S" or "S has maximum element 
and it is the only element such that S below it is stationary".
9. Length of S has sufficiently strong reflection properties relative to S.

To go beyond what is expressible with S, we mark some points on S that 
have sufficient (relative to S) degree of reflectiveness, analogously to 
how S is obtained by marking some ordinals with sufficient 
reflectiveness in V.  To go further, we then add a third type of 
marking, and in general, for each ordinal in S, we can assign a degree 
of reflectiveness through a function f:S?Ord\{0}. For a limit alpha, 
f(x)=alpha corresponds to x receiving all markings <alpha.  In the 
definition of reflectiveness, use f in place of S, min(dom(f)) in place 
of min(S) (and min(dom(T)) in place of min(T)), sup(dom(f)) in place of 
sup(S), and in place of S\alpha, use f restricted to Ord\alpha modified 
to make f(min(dom(f)\alpha)) equal 1.  Here are some notions. (Here 
S=dom(f).)

10. For a fixed ordinal alpha, f(sup(S))=alpha, and sup(S) is the only 
ordinal s with f(s)=alpha (and min(dom(f))>alpha).
11. f(sup(S))=sup(S), and sup(S) is the only ordinal kappa with 
f(kappa)=kappa.
12. To go beyond f(kappa)=kappa, we modify the condition -- that 
f(kappa)=alpha+1 implies that kappa has sufficiently strong reflection 
properties relative to f that is clipped to alpha -- by invoking an 
ordinal notation system (for ordinals >=kappa) or just a comparison 
method between f(lambda) and f(kappa), and considering f clipped below 
f(kappa) as defined through the comparison method.  A reasonable 
stopping point is the following: There is only one ordinal kappa with 
f(kappa)=(kappa^+)^HOD, and it equals sup(dom(f)).

An axiomatization of notion 9 and notion 12 is in my paper (linked 
below).  The axiomatizations avoid the inconsistency in my FOM posting 
"Axiomatization of Reflective Sequences" (12 Feb 2015), but their 
consistency remains unclear.  Here I will note a connection with inner 
model theory.

The type of f and many of its basic properties can be formalized using 
inner models.  For example, for notion 11 we can require that there is 
an inner model M of ZFC such that
  - M is an iterate of the minimal inner model with o(kappa)=kappa for 
some kappa (o refers to Mitchell order)
  - dom(f) is the set of measurable cardinals in M
  - for every kappa in the domain of f, f(kappa)=o(kappa)^M

The possibilities for iteration appear sufficiently rich to get any 
reasonable candidate for notion 11.  Also, one can similarly specify M 
for other notions.

If a function f satisfies the condition on M, then the theory of 
(L(f),in,f) is independent of f, which is one example of the Convergence 
Hypothesis, and moreover, we can study this theory to get a better 
understanding of f.

It is unclear how far the Convergence Hypothesis extends in V, and if it 
reaches that far, how to handle M with overlapping extenders.

For more details, see my paper:
http://web.mit.edu/dmytro/www/ReflectiveSequences.htm

Sincerely,
Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm

From robert at moir.net  Sun Mar  1 21:07:31 2015
From: robert at moir.net (Robert Moir)
Date: Sun, 1 Mar 2015 21:07:31 -0500
Subject: [FOM] Extended Deadline: ACMES Conference
Message-ID: <CAFRG=DWXZZ2Nj-fQ5BXy5xy7FDVK5TTJL8H9Mbfg4D8bQrU+6g@mail.gmail.com>

/* Abstract submission deadline extended to March 8 */
/* Apologies for cross-posting */

Conference on
Algorithms and Complexity in Mathematics, Epistemology and Science (ACMES)

London, Ontario, Canada, May 6-8, 2015

http://acmes.org

Philosophers of science are of course concerned with the reliability of
scientific inference and the use of mathematical models throughout
scientific practice. Increasingly, philosophers are becoming concerned
with the reliability of the methods of computation scientific methods
rely upon. This conference seeks a richer understanding by blending talks
by philosophers of science with talks by computational scientists, who
have developed many tools and techniques for the assessment of reliability.

We invite contributions to the study of reliability of methods of
scientific modeling, computing, and inference. Works in progress and
completed research projects are both acceptable.

Graduate students are particularly encouraged to contribute and attend.

ACMES will be held at Western University in London, ON, Canada from
May 6-8 2015 and will be held in conjunction with Southern Ontario
Numerical Analysis Day (SONAD).

// Invited Talks //

Confirmed Invited Speakers are:
  Philosophy of Science:
   * Anouk Barberousse (Universit? Lille 1, France)
   * Robert Batterman (University of Pittsburgh, USA)
   * Mark Wilson (University of Pittsburgh, USA)
  Computing Science
   * Max Gunzburger (Florida State University, USA)
   * Ursula Martin (Oxford University, UK)
   * Jes?s Sanz-Serna (Universidad Carlos III de Madrid, Spain)

// Important Dates //

   * Submission: March 8, 2015
   * Notification: March 15, 2015
   * Workshop: May 6-8, 2015

// Travel Support //

Limited travel support will be available to graduate student
contributors in philosophy and computational sciences.

// Submission Instructions //

Please submit an abstract (150-200 words) of your proposed talk to
the conference email below. The time allotted for contributed talks
will be 30 minutes including 5 minutes for questions. Supplementary
material may be submitted, and will be considered at the discretion
of the Program Committee.

acmes.conference at gmail.com

// Publication //

Depending on interest we may pursue opportunities for publication
in a high quality journal or series volume. In such an event,
a request will be made to selected contributors to submit full
papers, due at a date to be determined following the conference.

// Program Committee //

   * Robert Corless, Western (Applied Math) (Co-chair)
   * Nicolas Fillion, SFU (Philosophy) (Co-chair)
   * Chris Smeenk, Western (Philosophy) (Co-chair)
   * Robert Moir, Western (Applied Math)

// Poster //

A conference poster for distribution and display is available
here:

http://acmes.org/poster/acmes.pdf

// Funding //

ACMES is made possible by grants from the Fields Institute and the
Rotman Institute of Philosophy.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150301/fb8c14c0/attachment-0001.html>

From tennant.9 at osu.edu  Mon Mar  2 10:57:48 2015
From: tennant.9 at osu.edu (Tennant, Neil)
Date: Mon, 2 Mar 2015 15:57:48 +0000
Subject: [FOM] Translation of a paper by Maksimova?
Message-ID: <3188F1ACFDF24246BE3EB20656F10C5D3F2E1F7C@CIO-TNC-D2MBX04.osuad.osu.edu>

Does any fom-er know of an English translation of this paper?:

L. L. MAKSIMOVA [1977], The Craig theorem in superintuitionistic logics and amalgamated varieties of pseudo-Boolean algebras, Algebra and Logic 16 (1977), pp.
643-681 (in Russian).

There is textual reason to believe that her paper

L. L. MAKSIMOVA [1979], Interpolation properties of superintuitionistic logics, Studia
Logica 38 (1979), pp. 419-128

is not a translation of it.

Neil Tennant
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150302/953b9c89/attachment-0001.html>

From andrea.sereni at iusspavia.it  Mon Mar  2 11:10:44 2015
From: andrea.sereni at iusspavia.it (Andrea Sereni)
Date: Mon, 02 Mar 2015 17:10:44 +0100
Subject: [FOM] Origins and Varieties of Logicism - Workshop - IUSS Pavia -
 March 16th, 2015
Message-ID: <54F48B84.3030908@iusspavia.it>

*/Origins and Varieties o//f Logicism/*/
/*
**//Workshop//
//***/
/March 16th, 2015
*Institute for Advanced Study, IUSS, Pavia
Sala del Camino*

NEtS (Neurocognition, Epistemology and Theoretical Syntax) Centre @ IUSS
IUSS \ San Raffaele PhD Program in /Cognitive Neurosciences and 
Philosophy of Mind/

/Under the auspices of//
/Italian Network for the Philosophy of Mathematics, FilMat

Among traditional foundational programs in the philosophy of 
mathematics, logicism still attracts much attention. In recent times, 
also thanks to renewed interest in this philosophical endeavour, we 
witness a revival of studies on the origins of logicism and the various 
shapes it may assume, also in its relation to apparently rival views 
such as structuralism. By focusing on recent works on Frege, Dedekind, 
Peano, Russell, and Carnap, the workshop aims at both an historical and 
theoretical exploration of the main figures in the philosophical and 
mathematical milieu in which logicist views were first expounded, in 
order to shed new light on their legacy for contemporary philosophy of 
mathematics.


/*_Program_*//*_
_*/*
* 10.00-11.15
*Marco Panza* (CNRS, IHPST & Paris 1 Sorbonne)
/What makes Frege's and Dedekind's logicisms deeply different?////
/
11.15-12.30
*Erich **Reck* (University of California, Riverside)
/Dedekind's logicism reconsidered///

12.30-13.30 Lunch break

13.30-14.45
*Paola **Cant?* (Universit? Aix-Marseille / CNRS; CEPERC UMR 7304)
/Peano and his school between logicism and algebrism//
/
14.45-16.00
*Gre**gory **Landini* (University of Iowa)
/The Logic of Russell's Logicism////
/
16.00-16.30 Coffee break

16.30-17.45
*Georg Schiemer* (University of Vienna)
/Carnap on Logicism and Structuralism//
/
17.45-18.30
/Open discussion and Q&A//
/The discussion will be opened by: *
Eva Picardi* (University of Bologna)


/Invited discussants/
Massimilano Carrara (University of Padua)
Ciro De Florio (Catholic University, Milan)
Miriam Franchella (University of Milan)
Alessandro Giordani (Catholic University, MIlan)
Daniele Molinini (University of Roma Tre)
Francesco Orilia (University of Macerata)
Claudio Ternullo (University of Vienna, Kurt G?del Research Center for 
Mathematical Logic)



/Organized by://
/Francesca Boccuni (San Raffaele University, Milan)
Andrea Sereni (Institute for Advanced Study, IUSS, Pavia)


/Workshop Venue/
Institute for Advanced Study, IUSS, Pavia
Sala del Camino
Palazzo del Broletto
Piazza della Vittoria, 15
27100, Pavia, Italy


Everyone is invited

________________________________________________
How to reach the workshop venue: 
http://www.iusspavia.it/eng/index.php?id=40#.VFdrPfSG_YV
Info at: andrea.sereni at iusspavia.it
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150302/fee94589/attachment-0001.html>

From sereny at math.bme.hu  Mon Mar  2 17:19:42 2015
From: sereny at math.bme.hu (Gyorgy Sereny)
Date: Mon, 2 Mar 2015 23:19:42 +0100 (CET)
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
Message-ID: <alpine.DEB.2.02.1503022306020.59085@leibniz.math.bme.hu>


Dear Fomers,

I would like to inform you about a strange publication.
I have just come across a book newly published by
Oxford University Press:

    The Consistency of Arithmetic: And Other Essays Hardcover
    24 Jul 2014 by Storrs McCall (Author)

http://www.amazon.co.uk/The-Consistency-Arithmetic-Other-Essays/dp/0199316546

The last sentence of the review found in this page of amazon
about the book is intriguing:

    The eponymous first essay contains the proof of
    a fact that in 1931 Kurt G?del had claimed to be
    unprovable, namely that the set of arithmetic
    truths forms a consistent system.

And indeed, the first one in the collection of essays by the
philosopher Storrs McCall has the title: "The Consistency
of Arithmetic". I do not have the book at hand, so I was able
to look into it only as a Google book. The first lines of
the essay on p.8. are:

   Is Peano arithmetic (PA) consistent? This paper contains a
   proof that it is: a proof moreover, that does not lie in
   deducing its consistency as a theorem in a system with axioms
   and rules of inference. [...]  If there is to be a genuine
   proof of PA's consistency, it cannot be a proof
   relative to the consistency of some other, stronger system,
   but an absolute proof, such as the proof of consistency of
   two-valued propositional logic using truth-tables. Axiomatic
   proofs we may categorize as "syntactic", meaning that they
   concern only symbols and the derivation of one string of
   symbols from another, according to set rules. "Semantic"
   proofs, on the other hand, differ from syntactic proofs
   in being based not only on symbols but on a non-symbolic,
   non-linguistic component, a domain of objects. If the sole
   paradigm of "proof" in mathematics is "axiomatic proof",
   in which to prove a formula means to deduce it from axioms
   using specified rules of inference, then G?del indeed appears
   to have had the last word on the question of PA
   consistency. But in addition to axiomatic proofs there is
   another kind of proof. In this paper I give a proof of
   PA's consistency based on a formal semantics for PA. To my
   knowledge, no semantic consistency proof of Peano
   arithmetic has yet been constructed.

Later, at the bottom of the p.9, one can read the following
remark concerning the method of the proof:

   The proof constructed in this paper [...] is based on a
   non-linguistic component, [...] a _physical domain of
   three-dimensional cube-shaped blocks_.
   (emphasis mine)


Once more, this text was published by Oxford University Press.
It's bizarre, isn't it?


Gyorgy Sereny



From zaljohar at yahoo.com  Sun Mar  1 21:49:46 2015
From: zaljohar at yahoo.com (Zuhair Abdul Ghafoor Al-Johar)
Date: Mon, 2 Mar 2015 02:49:46 +0000 (UTC)
Subject: [FOM] One axiom for all sets.
Message-ID: <1164545383.1226372.1425264586407.JavaMail.yahoo@mail.yahoo.com>

Dear FOMers,

Mr. M.Randall Holmes had put forth a proof on his home page of existence of
a set of all sets hereditarily smaller than a given set, for every set, in ZF.
The proof is independent of choice. 

One consequence of such a result is that the following would be a theorem
of Morse-Kelley (MK) class theory (with any size limitation axiom sufficient
to prove replacement over sets)

Set(x) <-> Ez (Ayez(z<y) ^ Ayex(y<<TC(z)))  

where "<" stand for "is subnumerouse to" defined in the usual manner
(non strict subnumeroustiy); "TC" stands for "transitive closure of" 
defined as the minimal transitive superclass; and "<<" stands for
"hereditarily subnumerouse to".

The nice result is that adding that sentence to axioms of Extensionality,
Class comprehension and Pairing of MK results in a theory that proves
all other set axioms of a version of MK that is sufficient to interpret
ZFC. If we weaken the above axiom by just replacing "TC(z)" by "z", then
we'd get a theory that interpret Z.


Details at: 
http://zaljohar.tripod.com/shortmk.txt

Best Regards,

Zuhair Al-Johar
PS:"e" stands for membership; "E" for the existential quantifier and
"A" for the universal quantifier.

From tchow at alum.mit.edu  Mon Mar  2 21:51:47 2015
From: tchow at alum.mit.edu (Timothy Y. Chow)
Date: Mon, 2 Mar 2015 21:51:47 -0500 (EST)
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <mailman.442.1425346301.28387.fom@cs.nyu.edu>
References: <mailman.442.1425346301.28387.fom@cs.nyu.edu>
Message-ID: <alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>

Gyorgy Sereny wrote:
> I would like to inform you about a strange publication.
> I have just come across a book newly published by
> Oxford University Press:
>
>    The Consistency of Arithmetic: And Other Essays Hardcover
>    24 Jul 2014 by Storrs McCall (Author)

I have to agree with Gyorgy Sereny that the first article in this book is 
strange.  Here I mainly want to point out that the full text of the 
article is linked from the author's website:

http://www.mcgill.ca/philosophy/people/faculty/mccall

Or you can go directly to the Word document:

http://www.mcgill.ca/philosophy/files/philosophy/the_consistency_of_arithmetic_feb_10_2011.doc

Skimming through the paper, I do not see any interesting mathematical 
insight.  I would characterize it as an argument for the consistency of PA 
based on physical intuition.  Perhaps there is an audience for this sort 
of thing, but I'm having trouble imagining one.

Tim

From rlk at knighten.org  Tue Mar  3 01:16:59 2015
From: rlk at knighten.org (rlk at knighten.org)
Date: Mon, 2 Mar 2015 22:16:59 -0800
Subject: [FOM]  "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <alpine.DEB.2.02.1503022306020.59085@leibniz.math.bme.hu>
References: <alpine.DEB.2.02.1503022306020.59085@leibniz.math.bme.hu>
Message-ID: <21749.20955.566827.420800@zeus.knighten.org>

Just as a matter of information the essay "The Consistency of Arithmetic" can
be found on the author's web page:

http://www.mcgill.ca/philosophy/files/philosophy/the_consistency_of_arithmetic_feb_10_2011.doc

-- 
Robert L. Knighten
RLK at knighten.org

From beziau100 at gmail.com  Wed Mar  4 10:41:28 2015
From: beziau100 at gmail.com (jean-yves beziau)
Date: Wed, 4 Mar 2015 16:41:28 +0100
Subject: [FOM] Launching of the South American Journal of Logic
Message-ID: <CAMrL2xuVBD8K3Phd-kAaa4qW01y5FnhNPqzuz6zPaGBsKmrFjA@mail.gmail.com>

We are very glad to present the first issue of the South American Journal
of Logic, SAJL:
http://www.sa-logic.org/
This journal will be published both on-line (open access)  and in-print,
starting with 2 issues a year.

South America is a huge continent with many different countries.
Logic is strong in South America since many years.
There are some very active groups of logic in particular in Argentina,
Brazil, Chile, Colombia and Venezuela.
The objective of SAJL is to promote the  work of people working in
South-America and their interaction with the rest of the world.

This journal is dedicated to  logical research in all its aspects:
philosophical, mathematical, historical, computational and
is also aiming at encouraging relations with other fields like linguistics,
law, biology, information, etc.
In this first issue we have a collection of 14 high quality papers
reflecting this perspective.
Enjoy!
>------------------------------------------------------------------------------->
Jean-Yves Beziau
Federal University of Rio de Janeiro (UFRJ),
Brazilian Research Council (CNPq) and Brazilian Academy of Philosophy (ABF)

Marcelo Esteban Coniglio
State University of Campinas (UNICAMP)
and Brazilian Research Council (CNPq)
>------------------------------------------------------------------------------->
South American Journal of Logic
http://www.sa-logic.org/
 New Horizons for Logic
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150304/392985d6/attachment.html>

From silver_1 at mindspring.com  Wed Mar  4 15:34:03 2015
From: silver_1 at mindspring.com (Charlie)
Date: Wed, 4 Mar 2015 12:34:03 -0800
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>
References: <mailman.442.1425346301.28387.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>
Message-ID: <567FF5E7-23D4-4AC8-82E5-8DE22A2A79F3@mindspring.com>

	Consider the axioms of Peano Arithmetic minus the induction schema, plus Robinson?s axiom, which says that if x differs from 0, then it?s a successor.  Every one of these axioms is ?obviously true? for the natural numbers.   One might say ? I?m *not* saying this ?  that since every single axiom is obviously true of the natural numbers, plus since they do not *seem* to interfere with each other, the entire system (Q) *must be* consistent.   If we wished to dignify this reasoning by calling it a real proof, we could say it?s a proof of the consistency of Q ?by intuitive inspection?.


> On Mar 2, 2015, at 6:51 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
> 
> Gyorgy Sereny wrote:
>> I would like to inform you about a strange publication.
>> I have just come across a book newly published by
>> Oxford University Press:
>> 
>>   The Consistency of Arithmetic: And Other Essays Hardcover
>>   24 Jul 2014 by Storrs McCall (Author)
> 
> I have to agree with Gyorgy Sereny that the first article in this book is strange.  Here I mainly want to point out that the full text of the article is linked from the author's website:
> 
> http://www.mcgill.ca/philosophy/people/faculty/mccall
> 
> Or you can go directly to the Word document:
> 
> http://www.mcgill.ca/philosophy/files/philosophy/the_consistency_of_arithmetic_feb_10_2011.doc
> 
> Skimming through the paper, I do not see any interesting mathematical insight.  I would characterize it as an argument for the consistency of PA based on physical intuition.  Perhaps there is an audience for this sort of thing, but I'm having trouble imagining one.
> 
> Tim
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom


From richard_heck at brown.edu  Wed Mar  4 15:33:52 2015
From: richard_heck at brown.edu (Richard Heck)
Date: Wed, 04 Mar 2015 15:33:52 -0500
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>
References: <mailman.442.1425346301.28387.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>
Message-ID: <54F76C30.5060208@brown.edu>

On 03/02/2015 09:51 PM, Timothy Y. Chow wrote:
> Gyorgy Sereny wrote:
>> I would like to inform you about a strange publication.
>> I have just come across a book newly published by
>> Oxford University Press:
>>
>>    The Consistency of Arithmetic: And Other Essays Hardcover
>>    24 Jul 2014 by Storrs McCall (Author)
>
> I have to agree with Gyorgy Sereny that the first article in this book 
> is strange.  Here I mainly want to point out that the full text of the 
> article is linked from the author's website:
>
> http://www.mcgill.ca/philosophy/people/faculty/mccall
>
> Or you can go directly to the Word document:
>
> http://www.mcgill.ca/philosophy/files/philosophy/the_consistency_of_arithmetic_feb_10_2011.doc 
>
>
> Skimming through the paper, I do not see any interesting mathematical 
> insight.  I would characterize it as an argument for the consistency 
> of PA based on physical intuition.  Perhaps there is an audience for 
> this sort of thing, but I'm having trouble imagining one.

What I find hard to understand is what contrast exactly he thinks there 
is between the kind of argument he is giving and other sorts of proofs 
of Con(PA). He claims "no semantic consistency proof of Peano arithmetic 
has yet been constructed", but that is pretty obviously false. So the 
fact that the proof is semantic rather than syntactic isn't, by itself, 
all that significant.

The fact that the proof, as stated, isn't fomalized isn't all that 
interesting, either. Very few actual proofs are formalized, and there is 
no obvious bar to formalizing this one.

So we're being given an informal semantic consistency proof. But exactly 
what the cash value of the proof is, it seems to me, isn't obvious until 
we know exactly what sorts of assumptions it is employing. Somewhere, 
obviously, there are some strong assumptions being deployed. The fact 
that the presentation obscures where they are is not a virtue.

Here again, an early remark seems to me at best misleading: " But to 
deduce [PA's] consistency in some stronger system PA+ that includes PA 
is self-defeating, since if PA+ is itself inconsistent the proof of PA's 
consistency is worthless". That isn't the only option, and it isn't the 
usual reaction, it seems to me, to Gentzen's proof of Con(PA), either.

If there's something interesting here, it's the way the semantics he 
develops doesn't require there to be a single infinite model, but only a 
succession of every-larger finite models. There are antecedents to that 
sort of idea in modal structuralist views, I believe, of the sort 
developed by Hellman, and perhaps more than antecedents. Maybe there are 
more developed forms of this idea, too, and if so I'd be interested to 
know where.

Richard Heck

-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150304/713a2f31/attachment.html>

From kremer at uchicago.edu  Thu Mar  5 07:56:29 2015
From: kremer at uchicago.edu (Michael Kremer)
Date: Thu, 5 Mar 2015 12:56:29 +0000
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <54F76C30.5060208@brown.edu>
References: <mailman.442.1425346301.28387.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>,
	<54F76C30.5060208@brown.edu>
Message-ID: <ECAF14BB08F59040BA4E3788A495E387137DF939@XM-MBX-03-PROD.ad.uchicago.edu>

The following is related to Richard's points...

On a quick skim I find the final section of the paper confused. He wants to ward off the claim that his argument requires an infinite set. He does this by claiming that his argument does not require an infinite model; and that he does not have an unrestricted comprehension scheme which would allow one to construct the set of all models. But his argument requires that there is no upper bound on the size of his models. Whether this requires an infinite set or not, it surely requires in some sense that there are infinitely many blocks. (There is no finite bound on the number of blocks.)

Moreover, the argument makes use of an operation of concatenation which yield larger arrays of blocks out of smaller arrays of blocks. And this operation needs to satisfy some set of axioms with sufficient similarities to the axioms of arithmetic to make one suspicious that the theory of concatenation is at least as strong as arithmetic. If there are only finitely many blocks in the universe we would be in a position in which the concatenation operation could not always be applied. (If there are 1000 blocks in the universe, we could make an array of 600 blocks, and an array of 601 blocks but we could not concatenate them as they would always have overlap.)

Put it another way: Russell and Whitehead's Axiom of Infinity in Principia Mathematics would seem to be satisfied by McCall's blocks (iirc, it does not state that there is an infinite set; it states that for every inductive cardinal number n, there are n things).

Michael Kremer
________________________________
From: fom-bounces at cs.nyu.edu [fom-bounces at cs.nyu.edu] on behalf of Richard Heck [richard_heck at brown.edu]
Sent: Wednesday, March 04, 2015 2:33 PM
To: tchow at alum.mit.edu; Foundations of Mathematics
Subject: Re: [FOM] "Proof" of the consistency of PA published by Oxford UP

On 03/02/2015 09:51 PM, Timothy Y. Chow wrote:
Gyorgy Sereny wrote:
I would like to inform you about a strange publication.
I have just come across a book newly published by
Oxford University Press:

   The Consistency of Arithmetic: And Other Essays Hardcover
   24 Jul 2014 by Storrs McCall (Author)

I have to agree with Gyorgy Sereny that the first article in this book is strange.  Here I mainly want to point out that the full text of the article is linked from the author's website:

http://www.mcgill.ca/philosophy/people/faculty/mccall

Or you can go directly to the Word document:

http://www.mcgill.ca/philosophy/files/philosophy/the_consistency_of_arithmetic_feb_10_2011.doc

Skimming through the paper, I do not see any interesting mathematical insight.  I would characterize it as an argument for the consistency of PA based on physical intuition.  Perhaps there is an audience for this sort of thing, but I'm having trouble imagining one.

What I find hard to understand is what contrast exactly he thinks there is between the kind of argument he is giving and other sorts of proofs of Con(PA). He claims "no semantic consistency proof of Peano arithmetic has yet been constructed", but that is pretty obviously false. So the fact that the proof is semantic rather than syntactic isn't, by itself, all that significant.

The fact that the proof, as stated, isn't fomalized isn't all that interesting, either. Very few actual proofs are formalized, and there is no obvious bar to formalizing this one.

So we're being given an informal semantic consistency proof. But exactly what the cash value of the proof is, it seems to me, isn't obvious until we know exactly what sorts of assumptions it is employing. Somewhere, obviously, there are some strong assumptions being deployed. The fact that the presentation obscures where they are is not a virtue.

Here again, an early remark seems to me at best misleading: " But to deduce [PA's] consistency in some stronger system PA+ that includes PA is self-defeating, since if PA+ is itself inconsistent the proof of PA's consistency is worthless". That isn't the only option, and it isn't the usual reaction, it seems to me, to Gentzen's proof of Con(PA), either.

If there's something interesting here, it's the way the semantics he develops doesn't require there to be a single infinite model, but only a succession of every-larger finite models. There are antecedents to that sort of idea in modal structuralist views, I believe, of the sort developed by Hellman, and perhaps more than antecedents. Maybe there are more developed forms of this idea, too, and if so I'd be interested to know where.

Richard Heck

-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150305/40b48824/attachment.html>

From colin.mclarty at case.edu  Thu Mar  5 09:21:37 2015
From: colin.mclarty at case.edu (Colin McLarty)
Date: Thu, 5 Mar 2015 09:21:37 -0500
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <54F76C30.5060208@brown.edu>
References: <mailman.442.1425346301.28387.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>
	<54F76C30.5060208@brown.edu>
Message-ID: <CAOzx82oGaw_c-=7qO1FGQDNaoQLvyAzCF2U5+rAJ3qGdWd0FGw@mail.gmail.com>

On Wed, Mar 4, 2015 at 3:33 PM, Richard Heck <richard_heck at brown.edu> wrote:

>
> If there's something interesting here, it's the way the semantics he
> develops doesn't require there to be a single infinite model, but only a
> succession of every-larger finite models. There are antecedents to that
> sort of idea in modal structuralist views, I believe, of the sort developed
> by Hellman, and perhaps more than antecedents. Maybe there are more
> developed forms of this idea, too, and if so I'd be interested to know
> where.
>
>
> I don't know if Hellman talks about this but you might be thinking of
Shaughan Lavine. *Understanding the In?nite*.Harvard University Press,
Cambridge,, which gives philosophic treatment to

Mycielski, J., ?Analysis without actual infinity,? The Journal of Symbolic
Logic, vol. 46
(1981), pp. 625?33.

[14] Mycielski, J., ?Locally finite theories,? The Journal of Symbolic
Logic, vol. 51 (1986),
pp. 59?62.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150305/b73339bd/attachment.html>

From richard_heck at brown.edu  Thu Mar  5 15:35:20 2015
From: richard_heck at brown.edu (Richard Heck)
Date: Thu, 05 Mar 2015 15:35:20 -0500
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <ECAF14BB08F59040BA4E3788A495E387137DF939@XM-MBX-03-PROD.ad.uchicago.edu>
References: <mailman.442.1425346301.28387.fom@cs.nyu.edu>	<alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>,
	<54F76C30.5060208@brown.edu>
	<ECAF14BB08F59040BA4E3788A495E387137DF939@XM-MBX-03-PROD.ad.uchicago.edu>
Message-ID: <54F8BE08.5080406@brown.edu>

On 03/05/2015 07:56 AM, Michael Kremer wrote:
> The following is related to Richard's points...

Hi, Michael! Hope things are well with you.

> On a quick skim I find the final section of the paper confused. He 
> wants to ward off the claim that his argument requires an infinite 
> set. He does this by claiming that his argument does not require an 
> infinite model; and that he does not have an unrestricted 
> comprehension scheme which would allow one to construct the set of all 
> models. But his argument requires that there is no upper bound on the 
> size of his models. Whether this requires an infinite set or not, it 
> surely requires in some sense that there are infinitely many blocks. 
> (There is no finite bound on the number of blocks.)

Yes, that is clearly correct.

> Moreover, the argument makes use of an operation of concatenation 
> which yield larger arrays of blocks out of smaller arrays of blocks. 
> And this operation needs to satisfy some set of axioms with sufficient 
> similarities to the axioms of arithmetic to make one suspicious that 
> the theory of concatenation is at least as strong as arithmetic.

Your use of the term "concatenation" does rather remind me of Quine's 
paper "Concatenation as a Basis for Arithmetic". One could of course 
also prove the consistency of PA by modelling arithmetic in strings---as 
Hilbert more or less wanted to do.

> If there are only finitely many blocks in the universe we would be in 
> a position in which the concatenation operation could not always be 
> applied. (If there are 1000 blocks in the universe, we could make an 
> array of 600 blocks, and an array of 601 blocks but we could not 
> concatenate them as they would always have overlap.)

The argument for induction uses what McCall calls "finite descent", 
i.e., a version of the least number principle. So it seems clear enough 
that the principles used to reason about blocks are at least as strong 
as those of PA. Indeed, "finite descent" here is being used in 
connection with the predicate:
     ~F is true of a linear array of N blocks
I.e., he is reasoning inductively about models. So it appears that the 
theory in which he is working is roughly equivalent in strength to 
something like PA + a compositional truth theory with extended 
induction. And of course that theory is well known to prove Con(PA).

One thing I did learn from the paper concerns how McCall expresses 
induction, as the rule of inference:
     From |- F(0) and |- (x)(F(x) --> F(Sx)), infer |- (x)(F(x))
Note that this is really a "rule of proof", in the way that 
necessitation is a rule of proof: The premises of the inference have to 
be *theorems*. This is entirely different from formulating induction as 
a rule of inference:
     F(0), (x)(F(x) --> F(Sx)) |- (x)(F(x))
which is of course equivalent to the usual scheme. I was worried for a 
bit that this must be weaker than induction in the conditional form, but 
in fact it seems to be equivalent.

To prove induction as a conditional:
     F(0) & (x)(F(x) --> F(Sx)) --> (x)(F(x))
note that this is equivalent to:
     (z)[F(0) & (x)(F(x) --> F(Sx)) --> F(z)]
And we can now prove this by induction on z. So we have to prove:
     F(0) & (x)(F(x) --> F(Sx)) --> F(0)
which is trivial, and
     (z){[F(0) & (x)(F(x) --> F(Sx)) --> F(z)] --> [F(0) & (x)(F(x) --> 
F(Sx)) --> F(Sz)]}
Assume [F(0) & (x)(F(x) --> F(Sx)) --> F(z)] and F(0) & (x)(F(x) --> 
F(Sx)). The second in the antecedent of the first, so F(z); but then 
F(z) --> F(Sz), so F(Sz). QED.

McCall's proof of induction depends, as he notes (end of section 5), 
upon its being in this particular form.

Richard

-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150305/3560dd9d/attachment-0001.html>

From rda at lemma-one.com  Thu Mar  5 16:31:57 2015
From: rda at lemma-one.com (Rob Arthan)
Date: Thu, 5 Mar 2015 21:31:57 +0000
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <54F76C30.5060208@brown.edu>
References: <mailman.442.1425346301.28387.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>
	<54F76C30.5060208@brown.edu>
Message-ID: <89140FDF-A55B-4E64-B6CE-0EEFB781DA10@lemma-one.com>


On 4 Mar 2015, at 20:33, Richard Heck <richard_heck at brown.edu> wrote:

> So we're being given an informal semantic consistency proof. But exactly what the cash value of the proof is, it seems to me, isn't obvious until we know exactly what sorts of assumptions it is employing. Somewhere, obviously, there are some strong assumptions being deployed. The fact that the presentation obscures where they are is not a virtue.

I couldn?t agree more about the lack of virtue in the presentation.
The following points do seem to be clear:

1) McCall wants to give a semantics couched in some kind of naive physics,
in which certain properties of arrangements of congruent cubes
(referred to as ?blocks?)  in physical space are taken to be self-evident.

2) The semantics for the quantifiers assumes that an additional block
can be added to an arrangement of blocks without unwanted intersections.
So the naive physics assumes that the universe has infinite volume and
contains an unbounded amount of matter.

3) The proof of the induction axiom via what McCall calls the method
of ?finite descent? assumes that a process that removes blocks from
an arrangement one at a time will always terminate with an empty
arrangement. So the naive physics disallows the creation of infinite
arrangements despite the abundance of space and matter implied
by the semantics for quantifiers.

I, personally, find it fairly irritating that the article claims to be doing
something very new, when the naive physics adds nothing to the
naive arithmetic intuition that PA is obviously consistent, that one
gets, for example, by thinking about concrete decimal representations of
numbers and the arithmetic operations on them. 

Regards,

Rob.

-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150305/0b7f7a6c/attachment.html>

From martin at eipye.com  Thu Mar  5 17:47:35 2015
From: martin at eipye.com (Martin Davis)
Date: Thu, 5 Mar 2015 14:47:35 -0800
Subject: [FOM] Fwd: 3rd Workshop on Interpolation - Call for Papers
In-Reply-To: <20150305181608.96A611214F4@mcclellan.cs.miami.edu>
References: <20150305181608.96A611214F4@mcclellan.cs.miami.edu>
Message-ID: <CA+cpue+c2wCntz-DoA_tw6veMK1mqperNHOHrLajODVxacKguw@mail.gmail.com>

---------- Forwarded message ----------
From: Geoff Sutcliffe <geoff at cs.miami.edu>
Date: Thu, Mar 5, 2015 at 10:16 AM
Subject: 3rd Workshop on Interpolation - Call for Papers
To: davism at cs.nyu.edu


iPRA 2015 - THIRD WORKSHOP ON INTERPOLATION: FROM PROOFS TO APPLICATIONS

CALL FOR CONTRIBUTIONS

Date:     July 18, 2015
Location: San Francisco, CA (co-located with CAV 2015)
Web:      http://forsyte.at/interpolation/

IMPORTANT DATES

Submission deadline: May 7, 2015, AOE
Notification:        May 14, 2015
Workshop:            July 18, 2015

ORGANISATION AND COMMITTEE

Laura Kovacs and Georg Weissenbacher

SCOPE

Craig interpolation enjoys a continuing popularity in the field of
verification. Historically, Craig's interpolation theorem has received
ample attention in proof theory and mathematical logic as well as in
complexity theory. The aim of the workshop is to bring together
theoreticians and practitioners from different fields.

We solicit submissions in form of an abstract of at most one page in
PDF format. The authors of accepted abstracts are required to
present their work at the workshop. There will be no published
proceedings.

We encourage submissions presenting work in progress, tools under
development, as well as research of PhD students, such that the
workshop can become a forum for active dialog between the groups
involved in applications of interpolation. We also encourage
contributions from outside the verification community.

Presentations of recently published papers are also allowed and
encouraged, but please indicate on your submission where the paper was
published/presented.

Relevant topics include (but are not limited to) applications of
interpolation in:

- Interpolating decision procedures
- Proof theoretic approaches to interpolation
- Proof systems and calculi for interpolation
- Proof transformation techniques
- Inductive Proofs
- Logical Abduction
- Interpolation techniques based on constraint solving, linear
programming...
- Alternative techniques for interpolation
- Interpolation theorems (for theories and extensions, non-classical
  logic, ...)
- Interpolation-based/Inductive invariant generation
- Program analysis and verification
- Tools for interpolation
- Applications of Craig interpolation (verification, synthesis,
  automated reasoning, ...)
- Complexity results and limitations
...

SUBMISSION INSTRUCTIONS

Abstracts (at most one page in PDF format) have to be submitted until
May 7 via the EasyChair system:

https://easychair.org/conferences/?conf=ipra15

The authors will be notified on May 14, 2015.
There will be no formal workshop proceedings.

FORMAT

The workshop will feature

- an invited talk by Arie Gurfinkel (SEI/CMU),
- presentations (selected by a committee based on the submission of
  abstracts) by workshop participants, and
- discussion and panel sessions.

The program will be coordinated with the HCVS workshop,
which takes place on July 19.

REGISTRATION

Registration for the workshop will be possible via the CAV
registration site: http://i-cav.org/2015/

POSTER

A poster is available on http://forsyte.at/interpolation. We
kindly ask you to print and display a copy in your department/workplace.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150305/60577de7/attachment.html>

From tchow at alum.mit.edu  Fri Mar  6 18:11:45 2015
From: tchow at alum.mit.edu (Timothy Y. Chow)
Date: Fri, 6 Mar 2015 18:11:45 -0500 (EST)
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <54F8BE08.5080406@brown.edu>
References: <mailman.442.1425346301.28387.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>,
	<54F76C30.5060208@brown.edu>
	<ECAF14BB08F59040BA4E3788A495E387137DF939@XM-MBX-03-PROD.ad.uchicago.edu>
	<54F8BE08.5080406@brown.edu>
Message-ID: <alpine.LFD.2.11.1503061756130.24538@laurent.mit.edu>

Richard Heck wrote:

> If there's something interesting here, it's the way the semantics he 
> develops doesn't require there to be a single infinite model, but only a 
> succession of every-larger finite models. There are antecedents to that 
> sort of idea in modal structuralist views, I believe, of the sort 
> developed by Hellman, and perhaps more than antecedents. Maybe there are 
> more developed forms of this idea, too, and if so I'd be interested to 
> know where.

There is of course a long tradition in philosophy of distinguishing 
between "potential infinity" and "actual infinity."  In modern 
mathematics, this distinction doesn't seem to exist.  The closest thing to 
"potential infinity" seems to be an axiomatic system that lacks anything 
that could be identified as an explicit "axiom of infinity," yet admits 
only (actually) infinite models.  (PA would be an example.)  But for 
example, I've never seen anyone define two separate axioms and declare one 
of them to be an "axiom of potential infinity" and the other an "axiom of 
actual infinity."

I'm wondering if anyone can come up with (or has already come up with) 
candidates for two such axioms, the former demonstrably weaker than the 
latter, such that the consistency of PA can be proved using only the 
"axiom of potential infinity."  Doing this might be pleasing to those who 
not only perceive an important distinction between potential and actual 
infinity but go so far as to reject the latter while accepting the former. 
(McCall seems to be one such person, if I read him correctly.)

Tim

From aa at tau.ac.il  Fri Mar  6 05:29:37 2015
From: aa at tau.ac.il (Arnon Avron)
Date: Fri, 6 Mar 2015 12:29:37 +0200
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <567FF5E7-23D4-4AC8-82E5-8DE22A2A79F3@mindspring.com>
References: <mailman.442.1425346301.28387.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>
	<567FF5E7-23D4-4AC8-82E5-8DE22A2A79F3@mindspring.com>
Message-ID: <20150306102936.GA22368@localhost.localdomain>

I truly do not understand the point here.

I should admit a terrible sin: I am 100%  certain of the
consistency of full PA (not only Q). Worse: the reason
I am sure of it no less than I am sure about any theorem of Q is that
the axioms of PA are indeed *obviously true* in the natural
numbers. What better and more convincing proof can one want?

I would add that I *do not believe* people who pretend to
doubt the consistency of PA. As I have pointed out 
on FOM in the past, people who do not understand the 
natural numbers cannot understand the notions of formulas
and of formal proofs either (both being recursively defined).
Accordingly, they cannot even understand what PA is and what 
its consistency mean - let alone doubt its truth.

Arnon Avron



On Wed, Mar 04, 2015 at 12:34:03PM -0800, Charlie wrote:
> 	Consider the axioms of Peano Arithmetic minus the induction schema, plus Robinson???s axiom, which says that if x differs from 0, then it???s a successor.  Every one of these axioms is ???obviously true??? for the natural numbers.   One might say ??? I???m *not* saying this ???  that since every single axiom is obviously true of the natural numbers, plus since they do not *seem* to interfere with each other, the entire system (Q) *must be* consistent.   If we wished to dignify this reasoning by calling it a real proof, we could say it???s a proof of the consistency of Q ???by intuitive inspection???.
> 
> 
> > On Mar 2, 2015, at 6:51 PM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:
> > 
> > Gyorgy Sereny wrote:
> >> I would like to inform you about a strange publication.
> >> I have just come across a book newly published by
> >> Oxford University Press:
> >> 
> >>   The Consistency of Arithmetic: And Other Essays Hardcover
> >>   24 Jul 2014 by Storrs McCall (Author)
> > 
> > I have to agree with Gyorgy Sereny that the first article in this book is strange.  Here I mainly want to point out that the full text of the article is linked from the author's website:
> > 
> > http://www.mcgill.ca/philosophy/people/faculty/mccall
> > 
> > Or you can go directly to the Word document:
> > 
> > http://www.mcgill.ca/philosophy/files/philosophy/the_consistency_of_arithmetic_feb_10_2011.doc
> > 
> > Skimming through the paper, I do not see any interesting mathematical insight.  I would characterize it as an argument for the consistency of PA based on physical intuition.  Perhaps there is an audience for this sort of thing, but I'm having trouble imagining one.
> > 
> > Tim
> > _______________________________________________
> > FOM mailing list
> > FOM at cs.nyu.edu
> > http://www.cs.nyu.edu/mailman/listinfo/fom
> 
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom

From tchow at alum.mit.edu  Fri Mar  6 22:17:51 2015
From: tchow at alum.mit.edu (Timothy Y. Chow)
Date: Fri, 6 Mar 2015 22:17:51 -0500 (EST)
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
Message-ID: <alpine.LFD.2.11.1503062159110.25430@laurent.mit.edu>

Arnon Avron wrote:

> I would add that I *do not believe* people who pretend to doubt the 
> consistency of PA. As I have pointed out on FOM in the past, people who 
> do not understand the natural numbers cannot understand the notions of 
> formulas and of formal proofs either (both being recursively defined). 
> Accordingly, they cannot even understand what PA is and what its 
> consistency mean - let alone doubt its truth.

I agree with your criticism of those who claim to understand what PA is 
but also claim not to understand what the natural numbers are.

However, there is another possible avenue to doubting the consistency of 
PA.  Namely, having some grasp of the natural numbers is not the same as 
understanding every *property* of the natural numbers.  Certainly none of 
us is able to directly apprehend the *truth* of all first-order statements 
about the integers.  If we can be uncertain about the truth of (for 
example) the twin prime conjecture, then why can't we be uncertain about 
the truth of instances of the induction schema with arbitrarily many 
alternations of quantifiers?  The doubts here would not be about the 
natural numbers per se but about whether such incomprehensibly complex 
formulas coherently define a "property" of the natural numbers to which 
induction can be applied.

Although I don't have such doubts myself, I don't see anything incoherent 
about them.  After all, we know many ways of weakening the induction axiom 
to produce a system whose consistency doesn't imply the consistency of PA. 
Hence we have a fairly clear picture of what it could mean for the natural 
numbers to exist and to satisfy a weaker induction schema, yet for PA to 
be inconsistent.

Tim

From francoli at kqnet.pt  Sat Mar  7 07:40:34 2015
From: francoli at kqnet.pt (A J Franco de Oliveira)
Date: Sat, 07 Mar 2015 12:40:34 +0000
Subject: [FOM] "Proof" of the consistency of PA published by Oxford  UP
In-Reply-To: <alpine.LFD.2.11.1503061756130.24538@laurent.mit.edu>
References: <mailman.442.1425346301.28387.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>
	<54F76C30.5060208@brown.edu>
	<ECAF14BB08F59040BA4E3788A495E387137DF939@XM-MBX-03-PROD.ad.uchicago.edu>
	<54F8BE08.5080406@brown.edu>
	<alpine.LFD.2.11.1503061756130.24538@laurent.mit.edu>
Message-ID: <201503071240.t27Cei4B007970@mx2.cims.nyu.edu>

Hello
This is a response to you only.
How could one define "potential infinite"? I 
believe that there is a possible approximation of 
a "potential infinite" axiom in E. Nelson?s 
Internal Set Theory (1977): Internal set theory: 
A new approach to nonstandard analysis. Bulletin 
of the American Mathematical Society 83(6):1165?1198.
In this conservative extension of ZFC (a similar 
extension of PA exists) the definable collection 
of standard natural numbers  is not a 
set  (although it is said to be an external set) 
and so is not an infinite set but is a good 
approximation to a potential infinite set as one 
can get: the number 0 is standard and for every 
standard natural number n, the sucessor of n is 
standard. The said collection also satisfies an 
"external induction principle". In every infinite 
set we can obtain a copy of the standard natural 
numbers which is not a set. So what I propose is 
that the idea of "potential infinite" can be 
realized by any copy of the standard natural numbers.
Regards
ajfo
At 23:11 06-03-2015, you wrote:
>Richard Heck wrote:
>
>>If there's something interesting here, it's the 
>>way the semantics he develops doesn't require 
>>there to be a single infinite model, but only a 
>>succession of every-larger finite models. There 
>>are antecedents to that sort of idea in modal 
>>structuralist views, I believe, of the sort 
>>developed by Hellman, and perhaps more than 
>>antecedents. Maybe there are more developed 
>>forms of this idea, too, and if so I'd be interested to know where.
>
>There is of course a long tradition in 
>philosophy of distinguishing between "potential 
>infinity" and "actual infinity."  In modern 
>mathematics, this distinction doesn't seem to 
>exist.  The closest thing to "potential 
>infinity" seems to be an axiomatic system that 
>lacks anything that could be identified as an 
>explicit "axiom of infinity," yet admits only 
>(actually) infinite models.  (PA would be an 
>example.)  But for example, I've never seen 
>anyone define two separate axioms and declare 
>one of them to be an "axiom of potential 
>infinity" and the other an "axiom of actual infinity."
>
>I'm wondering if anyone can come up with (or has 
>already come up with) candidates for two such 
>axioms, the former demonstrably weaker than the 
>latter, such that the consistency of PA can be 
>proved using only the "axiom of potential 
>infinity."  Doing this might be pleasing to 
>those who not only perceive an important 
>distinction between potential and actual 
>infinity but go so far as to reject the latter 
>while accepting the former. (McCall seems to be 
>one such person, if I read him correctly.)
>
>Tim
>_______________________________________________
>FOM mailing list
>FOM at cs.nyu.edu
>http://www.cs.nyu.edu/mailman/listinfo/fom
>
>

Augusto J. Franco de Oliveira
Prof. Em?rito Univ. ?vora
CFCUL
ajfrancoli at gmail.com

(Este escriba n?o respeita o AO90.)

http://sites.google.com/site/tutasplace/
http://cfcul.fc.ul.pt/equipa/3_cfcul_elegiveis/franco_oliveira/afrancoliveira.htm
in http://cfcul.fc.ul.pt/
http://jnsilva.ludicum.org/Vasconcellos_web/Vasconcellos.html

R. Ant?nio Aleixo 24-26
Cotovia, 2970-298 Sesimbra
Tlm 919807966

O n?mero dos tolos e dos cegos continua sendo 
infinito, como nos tempos b?blicos. T?tulo da 
Parte V do livro de Tom?s da Fonseca (1887-1968), 
NA COVA DOS LE?ES, Lisboa,  955 (edi??o 
fac-simile com o t?tulo F?TIMA, A Bela e o Monstro, Lisboa, 2014)
Continuo ? procura dela (da esperan?a) para 
perceber onde ? que os meus filhos e netos ir?o viver.
A grande desvantagem de ser velho ? perceber que 
pouco ou nada muda. John Le Carr?, in Suplemento 
"Actual" do EXPRESSO de 27-04-2013.
You can only find truth with logic if you have 
already found truth without it. Gilbert Keith Chesterton (1874-1936)
Quem sabe, faz, quem compreende, ensina. Arist?teles  
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150307/26c1880b/attachment-0001.html>

From martdowd at aol.com  Sat Mar  7 11:11:33 2015
From: martdowd at aol.com (martdowd at aol.com)
Date: Sat, 7 Mar 2015 11:11:33 -0500
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <20150306102936.GA22368@localhost.localdomain>
References: <20150306102936.GA22368@localhost.localdomain>
Message-ID: <14bf501877d-6345-74ac@webprd-a17.mail.aol.com>



 


the axioms
of PA are indeed *obviously true* in the natural
numbers. What better and more
convincing proof can one want?


 

 

 

Arnon,

I would call this the standard classical position.  It's worth adding that there are progressions of theories, both the "natuarally occurring" ones (Q, E;ementaty arithmetix, PA, various subsystems of second order arithmetic, etc.), and artificia iterations (Turing, Feferman), all of which are consistent in the standard classical view, with stronger ones proving the consistency of weaker ones.

- Martin Dowd



-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150307/18d6a12c/attachment-0001.html>

From botocudo at gmail.com  Sat Mar  7 13:37:00 2015
From: botocudo at gmail.com (Joao Marcos)
Date: Sat, 7 Mar 2015 15:37:00 -0300
Subject: [FOM] axioms for potential infinity and for actual infinity?
Message-ID: <CAO6j_Lg29eztvTGfr2GinY1SCkYeDvUvCx3w+chBOUQksrbm0Q@mail.gmail.com>

Timothy Y. Chow wrote:
>
> There is of course a long tradition in philosophy of distinguishing
> between "potential infinity" and "actual infinity."  In modern
> mathematics, this distinction doesn't seem to exist.  The closest thing to
> "potential infinity" seems to be an axiomatic system that lacks anything
> that could be identified as an explicit "axiom of infinity," yet admits
> only (actually) infinite models.  (PA would be an example.)  But for
> example, I've never seen anyone define two separate axioms and declare one
> of them to be an "axiom of potential infinity" and the other an "axiom of
> actual infinity."
>
> I'm wondering if anyone can come up with (or has already come up with)
> candidates for two such axioms, the former demonstrably weaker than the
> latter, such that the consistency of PA can be proved using only the
> "axiom of potential infinity."  Doing this might be pleasing to those who
> not only perceive an important distinction between potential and actual
> infinity but go so far as to reject the latter while accepting the former.

Hummm... what if one replaced ZF's *Axiom of Infinity*, that
postulates the existence of inductive sets, by an axiom that
postulated the existence of
sets equipped with endofunctions that are injective but not
surjective?  It is well-known that the latter sets (known as
*Dedekind-infinite*) are infinite in the usual sense, while proving
the converse demands some extra technology, such as AC? (countable
choice).

This is all standard, but I wonder if anyone has tried to use it as a
way of rescuing the age-old philosophical distinction between actual
and potential infinity.  No doubt, if Dedekind-infinite is to mean
"actual", to give the traditional Axiom of Infinity a distinct
"potential" flavor it might make sense to move first to a *predicative
version of ZF* (which will often weaken also the powerset axiom to
some sort of "axiom of exponentiation"), so that the "existence"
postulated by the axiom is given a constructive reading.  Does anyone
know if *infinite* implies *Dedekind-infinite* in Constructive ZF with
Dependent Choice?

Joao Marcos

From hmflogic at gmail.com  Sat Mar  7 14:54:07 2015
From: hmflogic at gmail.com (Harvey Friedman)
Date: Sat, 7 Mar 2015 14:54:07 -0500
Subject: [FOM] 577: New Pi01 Incompleteness
Message-ID: <CACWi-GW+ZUFekKDx=kq-+6h9tJpPwsrsumdJYOKhT7=cYZRezg@mail.gmail.com>

We have previously given "perfect" implicitly Pi01 incompleteness
using maximal squares (roots, cliques) and projections. See
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#87. Here are two samples.

PROPOSITION. Every order invariant subset of Q[0,1]^k has a maximal
square whose projections at equal length subsequences of 1,1/2,...,1/k
agree below 1/k.

PROPOSITION. Every order invariant subset of Q[0,1]^2k has a maximal
root whose projections at equal length subsequences of 1,1/2,...,1/n
agree below the min of their collective terms.

I now think of the first of these two as the leadoff perfect statement.

There are a lot of variants involving graphs and cliques instead of
maximal squares and maximal roots, fixed small dimensions, additional
parameter for length of subsequences, and consecutive terms. These
statements are all implicitly Pi01 via straightforward applications of
Goedel's completeness theorem. They are all provably equivalent to
Con(SRP) over WKL_0. I regard them as perfect, in a sense that has yet
to be seriously analyzed.

We have another kind of perfect implicitly Pi01 incompleteness, this
time using bases and upper shifts (explained below). This is an
improved version of an earlier statement.

PROPOSITION 2.2. In every order invariant relation on Q^k, some basis
of some (E union {0})^k contains its upper shift.

By an application of Goedel's completeness theorem, this also can be
seen to be implicitly Pi01. We also get provable equivalence with
Con(SRP) over WKL_0. Here there are few competitive variants. Most
notably, we can fix a small k, and still obtained equivalence with
Con(SRP).

The biggest recent advance relates to the quest for EXPLICITLY Pi01
incompleteness. Here we have a major advance toward perfection.

PROPOSITION 4.1. In every order invariant relation on Q^k, some
nonempty finite set, point reducing its nonempty finite
{0,...,k}-images, is free with its upper shift.

The above is obviously explicitly finite. It can be construed as Pi01
first by bounding the cardinality of the set and then using well known
facts about the first order theory of (Q,<). For an explicitly Pi01
form, we use {0,...,k}-coordinate images. It is easily seen that the
finite {0,...,k}-images and the {0,...,k}-coordinate images of a
finite set are the same.

PROPOSITION 5.3. In every order invariant relation on Q^k, some
nonempty finite subset of Q[(8k)!)]^k, point reducing its nonempty
{0,...,k}-coordinate images, is free with its upper shift.

Proposition 5.3 is explicitly Pi01.

As usual, we work with order invariant relations on Q^k, which are
order invariant subsets of Q^2k. We will also work with order
theoretic relations with parameters 0,...,k that are subsets of
(Q^k)^k x Q^k. Recall that for fixed k, there are only finitely many
such.

1. Bases and Upper Shifts.
2. Implicitly Pi01 Incompleteness.
3. Reductions and Principal Images.
4. Explicitly Finite Incompleteness.
5. Explicitly Pi01 Incompleteness.
6. Extreme Implicitly Pi01 Incompleteness.
7. Further Considerations.

1. BASES

DEFINITION 1.1. Let R be a relation on Q^k. x strictly reduces to y if
and only if x R y and max(x) > max(y). S is a basis for E if and only
if E containedin Q^k and S = {x in E: x does not strictly reduce to
any y in S}.

DEFINITION 1.2. The upper shift of x in Q^k is the result of adding 1
to all nonnegative coordinates of x. The upper shift of S containedin
Q^k is the set of all upper shifts of elements of S.

2. IMPLICITLY Pi01 INCOMPLETENESS

THEOREM 2.1. In every relation on Q^k, every subset of Q^k with a well
ordered set of coordinates has a unique basis.

PROPOSITION 2.2. In every order invariant relation on Q^k, some basis
of some (E union {0})^k contains its upper shift.

There is a nice way to use Goedel's Completeness Theorem to see that
Proposition 2.2 is implicitly Pi01.

THEOREM 2.3. Proposition 2.2 is provably equivalent to Con(SRP) over WKL_0.

3. REDUCTIONS AND X-IMAGES

DEFINITION 3.1. Let R be a relation on Q^k. x reduces to y if and only
if x = y in Q^k or x strictly reduces to y.  S point reduces T if and
only if S,T containedin Q^k and some element of T reduces to some
element of S.

DEFINITION 3.2. Let R be a relation on Q^k. S is free if and only if S
containedin Q^k and no element of S strictly reduces to an element of
S. S is free with T if and only if S union T is free.

DEFINITION 3.3. The X-images of S containedin Q^k are the images of
S^k under order theoretic R containedin (Q^k)^k x Q^k with parameters
from X.

4. EXPLICITLY FINITE INCOMPLETENESS

PROPOSITION 4.1. In every order invariant relation on Q^k, some
nonempty finite set, point reducing its nonempty finite
{0,...,k}-images, is free with its upper shift.

THEOREM 4.2. Proposition 4.1 is provably equivalent to Con(SRP) over EFA.

Proposition 4.1 is obviously explicitly finite, We can put Proposition
4.1 in Pi01 form by the following considerations. First observe that
if S obeys the conclusion, then there is an S' containedin S obeying
the conclusion, with cardinality bounded by an exponential expression
in k. Next observe finite {0,...,k}-images of S' must use only
coordinates of elements of S' and 0,...,k. From these observations,
for each k, the conclusion is a corresponding first order sentence in
(Q,<) with parameters 0,...,k. From elimination of quantifiers in
(Q,<), we have put Proposition 4.1 in Pi01 form.

5. EXPLICITLY Pi01 INCOMPLETENESS

DEFINITION 5.1. Let S containedin Q^k. The X-coordinate images of S
are the X-images V of S^k such that every coordinate of any element of
V is a coordinate of some element of S or an element of X

Definition 5.1 is used only to obtain explicit Pi01 incompleteness. It
is easily seen that the finite X-images are the same as the
X-coordinate images (assuming S is finite).

PROPOSITION 5.1. In every order invariant relation on Q^k, some
nonempty finite set, point reducing its nonempty {0,...,k}-coordinate
images, is free with its upper shift.

PROPOSITION 5.2. In every order invariant relation on Q^k, some
nonempty finite sett with at most (8k)! elements, point reducing its
nonempty {0,...,k}-coordinate images, is free with its upper shift.

Propositions 5.1, 5.2 are explicitly Pi02. Proposition 5.2 is Pi01 via
immediate use of elimination of quantifiers for (Q,<).

DEFINITION 5.2. Q[n]  is the set of fractions whose numerator and
denominator are of magnitude at most n.

PROPOSITION 5.3. In every order invariant relation on Q^k, some
nonempty finite subset of Q[(8k)!)]^k, point reducing its nonempty
{0,...,k}-coordinate images, is free with its upper shift.

Proposition 5.3 is explicitly Pi01.

THEOREM 5.4. Propositions 5.1 - 5.3 are provably equivalent to
Con(SRP) over EFA.

6. EXTREME IMPLICITY Pi01 INCOMPLETENESS

DEFINITION 6.1. Let S containedin Q^k and x in Q^k. S^<= = {x in S:
x_1 <= ... <= x_k}. The lower projections of S are the sets {x:
S(p,x): max(x) < p}, p in Q.

DEFINITION 6.2. Let R be a relation on Q^k. S is a <=-basis for E if
and only if S containedin E and S^<= = {x in E^<=: x does not reduce
to any y in S}.

PROPOSITION 6.1. In every order invariant relation on Q^k, for some
<=-basis of some (E union {0})^k, the elements and lower projections
include those of its upper shift.

There is a nice way to use Goedel's Completeness Theorem to see that
Proposition 6.1 is implicitly Pi01.

THEOREM 6.2. Proposition 6.1 is provably equivalent to Con(HUGE) over EFA.

7. FURTHER CONSIDERATIONS

In all Propositions mentioned here, we can require that the sets
claimed in the conclusion are either

i. Definable in (Q,<,N). This is the same as being arithmetical.
2. Recursive in 0'. This is the same as being delta-0-2.

If we make this requirement, then we get statements that are provably
equivalent to the original forms over ACA_0. This provides a
straightforward but not satisfactorily mathematical way of obtaining
explicitly finite Incompleteness.

************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 577th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-527 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2014-August/018092.html

528: More Perfect Pi01  8/16/14  5:19AM
529: Yet more Perfect Pi01 8/18/14  5:50AM
530: Friendlier Perfect Pi01
531: General Theory/Perfect Pi01  8/22/14  5:16PM
532: More General Theory/Perfect Pi01  8/23/14  7:32AM
533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
534: Perfect Explicitly Pi01  8/27/14  10:40AM
535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
536: Pi01 Progress  9/1/14 11:31AM
537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
538: Progress Pi01 9/6/14  11:31PM
539: Absolute Perfect Naturalness 9/7/14  9:00PM
540: SRM/Comparability  9/8/14  12:03AM
541: Master Templates  9/9/14  12:41AM
542: Templates/LC shadow  9/10/14  12:44AM
543: New Explicitly Pi01  9/10/14  11:17PM
544: Initial Maximality/HUGE  9/12/14  8:07PM
545: Set Theoretic Consistency/SRM/SRP  9/14/14  10:06PM
546: New Pi01/solving CH  9/26/14  12:05AM
547: Conservative Growth - Triples  9/29/14  11:34PM
548: New Explicitly Pi01  10/4/14  8:45PM
549: Conservative Growth - beyond triples  10/6/14  1:31AM
550: Foundational Methodology 1/Maximality  10/17/14  5:43AM
551: Foundational Methodology 2/Maximality  10/19/14 3:06AM
552: Foundational Methodology 3/Maximality  10/21/14 9:59AM
553: Foundational Methodology 4/Maximality  10/21/14 11:57AM
554: Foundational Methodology 5/Maximality  10/26/14 3:17AM
555: Foundational Methodology 6/Maximality  10/29/14 12:32PM
556: Flat Foundations 1  10/29/14  4:07PM
557: New Pi01  10/30/14  2:05PM
558: New Pi01/more  10/31/14 10:01PM
559: Foundational Methodology 7/Maximality  11/214  10:35PM
560: New Pi01/better  11/314  7:45PM
561: New Pi01/HUGE  11/5/14  3:34PM
562: Perfectly Natural Review #1  11/19/14  7:40PM
563: Perfectly Natural Review #2  11/22/14  4:56PM
564: Perfectly Natural Review #3  11/24/14  1:19AM
565: Perfectly Natural Review #4  12/25/14  6:29PM
566: Bridge/Chess/Ultrafinitism 12/25/14  10:46AM
567: Counting Equivalence Classes  1/2/15  10:38AM
568: Counting Equivalence Classes #2  1/5/15  5:06AM
569: Finite Integer Sums and Incompleteness  1/515  8:04PM
570: Philosophy of Incompleteness 1  1/8/15 2:58AM
571: Philosophy of Incompleteness 2  1/8/15  11:30AM
572: Philosophy of Incompleteness 3  1/12/15  6:29PM
573: Philosophy of Incompleteness 4  1/17/15  1:44PM
574: Characterization Theory 1  1/17/15  1:44AM
575: Finite Games and Incompleteness  1/23/15  10:42AM
576: Game Correction/Simplicity Theory 1  1/27/15 10:39AM

Harvey Friedman

From hmflogic at gmail.com  Sat Mar  7 14:54:40 2015
From: hmflogic at gmail.com (Harvey Friedman)
Date: Sat, 7 Mar 2015 14:54:40 -0500
Subject: [FOM] 578: Provably Falsifiable Proposiitons
Message-ID: <CACWi-GWLQc8n-YUhip3REJAr875So6QG4Xs8csJ93cqh_E54VA@mail.gmail.com>

There is a well recognized key property of certain mathematical
propositions. Informally,

*if phi is false then it is automatically refutable*

We say that phi is "provably falsifiable". E.g., before FLT was
proved, it was well recognized that FLT is provably falsifiable. After
FLT was proved, FLT is of course seen to be provably falsifiable by
default.

DEFINITION 1. A sentence phi in the language of set theory is provably
falsifiable over ZFC if and only if the sentence "if phi is false then
phi is refutable in ZFC" is itself provable in ZFC.

Here is a stronger form.

DEFINITION 2. A sentence phi in the language of set theory is provably
falsifiable over ACA_0 if and only if the statement "if phi is false
then phi is refutable in ACA_0" is itself provable in ACA_0.

THEOREM 1. Let phi be a sentence in the language of ZFC. Suppose phi
is implicitly Pi01 in the sense that there is an algorithm alpha such
that "phi iff alpha goes on forever" is provable in ZFC (ACA_0). Then
phi is provably falsifiable over ZFC (ACA_0).

Theorem 1, even with ACA_0, applies to all of the propositions we have
presented recently in Concrete Mathematical Incompleteness, such as in
FOM posting #577. Thus they are all provably falsifiable over ACA_0.

The condition "provably falsifiable" is related to falsifiability of
physical theories. Generally speaking, physical theories that are not
falsifiable by observations have rather controversial reputations.
Such physical theories are often outright rejected as not being
meaningful by many physical scientists.

Will this kind of attitude be adopted by mathematicians? I.e., that in
order for a mathematical question to be regarded as truly significant,
must it be first seen to be provably falsifiable? This attitude has
already been perhaps arguably adopted by a significant segment of
applied mathematicians.

We can use the phrases

Provably Falsifiable Mathematical Incompleteness
Provably Falsifiable Concrete Mathematical Incompleteness

as alternatives to

Pi01 Mathematical Incompleteness.

Implicitly Pi01 sentences (over ZFC, ACA_0 respectively) are
automatically Provably Falsifiable (over ZFC, ACA_0 respectively). We
do not have to use Explicitly Pi01 sentences for Provable
Falsifiability.

The work on Concrete Mathematical Incompleteness provides the only
current examples of Provably Falsifiable Mathematical Incompleteness.
The first Provably Falsifiable Incompleteness is of course due to
Goedel.

************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 578th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-527 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2014-August/018092.html

528: More Perfect Pi01  8/16/14  5:19AM
529: Yet more Perfect Pi01 8/18/14  5:50AM
530: Friendlier Perfect Pi01
531: General Theory/Perfect Pi01  8/22/14  5:16PM
532: More General Theory/Perfect Pi01  8/23/14  7:32AM
533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
534: Perfect Explicitly Pi01  8/27/14  10:40AM
535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
536: Pi01 Progress  9/1/14 11:31AM
537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
538: Progress Pi01 9/6/14  11:31PM
539: Absolute Perfect Naturalness 9/7/14  9:00PM
540: SRM/Comparability  9/8/14  12:03AM
541: Master Templates  9/9/14  12:41AM
542: Templates/LC shadow  9/10/14  12:44AM
543: New Explicitly Pi01  9/10/14  11:17PM
544: Initial Maximality/HUGE  9/12/14  8:07PM
545: Set Theoretic Consistency/SRM/SRP  9/14/14  10:06PM
546: New Pi01/solving CH  9/26/14  12:05AM
547: Conservative Growth - Triples  9/29/14  11:34PM
548: New Explicitly Pi01  10/4/14  8:45PM
549: Conservative Growth - beyond triples  10/6/14  1:31AM
550: Foundational Methodology 1/Maximality  10/17/14  5:43AM
551: Foundational Methodology 2/Maximality  10/19/14 3:06AM
552: Foundational Methodology 3/Maximality  10/21/14 9:59AM
553: Foundational Methodology 4/Maximality  10/21/14 11:57AM
554: Foundational Methodology 5/Maximality  10/26/14 3:17AM
555: Foundational Methodology 6/Maximality  10/29/14 12:32PM
556: Flat Foundations 1  10/29/14  4:07PM
557: New Pi01  10/30/14  2:05PM
558: New Pi01/more  10/31/14 10:01PM
559: Foundational Methodology 7/Maximality  11/214  10:35PM
560: New Pi01/better  11/314  7:45PM
561: New Pi01/HUGE  11/5/14  3:34PM
562: Perfectly Natural Review #1  11/19/14  7:40PM
563: Perfectly Natural Review #2  11/22/14  4:56PM
564: Perfectly Natural Review #3  11/24/14  1:19AM
565: Perfectly Natural Review #4  12/25/14  6:29PM
566: Bridge/Chess/Ultrafinitism 12/25/14  10:46AM
567: Counting Equivalence Classes  1/2/15  10:38AM
568: Counting Equivalence Classes #2  1/5/15  5:06AM
569: Finite Integer Sums and Incompleteness  1/515  8:04PM
570: Philosophy of Incompleteness 1  1/8/15 2:58AM
571: Philosophy of Incompleteness 2  1/8/15  11:30AM
572: Philosophy of Incompleteness 3  1/12/15  6:29PM
573: Philosophy of Incompleteness 4  1/17/15  1:44PM
574: Characterization Theory 1  1/17/15  1:44AM
575: Finite Games and Incompleteness  1/23/15  10:42AM
576: Game Correction/Simplicity Theory 1  1/27/15 10:39AM
577: New Pi01 Incompleteness

Harvey Friedman

From W.Taylor at math.canterbury.ac.nz  Sun Mar  8 00:27:07 2015
From: W.Taylor at math.canterbury.ac.nz (W.Taylor at math.canterbury.ac.nz)
Date: Sun, 08 Mar 2015 18:27:07 +1300
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <alpine.LFD.2.11.1503062159110.25430@laurent.mit.edu>
References: <alpine.LFD.2.11.1503062159110.25430@laurent.mit.edu>
Message-ID: <20150308182707.88gfxhuftw04ck4c@secure.math.canterbury.ac.nz>

Quoting "Timothy Y. Chow" <tchow at alum.mit.edu>:

> Arnon Avron wrote:
>
>> I would add that I *do not believe* people who pretend to doubt the  
>>  consistency of PA.

I would have said the same, up to the point where I got a private
communication from Edward Nelson about this point, perhaps 10 years ago or so.
I had politely suggested (something like) that his stance was a posture
of principle, and that he didn't really expect to find PA inconsistent.
His reply was that no, he fully expected to find such an inconsistency,
and that he was working continuously to dig one up.

> However, there is another possible avenue to doubting the consistency
> of PA.  Namely, having some grasp of the natural numbers is not the
> same as understanding every *property* of the natural numbers.

Excellent point.    I like to say that N is a crystal-clear model;
but that all of its properties are not; the latter is just set theory,
whose intended model is clear to many, and unclear to many.

> Certainly none of us is able to directly apprehend the *truth* of all
> first-order statements about the integers.  If we can be uncertain
> about the truth of (for example) the twin prime conjecture, then why
> can't we be uncertain about the truth of instances of the induction
> schema with arbitrarily many alternations of quantifiers?

I agree with the conclusion, but not necessarily the antecedent.
I recall raising a storm on a math forum by suggesting that
(for math reasons) we *knew* TPC was true, but we didn't have a proof.
Someone else made a supporting statement to counter the howls of derision,
that surely we *know* that chess with black starting with a queen off
was a forced win for white, but we have no proof of it?

> The doubts
> here would not be about the natural numbers per se but about whether
> such incomprehensibly complex formulas coherently define a "property"
> of the natural numbers to which induction can be applied.

Exactly so.  As in the chess case, a proof could be so horrible,
that we can never "behold" it all, even with computer help.
Such things are well known in math logic OC - the existence of
infeasibly long irreducible proofs.

For the record, I, like Arnon, feel that N is an obviously correct model
of PA, and that serious ontological doubt is almost ludicrous;
(while realizing that similar things have been said in the past
about cases that seemed clear-cut at the time).

Bill Taylor


----------------------------------------------------------------
This message was sent using IMP, the Internet Messaging Program.


From maarten.jordens at canterbury.ac.nz  Sun Mar  8 16:12:04 2015
From: maarten.jordens at canterbury.ac.nz (Maarten McKubre-Jordens)
Date: Sun, 08 Mar 2015 20:12:04 +0000
Subject: [FOM] axioms for potential infinity and for actual infinity?
Message-ID: <FBFAC81B7E2F7142A12EE91AA5FDB43131E63888@UCEXMBX02-D.canterbury.ac.nz>

Joao Marcos wrote:

> Hummm... what if one replaced ZF's *Axiom of Infinity*, that
> postulates the existence of inductive sets, by an axiom that
> postulated the existence of
> sets equipped with endofunctions that are injective but not
> surjective?  It is well-known that the latter sets (known as
> *Dedekind-infinite*) are infinite in the usual sense, while proving
> the converse demands some extra technology, such as AC? (countable
> choice).
>
> This is all standard, but I wonder if anyone has tried to use it as a
> way of rescuing the age-old philosophical distinction between actual
> and potential infinity.  No doubt, if Dedekind-infinite is to mean
> "actual", to give the traditional Axiom of Infinity a distinct
> "potential" flavor it might make sense to move first to a *predicative
> version of ZF* (which will often weaken also the powerset axiom to
> some sort of "axiom of exponentiation"), so that the "existence"
> postulated by the axiom is given a constructive reading.  Does anyone
> know if *infinite* implies *Dedekind-infinite* in Constructive ZF with
> Dependent Choice?

I don't know if this is a help, but there's a paper from Lubarsky and Rathjen where they construct a Kripke model that shows that, without Dependent Choice, the Dedekind reals form a proper class whereas the Cauchy reals form a set. The equivalence of the latter two in Bishop-style constructive mathematics can be explained by the acceptance of Dependent Choice in that theory. The reference is:

Robert S. Lubarsky & Michael Rathjen, 'On the constructive Dedekind reals', Logic and Analysis 1 (2):131-152 (2008).

Based on that, my guess would be that infinite does imply Dedekind-infinite provided one has Dependent Choice.

Best,
Maarten


This email may be confidential and subject to legal privilege, it may
not reflect the views of the University of Canterbury, and it is not
guaranteed to be virus free. If you are not an intended recipient,
please notify the sender immediately and erase all copies of the message
and any attachments.

Please refer to http://www.canterbury.ac.nz/emaildisclaimer for more
information.

From aa at tau.ac.il  Mon Mar  9 04:56:22 2015
From: aa at tau.ac.il (Arnon Avron)
Date: Mon, 9 Mar 2015 10:56:22 +0200
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <alpine.LFD.2.11.1503062159110.25430@laurent.mit.edu>
References: <alpine.LFD.2.11.1503062159110.25430@laurent.mit.edu>
Message-ID: <20150309085622.GC2866@localhost.localdomain>

On Fri, Mar 06, 2015 at 10:17:51PM -0500, Timothy Y. Chow wrote:
 
> However, there is another possible avenue to doubting the
> consistency of PA.  Namely, having some grasp of the natural numbers
> is not the same as understanding every *property* of the natural
> numbers.  Certainly none of us is able to directly apprehend the
> *truth* of all first-order statements about the integers.  If we can
> be uncertain about the truth of (for example) the twin prime
> conjecture, then why can't we be uncertain about the truth of
> instances of the induction schema with arbitrarily many alternations
> of quantifiers?  

The comparison is inadequate. Already Euclide distinguished between self-evident
propositions and those that are not, and realized that we can be sure about the 
truth of a proposition of the latter type  only
via a valid proof of it from the self-evident propositions.

Unilike TPC, derstanding the validity of the induction principle is inherent in
the understanding of the natural numbers. As Feferman has put it, it is a principle
that anyone that accepts the natural number should accept as well.

> The doubts here would not be about the natural
> numbers per se but about whether such incomprehensibly complex
> formulas coherently define a "property" of the natural numbers to
> which induction can be applied.

This is a better point. Here, if I understand you correctly, you do not
compare the induction principle to the TPC, or doubt its validity,
but point out that applying it depends on what we accept as a 
"property" of the natural numbers. I agree. So the question boils
down to whether any formula in the language of PA
defines a definite property of the natural numbers. Assuming that nobody
questions this in the case of the atomic formulas of that language,
the answer is affirmative -  provided we agree that if A and B define
such properties then so do -A, A&B, and \exists x.A (say). Once one accepts 
this then BY INDUCTION(!) every formula of the language of PA
defines a definite property. (And there is a really no difference here 
between formulas with  5 alternations of quantifiers and formula 
with 6 alternations of quantifiers... If you think otherwise, can you point
out what is the minimal number of alternations that makes formulas doubtful,
and what is special about that number?). Now I do not think that one can
truly and honestly doubt this induction step. In particular: assuming that all
free variables in A are assigned values in N, "\exists x.A"
is true if by trying to substitute 0 for x, then 1 for x, then 2 for x etc
we eventually get a true sentence, and "\exists x.A" is false otherwise. 
If somebody really does not undersrtand this, then I can do nothing about it, 
but for me (and I strongly believe that deep inside by everyone else) this is crystal 
clear. Moreover: for this a *potential* understanding of the
quantifiers and the infinity of N suffices!

> Although I don't have such doubts myself, I don't see anything
> incoherent about them.  After all, we know many ways of weakening
> the induction axiom to produce a system whose consistency doesn't
> imply the consistency of PA. Hence we have a fairly clear picture of
> what it could mean for the natural numbers to exist and to satisfy a
> weaker induction schema, yet for PA to be inconsistent.

???

I do not see the connection  between the two parts of this paragraph (before
and after the "Hence..."). Yes, I do know "many ways of weakening
the induction axiom to produce a system whose consistency doesn't
imply the consistency of PA". The simplest of them is to throw induction away 
altogether (like in Q). Still, I do not have any picture (let alone "fairly clear" one)
what it could mean for the *natural numbers* to exist, yet for PA to be inconsistent.

Arnon

From aa at tau.ac.il  Mon Mar  9 05:40:41 2015
From: aa at tau.ac.il (Arnon Avron)
Date: Mon, 9 Mar 2015 11:40:41 +0200
Subject: [FOM] Potential infinity
In-Reply-To: <alpine.LFD.2.11.1503061756130.24538@laurent.mit.edu>
References: <mailman.442.1425346301.28387.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>
	<54F76C30.5060208@brown.edu>
	<ECAF14BB08F59040BA4E3788A495E387137DF939@XM-MBX-03-PROD.ad.uchicago.edu>
	<54F8BE08.5080406@brown.edu>
	<alpine.LFD.2.11.1503061756130.24538@laurent.mit.edu>
Message-ID: <20150309094041.GD2866@localhost.localdomain>

On Fri, Mar 06, 2015 at 06:11:45PM -0500, Timothy Y. Chow wrote:
> 
> There is of course a long tradition in philosophy of distinguishing
> between "potential infinity" and "actual infinity."  In modern
> mathematics, this distinction doesn't seem to exist.  

I disagree.

The terminology used by many might indeed have changed, but not the need to
distinguish between the two types of infinity. Thus in set theory we
do distinguish between infinite collections like N, which are objects
themselves, and so are taken as "complete", and proper classes like the
universe of ZF (whatever this means) which is not a "complete"
object and is "absolutely infinite" (in Cantor's terminology). So instead
of talking about potential versus actual infinity, platonists speak now
about non-absolute versus absolute infinity, and push the limit
strongly higher. Still, I believe that many (perhaps most) mathematicians
view V not as a well-determined object, but as a potential collection
of actual objects (exactly like the way most predicativists see N).

I take this opportunity to note about the false claim that quantifying 
over N in general, and PA in particular, are justified only if one
accepts N as a complete object which is actually infinite.
According to this "logic", the quantification
that is done in ZF over the universe of sets  is
justified  only if one accepts V as a complete object which is absolutely infinite.

As I have hinted in my previous posting, for seeing  the validity of the axioms of PA 
a potential understanding of the collection of the natural numbers suffices.

Arnon

From fabio.zanasi at ens-lyon.fr  Mon Mar  9 06:06:43 2015
From: fabio.zanasi at ens-lyon.fr (fabio.zanasi at ens-lyon.fr)
Date: Mon,  9 Mar 2015 11:06:43 +0100 (CET)
Subject: [FOM] CALCO 2015 : Last Call for Papers
Message-ID: <20150309100643.42AEAA3B93@jabiru.ens-lyon.fr>


=========================================================

              CALL FOR PAPERS:  CALCO 2015

6th International Conference on Algebra and Coalgebra in Computer Science

	     In cooperation with ACM SIGLOG

                 June 24 - 26, 2015

                Nijmegen, Netherlands

               http://coalg.org/calco15/

==========================================================

       Abstract submission:    March 22, 2015
       Paper submission:       April 2, 2015
       Author notification:    May 6, 2015
       Final version due:      June 3, 2015

==========================================================

-- SCOPE --

CALCO aims to bring together researchers and practitioners with
interests in foundational aspects, and both traditional and emerging
uses of algebra and coalgebra in computer science.

It is a high-level, bi-annual conference formed by joining the forces 
and reputations of CMCS (the International Workshop on Coalgebraic 
Methods in Computer Science), and WADT (the Workshop on Algebraic 
Development Techniques). Previous CALCO editions took place in 
Swansea (Wales, 2005), Bergen (Norway, 2007), Udine (Italy, 2009), 
Winchester (UK, 2011) and Warsaw (Poland, 2013). The sixth edition will 
be held in Nijmegen, the Netherlands, colocated with MFPS XXXI.

-- INVITED SPEAKERS --

Andy Pitts - University of Cambridge, UK (joint with MFPS)
Chris Heunen - University of Oxford, UK
Matteo Mio - CNRS, ENS Lyon, FR
Daniela Petrisan - Radboud University, Nijmegen, NL

-- TOPICS OF INTEREST --

We invite submissions of technical papers that report results of
theoretical work on the mathematics of algebras and coalgebras, the
way these results can support methods and techniques for software
development, as well as experience with the transfer of the resulting
technologies into industrial practice. We encourage submissions in
topics included or related to those listed below.

 * Abstract models and logics
   - Automata and languages
   - Categorical semantics
   - Modal logics
   - Relational systems
   - Graph transformation
   - Term rewriting

 * Specialised models and calculi
   - Hybrid, probabilistic, and timed systems
   - Calculi and models of concurrent, distributed, mobile, and
     context-aware computing
   - General systems theory and computational models (chemical,
     biological, etc.)

 * Algebraic and coalgebraic semantics
   - Abstract data types
   - Inductive and coinductive methods
   - Re-engineering techniques (program transformation)
   - Semantics of conceptual modelling methods and techniques
   - Semantics of programming languages

 * System specification and verification
   - Algebraic and coalgebraic specification
   - Formal testing and quality assurance
   - Validation and verification
   - Generative programming and model-driven development
   - Models, correctness and (re)configuration of
     hardware/middleware/architectures,
   - Process algebra

 * Corecursion in Programming Languages
    - Corecursion in logic / constraint / functional / answer set
      programming
    - Corecursive type inference
    - Coinductive methods for proving program properties
    - Implementing corecursion
    - Applications

 * Algebra and Coalgebra in quantum computing
    - Categorical semantics for quantum computing
    - Quantum calculi and programming languages
    - Foundational structures for quantum computing
    - Applications of quantum algebra

-- NEW TOPIC --

This edition of CALCO will feature a new topic, and submission of papers
in this area is particularly encouraged.

* String Diagrams and Network Theory
    - Combinatorial approaches 
    - Theory of PROPs and operads
    - Rewriting problems and higher-dimensional approaches
    - Automated reasoning with string diagrams
    - Applications of string diagrams
    - Connections with Control Theory, Engineering and Concurrency


-- SUBMISSION GUIDELINES --

Prospective authors are invited to submit full papers in English
presenting original research. Submitted papers must be unpublished and
not submitted for publication elsewhere. Experience papers are
welcome, but they must clearly present general lessons learned that
would be of interest and benefit to a broad audience of both
researchers and practitioners. Starting with CALCO 2015, proceedings
will be published in the Dagstuhl LIPIcs???Leibniz International Proceedings 
in Informatics series. Final papers should be no more than 15 pages long 
in the format specified by LIPIcs 
(http://www.dagstuhl.de/publikationen/lipics/anleitung-fuer-autoren/).
It is recommended that submissions adhere to that format and
length. Submissions that are clearly too long may be rejected
immediately. Proofs omitted due to space limitations may be included
in a clearly marked appendix. Both an abstract and the full paper must
be submitted by their respective submission deadlines.

A special issue of the open access journal Logical Methods in Computer
Science (http://www.lmcs-online.org), containing extended versions of
selected papers, is also being planned.

Submissions will be handled via EasyChair
https://www.easychair.org/conferences/?conf=calco2015

-- BEST PAPER AND BEST PRESENTATION AWARDS --

Following from the successful trial at CALCO 2013, this edition of 
CALCO will feature two awards: a best paper award whose recipients will 
be selected by the PC before the conference and a best presentation 
award, elected by the participants.

-- IMPORTANT DATES --

Abstract submission:    March 22, 2015
Paper submission:       April 2, 2015
Author notification:    May 6, 2015
Final version due:      June 3, 2015

-- PROGRAMME COMMITTEE --

Samson Abramsky, University of Oxford, UK
Andrej Bauer, University of Ljubljana, SLO
Filippo Bonchi, CNRS and ENS Lyon, FR
Corina Cirstea, University of Southampton, UK
Andrea Corradini, University of Pisa, IT
Ross Duncan, University of Strathclyde, UK
Mart??n Escard??, University of Birmingham, UK
Dan Ghica, University of Birmingham, UK
Helle Hansen, Radboud University Nijmegen and CWI, NL
Ichiro Hasuo, University of Tokyo, JP
Bart Jacobs, Radboud University Nijmegen, NL
Bartek Klin, University of Warsaw, PL
Barbara K??nig, University of Duisburg-Essen, D
Dexter Kozen, Cornell, US
Alexander Kurz, University of Leicester, UK
Paul-Andr?? Melli??s, CNRS and University Paris VII, FR
Stefan Milus, University of Erlangen-N??rnberg, D
Larry Moss (co-chair), Indiana University, US
Dusko Pavlovic, University of Hawaii, US
Daniela Petrisan, ENS Lyon, FR
Damien Pous, ENS Lyon, FR
John Power, University of Bath, UK
Jan Rutten, Radboud University Nijmegen and CWI, NL
Lutz Schroeder, University of Erlangen-Nuernberg, D
Monika Seisenberger, University of Swansea, UK
Alexandra Silva, Radboud University Nijmegen, NL
Pawel Sobocinski (co-chair), University of Southampton, UK
Ana Sokolova, University of Salzburg, AT
Andrzej Tarlecki, University of Warsaw, PL 

-- ORGANISING COMMITTEE --

Alexandra Silva
Bart Jacobs
Nicole Messink
Sam Staton 

-- PUBLICITY --

Fabio Zanasi

-- LOCATION --

Nijmegen is the oldest city in the Netherlands and celebrated its 
2,000th year of existence in 2005. It is situated in the eastern 
province of Gelderland, quite near to the German border. The latin 
name for Nijmegen, `Noviomagus??, is a reminder of its Roman past. 
`Noviomagus?? means `new market?? and refers to the right to hold a 
market as granted by the Romans. In the days of Charlemagne, the 
city was called `Numaga??; later on, this became `Nieumeghen?? and 
`Nimmegen??. However, citizens born and bred in Nijmegen speak 
affectionately of `Nimwegen??.

Nijmegen is one of the warmest cities of the Netherlands, especially 
during summer, when the highest temperatures in the country are 
usually measured in the triangle Roermond ??? Nijmegen ??? Eindhoven. 
The lack of north-south oriented mountain ranges in Europe make this 
area prone to sudden shifts in weather, giving the region a 
semi-continental climate.

-- SATELLITE WORKSHOPS --

The workshop is intended to enable presentation of work in progress and 
original research proposals. PhD students and young researchers are 
particularly encouraged to contribute. CALCO 2015 will run together 
with the CALCO Early Ideas Workshop, with dedicated Early Ideas sessions 
at the end of each conference day.

-- CALCO Early Ideas Overview --

The CALCO Early Ideas Workshop invites submissions on the same topics
as the CALCO conference: reporting results of theoretical work on the
mathematics of algebras and coalgebras, the way these results can
support methods and techniques for software development, as well as
experience with the transfer of the resulting technologies into
industrial practice. The list of topics of particular interest is
shown on the main CALCO 2015 page.

CALCO Early Ideas presentations will be selected according to originality,
significance, and general interest, on the basis of submitted 2-page short
contributions.  It can be work in progress, a summary of work submitted to a
conference or workshop elsewhere, or work that in some other way might be 
interesting to the CALCO audience. A booklet with the accepted short 
contributions will be made available.

We encourage PhD students and young researchers to submit Early Ideas papers 
to the CALCO 2015 Easychair site as for ordinary submissions, mentioning in 
the abstract that the paper is to be considered for an Early Ideas talk.

===========
SIGLOG Anti-harassment Policy

The open exchange of ideas and the freedom of thought and expression
are central to the values and goals of SIGLOG. They require an
environment that recognizes the inherent worth of every person and
group. They flourish in communities that foster mutual understanding
and embrace diversity. For these reasons, SIGLOG is committed to
providing a harassment-free conference experience, and implements the ACM
policy against harassment (see http://www.acm.org/sigs/volunteer_resources/officers_manual/anti-harassment-policy).

Conference participants violating these standards may be sanctioned or
expelled from the meeting, at the discretion of the conference
organizers. Conference organizers are requested to report serious
incidents to the SIGLOG Chair.
=========


From aa at tau.ac.il  Mon Mar  9 10:06:47 2015
From: aa at tau.ac.il (Arnon Avron)
Date: Mon, 9 Mar 2015 16:06:47 +0200
Subject: [FOM] 578: Provably Falsifiable Proposiitons
In-Reply-To: <CACWi-GWLQc8n-YUhip3REJAr875So6QG4Xs8csJ93cqh_E54VA@mail.gmail.com>
References: <CACWi-GWLQc8n-YUhip3REJAr875So6QG4Xs8csJ93cqh_E54VA@mail.gmail.com>
Message-ID: <20150309140647.GA29315@localhost.localdomain>

As is well-known, Hilbert has divided mathematical
propositions into two types: the `real' ones and the
`ideal" ones, taking only the real ones as really 
meaningful.

  What is the *essential* difference, if any, between
Hilbert's real propositions and the "provably falsifiable" ones?
(Except of course that the latter is rigorously defined,
while the concept of real proposition was not, and that
"provably falsifiable" is relative to a theory, while
"real proposition" was not supposed to be). 

Was not "that in  order for a mathematical question to be regarded 
as truly significant, it must be first seen to be provably falsifiable"
exactly Hilbert's view, on which his program was based?

Arnon


On Sat, Mar 07, 2015 at 02:54:40PM -0500, Harvey Friedman wrote:
> There is a well recognized key property of certain mathematical
> propositions. Informally,
> 
> *if phi is false then it is automatically refutable*
> 
> We say that phi is "provably falsifiable". E.g., before FLT was
> proved, it was well recognized that FLT is provably falsifiable. After
> FLT was proved, FLT is of course seen to be provably falsifiable by
> default.
> 
> DEFINITION 1. A sentence phi in the language of set theory is provably
> falsifiable over ZFC if and only if the sentence "if phi is false then
> phi is refutable in ZFC" is itself provable in ZFC.
> 
> Here is a stronger form.
> 
> DEFINITION 2. A sentence phi in the language of set theory is provably
> falsifiable over ACA_0 if and only if the statement "if phi is false
> then phi is refutable in ACA_0" is itself provable in ACA_0.
> 
> THEOREM 1. Let phi be a sentence in the language of ZFC. Suppose phi
> is implicitly Pi01 in the sense that there is an algorithm alpha such
> that "phi iff alpha goes on forever" is provable in ZFC (ACA_0). Then
> phi is provably falsifiable over ZFC (ACA_0).
> 
> Theorem 1, even with ACA_0, applies to all of the propositions we have
> presented recently in Concrete Mathematical Incompleteness, such as in
> FOM posting #577. Thus they are all provably falsifiable over ACA_0.
> 
> The condition "provably falsifiable" is related to falsifiability of
> physical theories. Generally speaking, physical theories that are not
> falsifiable by observations have rather controversial reputations.
> Such physical theories are often outright rejected as not being
> meaningful by many physical scientists.
> 
> Will this kind of attitude be adopted by mathematicians? I.e., that in
> order for a mathematical question to be regarded as truly significant,
> must it be first seen to be provably falsifiable? This attitude has
> already been perhaps arguably adopted by a significant segment of
> applied mathematicians.
> 
> We can use the phrases
> 
> Provably Falsifiable Mathematical Incompleteness
> Provably Falsifiable Concrete Mathematical Incompleteness
> 
> as alternatives to
> 
> Pi01 Mathematical Incompleteness.
> 
> Implicitly Pi01 sentences (over ZFC, ACA_0 respectively) are
> automatically Provably Falsifiable (over ZFC, ACA_0 respectively). We
> do not have to use Explicitly Pi01 sentences for Provable
> Falsifiability.
> 
> The work on Concrete Mathematical Incompleteness provides the only
> current examples of Provably Falsifiable Mathematical Incompleteness.
> The first Provably Falsifiable Incompleteness is of course due to
> Goedel.
> 
> ************************************************************
> My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
> https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
> This is the 578th in a series of self contained numbered
> postings to FOM covering a wide range of topics in f.o.m. The list of
> previous numbered postings #1-527 can be found at the FOM posting
> http://www.cs.nyu.edu/pipermail/fom/2014-August/018092.html
> 
> 528: More Perfect Pi01  8/16/14  5:19AM
> 529: Yet more Perfect Pi01 8/18/14  5:50AM
> 530: Friendlier Perfect Pi01
> 531: General Theory/Perfect Pi01  8/22/14  5:16PM
> 532: More General Theory/Perfect Pi01  8/23/14  7:32AM
> 533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
> 534: Perfect Explicitly Pi01  8/27/14  10:40AM
> 535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
> 536: Pi01 Progress  9/1/14 11:31AM
> 537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
> 538: Progress Pi01 9/6/14  11:31PM
> 539: Absolute Perfect Naturalness 9/7/14  9:00PM
> 540: SRM/Comparability  9/8/14  12:03AM
> 541: Master Templates  9/9/14  12:41AM
> 542: Templates/LC shadow  9/10/14  12:44AM
> 543: New Explicitly Pi01  9/10/14  11:17PM
> 544: Initial Maximality/HUGE  9/12/14  8:07PM
> 545: Set Theoretic Consistency/SRM/SRP  9/14/14  10:06PM
> 546: New Pi01/solving CH  9/26/14  12:05AM
> 547: Conservative Growth - Triples  9/29/14  11:34PM
> 548: New Explicitly Pi01  10/4/14  8:45PM
> 549: Conservative Growth - beyond triples  10/6/14  1:31AM
> 550: Foundational Methodology 1/Maximality  10/17/14  5:43AM
> 551: Foundational Methodology 2/Maximality  10/19/14 3:06AM
> 552: Foundational Methodology 3/Maximality  10/21/14 9:59AM
> 553: Foundational Methodology 4/Maximality  10/21/14 11:57AM
> 554: Foundational Methodology 5/Maximality  10/26/14 3:17AM
> 555: Foundational Methodology 6/Maximality  10/29/14 12:32PM
> 556: Flat Foundations 1  10/29/14  4:07PM
> 557: New Pi01  10/30/14  2:05PM
> 558: New Pi01/more  10/31/14 10:01PM
> 559: Foundational Methodology 7/Maximality  11/214  10:35PM
> 560: New Pi01/better  11/314  7:45PM
> 561: New Pi01/HUGE  11/5/14  3:34PM
> 562: Perfectly Natural Review #1  11/19/14  7:40PM
> 563: Perfectly Natural Review #2  11/22/14  4:56PM
> 564: Perfectly Natural Review #3  11/24/14  1:19AM
> 565: Perfectly Natural Review #4  12/25/14  6:29PM
> 566: Bridge/Chess/Ultrafinitism 12/25/14  10:46AM
> 567: Counting Equivalence Classes  1/2/15  10:38AM
> 568: Counting Equivalence Classes #2  1/5/15  5:06AM
> 569: Finite Integer Sums and Incompleteness  1/515  8:04PM
> 570: Philosophy of Incompleteness 1  1/8/15 2:58AM
> 571: Philosophy of Incompleteness 2  1/8/15  11:30AM
> 572: Philosophy of Incompleteness 3  1/12/15  6:29PM
> 573: Philosophy of Incompleteness 4  1/17/15  1:44PM
> 574: Characterization Theory 1  1/17/15  1:44AM
> 575: Finite Games and Incompleteness  1/23/15  10:42AM
> 576: Game Correction/Simplicity Theory 1  1/27/15 10:39AM
> 577: New Pi01 Incompleteness
> 
> Harvey Friedman
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom

From pisanoraffaele at iol.it  Mon Mar  9 08:15:19 2015
From: pisanoraffaele at iol.it (pisanoraffaele at iol.it)
Date: Mon, 9 Mar 2015 13:15:19 +0100 (CET)
Subject: [FOM] Agassi | Kragh | Maxwell | Radelet de-Grave | Rahman in
 Lille: His. & Phil. Sci., Education, Summer School
Message-ID: <1424177706.2097861425903319612.JavaMail.httpd@webmail-35.iol.local>

(With apologies for cross-posting)  Of possible interest to some Member, 
   Delighted to have Emeriti Professors as Lecturers &amp; Keynote Speakers: 

Joseph Agassi (Israel)
Helge Kragh (Denmark)
Nicholas Maxwell (United Kingdom)
Patricia Radelet de-Grave (Belgium) 

and 
Shahid Rahman (France) 

in History and Philosophy of Science at the: 
1st International Summer School for Sciences, History and Philosophy of Sciences &amp; Science Education. New Educational and Fundamental Insights for Sciences and History-Epistemology-Philosophy of Sciences, &amp; Science Education  June 22nd-26th, 2015 | MESHS, Lille, France&amp;
  Roundtable &amp; Open Debate  Exploring
 Changes in How the Histories and Philosophies of Sciences Have Been 
Written: Interpreting the Dynamics of Change in these Sciences and 
Interrelations Amongst Them?Past Problems, Future Cures?June 26th 2015 | MESHS, Lille, France
You are welcome, join us!        http://summerschoollille2015.historyofscience.it 

Both events hosted by  Inter-Divisional Teaching Commission (IDTC)   and  Maison Europ?enne des Sciences de l'Homme et de la Soci?t? (MESHS).  
Patronage/Promoted/Collaboration by: IDTC, MESHS, UFR de Physique-Lille 1 University, DHST, DLMPS, HOPOS, IHPST, AAHPSSS

Delighted to have distinguished professors as Lecturers, as well:Abdelkader  Anakkar (France)Romano Gatto (Italy) 
Snezana Lawrence (United Kingdom)Michal Kokowski (Poland) 
Peter Heering &amp; Martin Panusch* (Germany)Mo?ra  M?ler (Spain)Agamenon Oliveira (Brazil) Bill  Palmer (Australia)Denise  Orange-Ravachol (France)Hayo  Siemsen (Germany)

et al..read more..Lecturers, Participants &amp; Roundtable Event Please visit our International Scientific Committee: 
http://summerschoollille2015.historyofscience.it/index.php/en/scientific-commitee


----------------


The Summer School is devoted to Sciences, History and Philosophy of Sciences &amp; Science Education. Admission is based solely on the individual merits of each Elegible candidate:
Young scholarsPostgraduatePh.D. candidatesTeachers (from other disciplines are welcome)No limitation concerning age, country or status/position
Fees per Elegible Participants: 250 Euros (5 days, all inclusive - except travel-roundtrip):
Five Days ISSHPSE-2015 International Summer School Enrolment.Accommodation in a Selected Hotel in
 the heart of Lille for five nights (22-26 June 2015)...Lunch....Early Registration, from  January 30th, 2015: 250 EuroRegistration from March 30th, 2015: 300 EuroClosing Payments:  April 30th, 2015....read more
The Roundtable is part of the Summer School, but organized as a standalone open &amp; free debate-colloquium in History, Epistemology and Philosophy of Science.

No fees for participants; free entry. 
Booking is mandatory (raffaele.pisano at gmail.com)



Please spread ithe events among your Ph.D. candidates and post doc scholars.  Thank you! 


Thank you for your consideration and my apologies for cross-posting if your email work with 
multiple lists.

Looking forward to meeting you in Lille.


Yours sincerely,On Behalf of the ISSHPSE?2015 Organizing Committee, Summer School DirectorRaffaele Pisano     ---------------------------------------------
Raffaele Pisano, Ph.D.
Lecturer-Researcher in History of Physics and History and Epistemology of Science
Vice President of the Inter-divisional Teaching Commission of the DLMPS/IUHPST http://idtc-iuhps.org/officers
University of Lille 1, Department of Physics |  B?t P5 bis, Bureau 168 
59655 Villeneuve d'Ascq Cedex | France

Office: +33 (0)3.20.43.46.49   |  French Mobile: +33 (0)6 70.64.32.07    |    Italian Mobile:  +39 339-31.64.592  
E-mail: pisanoraffaele at iol.it   |  Skype: pisanoraffaele
Past Officer (2006-2012) The European Society for the History of Science

Gillispie CC, Pisano R (2014) Lazare and Sadi Carnot. A Scientific and Filial Relationship. Springer, Dordrecht
http://www.springer.com/engineering/book/978-94-017-8010-0

LinkedIn Profile
fr.linkedin.com/pub/raffaele-pisano/6/a11/a61/     

"Thought does not respect national frontiers. Yet scientific ideas are far from stateless citizens". (Thackray, A. (1970). Atoms and Powers. Cambridge MA: The Harvard Univ. Press, p. 4, line 4).
--------------------------------------------------------------
Informativa: legge sulla privacy: Ai sensi dell'art. 10 della legge 31 dicembre 1996 n. 675, il vostro indirizzo e-mail ? utilizzato per ricevere alcune nostre news su eventi di carattere culturale. Qualora desideraste essere eliminati dalla nostra mailing list comunicatelo via e-mail a pisanoraffaele at iol.it indicando "unsubscribe" nel subject. 
--------------------------------------------------------------




-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150309/37617262/attachment.html>

From tchow at alum.mit.edu  Mon Mar  9 11:55:14 2015
From: tchow at alum.mit.edu (Timothy Y. Chow)
Date: Mon, 9 Mar 2015 11:55:14 -0400 (EDT)
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <20150309085622.GC2866@localhost.localdomain>
References: <alpine.LFD.2.11.1503062159110.25430@laurent.mit.edu>
	<20150309085622.GC2866@localhost.localdomain>
Message-ID: <alpine.LFD.2.11.1503091134520.14548@laurent.mit.edu>

On Mon, 9 Mar 2015, Arnon Avron wrote:
> (And there is a really no difference here between formulas with 5 
> alternations of quantifiers and formula with 6 alternations of 
> quantifiers... If you think otherwise, can you point out what is the 
> minimal number of alternations that makes formulas doubtful, and what is 
> special about that number?).

Again to play devil's advocate, suppose that one accepts the induction 
schema outright only for quantifier-free formulas?  Specific formulas with 
quantifiers could also be accepted on a case-by-case basis, but only after 
one has gone through the mental process of thinking about the formula and 
convincing oneself that induction for that property is "obvious."

> I do not see the connection between the two parts of this paragraph 
> (before and after the "Hence..."). Yes, I do know "many ways of 
> weakening the induction axiom to produce a system whose consistency 
> doesn't imply the consistency of PA". The simplest of them is to throw 
> induction away altogether (like in Q). Still, I do not have any picture 
> (let alone "fairly clear" one) what it could mean for the *natural 
> numbers* to exist, yet for PA to be inconsistent.

"Fairly clear" might be an overstatement, but as an analogy, let's 
consider the power set axiom P of ZF.  Someone might feel that they can 
grasp the set-theoretic universe (including of course the set of natural 
numbers) clearly enough to be convinced that ZF - P is consistent, yet not 
grasp the concept of an "arbitrary subset" with enough clarity to rule out 
the possibility that ZF is inconsistent.

Do you agree with that?  If so, then returning to PA, someone might feel 
that they understand what is meant by "1, 2, 3, 4, and so on indefinitely" 
yet not understand what an "arbitrary first-order property" of the natural 
numbers is with enough clarity to rule out the possibility that PA is 
inconsistent.  The induction that you describe for first-order properties 
is conceptually more complicated than the induction needed to grasp the 
successor operation.  And of course, we have technically precise ways of 
demonstrating that the greater conceptual complexity is real and not 
illusory.  Given this, I don't find it convincing to argue that induction 
for first-order formulas is part of the *fundamental meaning* of the 
natural numbers, any more than I find it convincing to argue that the 
power set axiom is part of the *fundamental meaning* of the set-theoretic 
universe.

Tim

From tchow at alum.mit.edu  Mon Mar  9 12:01:08 2015
From: tchow at alum.mit.edu (Timothy Y. Chow)
Date: Mon, 9 Mar 2015 12:01:08 -0400 (EDT)
Subject: [FOM] Potential infinity
In-Reply-To: <20150309094041.GD2866@localhost.localdomain>
References: <mailman.442.1425346301.28387.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>
	<54F76C30.5060208@brown.edu>
	<ECAF14BB08F59040BA4E3788A495E387137DF939@XM-MBX-03-PROD.ad.uchicago.edu>
	<54F8BE08.5080406@brown.edu>
	<alpine.LFD.2.11.1503061756130.24538@laurent.mit.edu>
	<20150309094041.GD2866@localhost.localdomain>
Message-ID: <alpine.LFD.2.11.1503091157000.14548@laurent.mit.edu>

On Mon, 9 Mar 2015, Arnon Avron wrote:
> As I have hinted in my previous posting, for seeing the validity of the 
> axioms of PA a potential understanding of the collection of the natural 
> numbers suffices.

Can you be more formal about this point?  What I mean is, suppose I asked 
you to formalize the argument in your previous posting.  Would it 
correspond to one of the standard proofs of the consistency of PA?  Which 
one?  Presumably it would not be a ZF-proof, since ZF explicitly assumes 
the set of natural numbers as a complete infinity, and you're claiming 
that your proof employs no such assumption.

Tim

From leivant at indiana.edu  Mon Mar  9 20:04:59 2015
From: leivant at indiana.edu (Daniel Leivant)
Date: Mon, 09 Mar 2015 20:04:59 -0400
Subject: [FOM] Potential infinity
In-Reply-To: <20150309094041.GD2866@localhost.localdomain>
References: <mailman.442.1425346301.28387.fom@cs.nyu.edu>	<alpine.LFD.2.11.1503022137550.15707@laurent.mit.edu>	<54F76C30.5060208@brown.edu>	<ECAF14BB08F59040BA4E3788A495E387137DF939@XM-MBX-03-PROD.ad.uchicago.edu>	<54F8BE08.5080406@brown.edu>	<alpine.LFD.2.11.1503061756130.24538@laurent.mit.edu>
	<20150309094041.GD2866@localhost.localdomain>
Message-ID: <54FE352B.3010000@indiana.edu>

A formalism related to potential infinity is described in
http://www.sciencedirect.com/science/article/pii/S0890540184710388
One starts there with a comprehension principle well below PRA.

Daniel

On 03/09/2015 05:40 AM, Arnon Avron wrote:
> On Fri, Mar 06, 2015 at 06:11:45PM -0500, Timothy Y. Chow wrote:
>> There is of course a long tradition in philosophy of distinguishing
>> between "potential infinity" and "actual infinity."  In modern
>> mathematics, this distinction doesn't seem to exist.
> I disagree.
>
> The terminology used by many might indeed have changed, but not the need to
> distinguish between the two types of infinity. Thus in set theory we
> do distinguish between infinite collections like N, which are objects
> themselves, and so are taken as "complete", and proper classes like the
> universe of ZF (whatever this means) which is not a "complete"
> object and is "absolutely infinite" (in Cantor's terminology). So instead
> of talking about potential versus actual infinity, platonists speak now
> about non-absolute versus absolute infinity, and push the limit
> strongly higher. Still, I believe that many (perhaps most) mathematicians
> view V not as a well-determined object, but as a potential collection
> of actual objects (exactly like the way most predicativists see N).
>
> I take this opportunity to note about the false claim that quantifying
> over N in general, and PA in particular, are justified only if one
> accepts N as a complete object which is actually infinite.
> According to this "logic", the quantification
> that is done in ZF over the universe of sets  is
> justified  only if one accepts V as a complete object which is absolutely infinite.
>
> As I have hinted in my previous posting, for seeing  the validity of the axioms of PA
> a potential understanding of the collection of the natural numbers suffices.
>
> Arnon
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom

-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150309/f08a390b/attachment-0001.html>

From m.mostowski at uw.edu.pl  Mon Mar  9 23:52:32 2015
From: m.mostowski at uw.edu.pl (Marcin Mostowski)
Date: Tue, 10 Mar 2015 04:52:32 +0100 (CET)
Subject: [FOM] Potential and actual infinities
In-Reply-To: <alpine.LFD.2.11.1503091157000.14548@laurent.mit.edu>
Message-ID: <817670404.30842459.1425959552691.JavaMail.root@zp1.poczta.uw.edu.pl>



Timothy Y. Chow observed:

I've never seen anyone define two separate axioms and declare one 
of them to be an "axiom of potential infinity" and the other an "axiom of 
actual infinity." 

No wonder, potential and actual infinity are views on mathematical truths, but on nature of mathematical reality.

Potential infinity appears in our computational experience. Always we have finite amount of memory and other given resources, but we can extend them ? still by a finite amount.

Actual infinity appears when we try to comprehend our computational practice, when we try to find axioms correctly describing our computational experience. The problem is that we want to know possible results of extendinding of extending of our experience.

So potential infinity can be described by the following claims: 

the world is finite,

but it is greater than any given finite amount.

Of course this is inconsistent, when we try to comprehend the situation.


I investigated the case from logical point of view in: http://www.impan.pl/~kz/KR/talks/Truth_in_the_limit.pdf



Another relevant point is meaning of mathematical proofs. We search for mathematical truths, 
THE BEST ARE PROOFS, BUT THEY ARE NOT ONLY SOURCES OF MATHEMATICAL TRUTHS.

We accept PA consistency, imposibility of easy computing of discrete logarithms, and many other claims which cannot be decided on the basis of mathematical proofs.

Marcin Mostowski


From rfl.urbaniak at gmail.com  Tue Mar 10 12:50:19 2015
From: rfl.urbaniak at gmail.com (Rafal Urbaniak)
Date: Tue, 10 Mar 2015 17:50:19 +0100
Subject: [FOM] Potential infinity
Message-ID: <CANLj=G454AsKkmQAOxFSxKodb-5UEZwmqXNLV0MKGr052RL6Zw@mail.gmail.com>

There is some work being done on making a formal sense of potential
infinity by Marcin Mostowski and some other logicians in Warsaw. A short
paper where you can get the gist of the idea is here:

http://www.frontiersinai.com/turingfiles/May/mostowski.pdf

and some other papers are:

Michal Krynicki, Marcin Mostowski, Konrad Zdanowski:
Finite Arithmetics. Fundam. Inform. 81(1-3): 183-202 (2007)

Marcin Mostowski:
Potential Infinity and the Church Thesis. Fundam. Inform. 81(1-3): 241-248
(2007)

Marcin Mostowski
On Representing Concepts in Finite Models. Math. Log. Quart. 47 (2001) 4,
513=523

Marcin Mostowski, Konrad Zdanowski:
FM-Representability and Beyond. CiE 2005: 358-367

Marcin Mostowski, Konrad Zdanowski:
Coprimality in Finite Models. CSL 2005: 263-275

Marcin Mostowski, Anna Wasilewska:
Arithmetic of divisibility in finite models. Math. Log. Q. 50(2): 169-174
(2004)

Also, there's Marek Czarnecki's work on what happens with truth definitions
in this framework

http://students.mimuw.edu.pl/~mc208417/referaty/semantics%20in%20coprimality[en].pdf


Although, this might be not what you're looking for, because the approach
is model-theoretic.
The idea is that instead of taking the omega-sequence, one looks at its
initial segments, evaluates formulas in initial segments, and
then generally says that a formula is FM-true if there is an initial
segment such that the formula is true at that segment and all longer
segments.

The set of FM-true sentences is different than the set of arithmetical
sentences true in the standard model of arithmetic. For instance,
"there is a greatest number" comes out FM-true, while there is no
particular number n of which "n is the greatest number" is FM-true.

While it's technically interesting what happens when you play around with
this setup, I'm not sure how convincing philosophically this is. For
it seems that the quantification in the definition of FM-truth is still
quantification over an infinite set (of all initial segments).

Best regards,
Rafal Urbaniak
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150310/2b3743e8/attachment-0001.html>

From martin at eipye.com  Wed Mar 11 11:53:24 2015
From: martin at eipye.com (Martin Davis)
Date: Wed, 11 Mar 2015 08:53:24 -0700
Subject: [FOM] Fwd: CFP: International Conference on Artificial Intelligence
 (ICAI'15: July 27-30, 2015, Las Vegas, USA); Submission Deadline: March 31
In-Reply-To: <20150311091948.9C33E3E77A45@world-comp.org>
References: <20150311091948.9C33E3E77A45@world-comp.org>
Message-ID: <CA+cpueJCn1xjP46PS9d_ee+PLuzC5zq5T+c4p-h30+SNpzVgew@mail.gmail.com>

---------- Forwarded message ----------
From: AI <wcom at world-comp.org>
Date: Wed, Mar 11, 2015 at 2:19 AM
Subject: CFP: International Conference on Artificial Intelligence (ICAI'15:
July 27-30, 2015, Las Vegas, USA); Submission Deadline: March 31
To: martin at eipye.com



                         CALL  FOR  PAPERS

              Paper Submission Deadline: March 31, 2015

                              ICAI'15
     The 2015 International Conference on Artificial Intelligence

         July 27-30, 2015, Monte Carlo Resort, Las Vegas, USA
 http://www.worldacademyofscience.org/worldcomp15/ws/conferences/icai15
========================================================================

SCOPE: TOPICS OF INTEREST INCLUDE, BUT ARE NOT LIMITED TO, THE FOLLOWING:
   Prologue: Artificial Intelligence (AI) is the science and engineering
   of making intelligent machines and systems. This is an important
   multi-disciplinary field which is now an essential part of technology
   industry, providing the heavy lifting for many of the most challenging
   problems in computer science. Since Machine Learning has strong ties
   with AI, the conference also covers the field of Machine Learning. The
   list of topics below is by no means meant to be exhaustive.

o  Artificial Intelligence:
   - Brain models, Brain mapping, Cognitive science
   - Natural language processing
   - Fuzzy logic and soft computing
   - Software tools for AI
   - Expert systems
   - Decision support systems
   - Automated problem solving
   - Knowledge discovery
   - Knowledge representation
   - Knowledge acquisition
   - Knowledge-intensive problem solving techniques
   - Knowledge networks and management
   - Intelligent information systems
   - Intelligent data mining and farming
   - Intelligent web-based business
   - Intelligent agents
   - Intelligent networks
   - Intelligent databases
   - Intelligent user interface
   - AI and evolutionary algorithms
   - Intelligent tutoring systems
   - Reasoning strategies
   - Distributed AI algorithms and techniques
   - Distributed AI systems and architectures
   - Neural networks and applications
   - Heuristic searching methods
   - Languages and programming techniques for AI
   - Constraint-based reasoning and constraint programming
   - Intelligent information fusion
   - Learning and adaptive sensor fusion
   - Search and meta-heuristics
   - Multisensor data fusion using neural and fuzzy techniques
   - Integration of AI with other technologies
   - Evaluation of AI tools
   - Social intelligence (markets and computational societies)
   - Social impact of AI
   - Emerging technologies
   - Applications (including: computer vision, signal processing, military,
     surveillance, robotics, medicine, pattern recognition, face
recognition,
     finger print recognition, finance and marketing, stock market,
     education, emerging applications, ...)

o  Machine Learning; Models, Technologies and Applications:
   - Statistical learning theory
   - Unsupervised and Supervised Learning
   - Multivariate analysis
   - Hierarchical learning models
   - Relational learning models
   - Bayesian methods
   - Meta learning
   - Stochastic optimization
   - Simulated annealing
   - Heuristic optimization techniques
   - Neural networks
   - Reinforcement learning
   - Multi-criteria reinforcement learning
   - General Learning models
   - Multiple hypothesis testing
   - Decision making
   - Markov chain Monte Carlo (MCMC) methods
   - Non-parametric methods
   - Graphical models
   - Gaussian graphical models
   - Bayesian networks
   - Particle filter
   - Cross-Entropy method
   - Ant colony optimization
   - Time series prediction
   - Fuzzy logic and learning
   - Inductive learning and applications
   - Grammatical inference
   - Graph kernel and graph distance methods
   - Graph-based semi-supervised learning
   - Graph clustering
   - Graph learning based on graph transformations
   - Graph learning based on graph grammars
   - Graph learning based on graph matching
   - Information-theoretical approaches to graphs
   - Motif search
   - Network inference
   - Aspects of knowledge structures
   - Computational Intelligence
   - Knowledge acquisition and discovery techniques
   - Induction of document grammars
   - General Structure-based approaches in information retrieval,
     web authoring, information extraction, and web content mining
   - Latent semantic analysis
   - Aspects of natural language processing
   - Intelligent linguistic
   - Aspects of text technology
   - Biostatistics
   - High-throughput data analysis
   - Computational Neuroscience
   - Computational Statistics

The conference is composed of a number of tracks, tutorials, sessions,
workshops, poster and panel discussions; all will be held simultaneously,
same location and dates: July 27-30, 2015.


INVITATION:

You are invited to submit a paper for consideration. All accepted papers
will be published in printed conference books/proceedings (ISBN) and will
also be made available online. The proceedings will be indexed in science
citation databases that track citation frequency/data. In addition, like
prior years, extended versions of selected papers (about 35%) of the
conference will appear in journals and edited research books (publishers
include: Springer, Elsevier, BMC, and others); some of these books and
journal special issues have already received the top 25% downloads in
their respective fields. See the link below for a very small subset of
the books published mostly based on extended versions of the accepted
papers of this congress:
http://www.worldacademyofscience.org/worldcomp15/ws/publications
The titles of proceedings of the federated congress have been indexed into
the ACM Digital Library ( http://dl.acm.org/ ) which includes bibliographic
citations from major publishers in computing.


IMPORTANT DATES:

March 31, 2015:    Submission of full papers (about 7 pages)
April 24, 2015:    Notification of acceptance (+/- two days)
May 15, 2015:      Final papers + Copyright + Registration
July 27-30, 2015:  The 2015 International Conference on Artificial
                   Intelligence (ICAI'15)


SUBMISSION OF DRAFT PAPERS FOR EVALUATION:

Prospective authors are invited to submit their papers by uploading them
to the evaluation web site at:  http://world-comp.org . Submissions must
be uploaded by the due date (see IMPORTANT DATES) and must be in either
MS doc or pdf formats (about 7 pages including all figures, tables, and
references.) All reasonable typesetting formats are acceptable (later,
the authors of accepted papers will be asked to follow a particular
typesetting format to prepare their final papers for publication; these
formatting instructions appear at: http://world-comp.org and they
conform to the two-column IEEE style format). Papers must not have been
previously published or currently submitted for publication elsewhere.
The first page of the paper should include: title of the paper, name,
affiliation, postal address, and email address for each author as well
as a maximum of 5 topical keywords that would best represent the content
of the paper. The first page should also identify the name of the
Contact/Corresponding author together with his/her professional email
address. A 100 to 150-word abstract should appear on the first page.
Authors are to conform to the common CODE OF ETHICS FOR AUTHORS (The
document for the Code of Ethics is available on the submission web site.)

Each paper will be peer-reviewed by two experts in the field for
originality, significance, clarity, impact, and soundness. In cases of
contradictory recommendations, a member of the conference program
committee would be charged to make the final decision (accept/reject);
often, this would involve seeking help from additional referees. Papers
whose authors include a member of the conference program committee will
be evaluated using the double-blinded review process. (Essay/philosophical
papers will not be refereed but may be considered for discussion/panels).

The proceedings will be published in printed conference books (ISBN) and
will also be made available online. The printed proceedings/books will
be available for distribution on site at the conference. The proceedings
will be indexed in science citation databases that track citation
frequency/data. The proceedings/books of the federated congress that
this conference is part of have been evaluated for inclusion into major
science citation index databases. We are happy to report that so far,
the evaluation board of science citation index databases have approved
the indexing, integrating, and inclusion of the following conference
tracks into relevant indexing databases (indexing databases include,
among others: Scopus, Engineering Village, EMBASE, and others):
BIOCOMP, DMIN, GCA, ICAI, ICOMP, ICWN, IKE, IPCV, PDPTA, and SAM. All
proceedings are also approved for inclusion into EBSCO (www.ebsco.com),
one of the largest subject index systems. The titles of proceedings of the
federated congress have been indexed into the ACM Digital Library
( http://dl.acm.org/ ) which includes bibliographic citations from major
publishers in computing.

In addition to the above, we have arranged two new book series (multiple
books in each series); one with Elsevier publishers (Emerging Trends in
Computer Science and Applied Computing) and another with Springer
publishers (Transactions of Computational Science and Computational
Intelligence). After the conference (the process may take 12 to 18
months), a significant number of authors of accepted papers of our
congress, will be given the opportunity to submit the extended version
of their papers for publication consideration in these books. We
anticipate having between 10 and 20 books a year in each of these
book series projects. Each book in each series will be subject to
Elsevier and Springer science indexing products (which includes:
Scopus, www.info.scopus.com; SCI Compendex, Engineering Village,
www.ei.org; EMBASE, www.info.embase.com; and others). For a recent and
a very small subset of the books (and journal special issues) that have
been published based on the extended versions of our congress papers,
see below:

http://www.worldacademyofscience.org/worldcomp15/ws/books_journals

Some of these books and journal special issues have already received
the top 25% downloads in their respective fields - we already have a
number of Elsevier and Springer books in the pipeline.

SUBMISSION OF POSTER PAPERS:

Poster papers can be 2 pages long. Authors are to follow the same
instructions that appear in section SUBMISSION OF DRAFT PAPERS FOR
EVALUATION - except that the submission is limited to 2 pages. On the
first page, the author should state that "This paper is being submitted
as a poster". Poster papers (if accepted) will be published as such, if
and only if the author of the accepted poster wishes to do so.


MEMBERS OF PROGRAM AND ORGANIZING COMMITTEES:

Currently being finalized. The members of the Steering Committee of the
federated congress that this conference is part of included:
Dr. Selim Aissi, (formerly: Chief Strategist - Security, Intel Corporation,
USA) Vice President, Global Information Security, Visa Inc., USA;
Prof. Nizar Al-Holou, Professor and Department Chair, and Vice Chair of
IEEE/SEM-Computer Chapter, University of Detroit Mercy, Detroit, Michigan,
USA; Prof. Hamid R. Arabnia, Professor of Computer Science, Elected Fellow
of ISIBM, Editor-in-Chief of Journal of Supercomputing (Springer),
University of Georgia, USA; Prof. Mary Mehrnoosh Eshaghian-Wilner,
Professor of Engineering Practice, University of Southern California,
USA (and Adjunct Professor, University of California Los Angeles, UCLA,
USA); Prof. Shiuh-Jeng Wang, Department of Information Management, Central
Police University, Taiwan and Program Chair, Security & Forensics, Taiwan
and Director, Information Crypto and Construction Lab (ICCL) & ICCL-FROG;
Prof. Michael Panayiotis Bekakos, Professor of Computer Systems and
Director of Laboratory of Digital Systems and Head of Parallel Algorithms
and architectures Research Group, Democritus University of Thrace, Greece;
Prof. Kevin Daimi, Professor of Computer Science and Director of Computer
Science and Software Engineering Programs, University of Detroit Mercy,
Detroit, Michigan, USA; Prof. Patrick S. P. Wang, Fellow of IAPR, ISIBM,
WASE and Professor of Computer and Information Science, Northeastern
University, Boston, Massachusetts, USA and Otto-von-Guericke Distinguished
Guest Professor, University Magdeburg, Germany; Prof. George Jandieri,
Georgian Technical University, Tbilisi, Georgia and Chief Scientist at
The Institute of Cybernetics, Georgian Academy of Science, Georgia;
Prof. D. V. Kodavade, Head of Computer Science and Engineering, DKTE
Institute, India; Prof. George Markowsky, Professor and Associate Director,
School of Computing and Information Science, Chair Int'l Advisory Board
of IEEE IDAACS, Director 2013 Northeast Collegiate Cyber Defense
Competition, University of Maine, Orono, Maine, USA; Prof. G. N. Pandey,
Vice-Chancellor, Arunachal University of Studies, India (and Adjunct
Professor, Indian Institute of Information Technology, India);
Prof. James J. (Jong Hyuk) Park, Professor of Computer Science and
Engineering, Seoul, Korea and President of KITCS, President of FTRA,
Editor-in-Chief of HCIS, JoC and IJITCC Journals; Prof. Fernando G.
Tinetti, Universidad Nacional de La Plata, Argentina, Co-editor, Journal
of CS and Technology (JCS&T); Dr. Predrag Tosic, Microsoft, Washington,
USA; Prof. Vladimir Volkov, The Bonch-Bruevich State University of
Telecommunications, Saint-Petersburg, Russia; Dr. Michael R. Grimaila,
Air Force Institute of Technology, Fellow of ISSA, CISM, CISSP, IAM/IEM,
Air Force Center of Cyberspace Research, Advisor to the Prince of Wales
Fellows & Prince Edward Fellows at MIT and Harvard Universities and PC
member of NATO Cooperative Cyber Defence Centre of Excellence (CCD COE);
Prof. Victor Malyshkin, Head of Supercomputer Software Department,
Russian Academy of Sciences, Russia; Prof. Andy Marsh, Director of HoIP,
Director of HoIP Telecom, UK, and Secretary-General of WABT and
Vice-president of ICET, Visiting Professor, University of Westminster,
UK; Ashu M. G. Solo, Fellow of British Computer Society, Principal R&D
Engineer, Maverick Technologies America; Prof. Sang C. Suh, Head and
Professor of Computer Science, Vice President, of Society for Design
and Process Science (SDPS), Director of Intelligent Cyberspace
Engineering (ICEL), Texas A&M University, Com., Texas, USA; Prof. Layne
T. Watson, IEEE Fellow, NIA Fellow, ISIBM Fellow, Fellow of The National
Institute of Aerospace, Virginia Polytechnic Institute & State University,
Virginia, USA; Prof. Mary Q. Yang, Director, Mid-South Bioinformatics
Center and Joint Bioinformatics Ph.D. Program, University of Arkansas,
USA; Prof. Byung-Gyu Kim, Multimedia Processing Communications Lab.(MPCL),
SunMoon University, South Korea; Prof. Young-Sik Jeong, Editor-in-Chief
of Journal of Information Processing Systems (JIPS), Dongguk University,
Seoul, South Korea; and others.

The 2015 Program Committee for individual conferences are currently being
compiled. Many who have already joined the committees are renowned leaders,
scholars, researchers, scientists and practitioners of the highest ranks;
many are directors of research labs., fellows of various societies, heads/
chairs of departments, program directors of research funding agencies, as
well as deans and provosts.

Program Committee members are expected to have established a strong and
documented research track record. Those interested in joining the Program
Committee should email editor at world-comp.org  the following information
for consideration: Name, affiliation and position, complete mailing address,
email address, a one-page biography that includes research expertise & the
name of this conference.


GENERAL INFORMATION:

The federated Congress that this conference is part of is composed of
research presentations, keynote lectures, invited presentations,
tutorials, panel discussions, and poster presentations. In recent past,
keynote/tutorial/panel speakers have included:
Prof. David A. Patterson (pioneer, architecture, U. of California,
Berkeley), Dr. K. Eric Drexler (known as Father of Nanotechnology),
Prof. John H. Holland (known as Father of Genetic Algorithms; U. of
Michigan), Prof. Ian Foster (known as Father of Grid Computing; U. of
Chicago & ANL), Prof. Ruzena Bajcsy (pioneer, VR, U. of California,
Berkeley), Prof. Barry Vercoe (Founding member of MIT Media Lab, MIT),
Dr. Jim Gettys (known as X-man, developer of X Window System, xhost;
OLPC), Prof. John Koza (known as Father of Genetic Programming, Stanford
U.), Prof. Brian D. Athey (NIH Program Director, U. of Michigan),
Prof. Viktor K. Prasanna (pioneer, U. of Southern California), Dr. Jose
L. Munoz (NSF Program Director and Consultant), Prof. Jun Liu (pioneer,
Broad Institute of MIT & Harvard U.), Prof. Lotfi A. Zadeh (Father of
Fuzzy Logic), Dr. Firouz Naderi (Head, NASA Mars Exploration Program/
2000-2005 and Associate Director, Project Formulation & Strategy, Jet
Propulsion Lab, CalTech/NASA; Director, NASA's JPL Solar System
Exploration), Prof. David Lorge Parnas (Fellow of IEEE, ACM, RSC, CAE, GI;
Dr.h.c.: ETH Zurich, Prof. Emeritus, McMaster U. and U. of Limerick),
Prof. Eugene H. Spafford (Executive Director, CERIAS and Professor, Purdue
University), Dr. Sandeep Chatterjee (Vice President & Chief Technology
Officer, SourceTrace Systems, Inc.), Prof. Haym Hirsh (Dean, Cornell
University - formerly at Rutgers University, New Jersey, USA and former
director of Division of Information and Intelligent Systems, National
Science Foundation, USA), Dr. Flavio Villanustre (Vice-President, HPCC
Systems), Prof. Victor Raskin (Distinguished Professor, Purdue University,
USA); Prof. Alfred Inselberg (School of Mathematical Sciences, Tel Aviv
University, Israel; Senior Fellow, San Diego Supercomputing Center;
Inventor of the multidimensional system of Parallel Coordinates and
author of textbook); Prof. H. J. Siegel (Abell Endowed Chair Distinguished
Professor of ECE and Professor of CS; Director, CSU Information Science
and Technology Center (ISTeC), Colorado State University, Colorado, USA);
Prof. Amit Sheth (Fellow of IEEE and LexisNexis Eminent Scholar;
Founder/Executive Director, Ohio Center of Excellence in Knowledge-enabled
Computing (Kno.e.sis) and Professor of Computer Science, Wright State
University, Ohio, USA); Dr. Leonid I. Perlovsky (Harvard University,
Cambridge, Massachusetts, USA and School of Engineering and Applied
Sciences and Medical School Athinoula Martinos Brain Imaging Center; and
The US Air Force Research Lab., USA; CEO, LP Information Technology, USA
and Chair of IEEE Task Force on The Mind and Brain; Recipient of John
McLucas Award, the highest US Air Force Award for basic research);
and many other distinguished speakers. The Congress is among the top five
largest annual gathering of researchers in computer science, computer
engineering and applied computing. We anticipate to have attendees from
about 85 countries/territories. To get a feeling about the conferences'
atmosphere, see some delegates photos available at:

http://worldcomp.phanfare.com/
http://worldcomp.phanfare.com/6626396 (to see a slide show, click on
"Start SlideShow" tab at the url above.)

An important mission of The Congress is "Providing a unique platform for
a diverse community of constituents composed of scholars, researchers,
developers, educators, and practitioners. The Congress makes concerted
effort to reach out to participants affiliated with diverse entities
(such as: universities, institutions, corporations, government agencies,
and research centers/labs) from all over the world. The congress also
attempts to connect participants from institutions that have teaching as
their main mission with those who are affiliated with institutions that
have research as their main mission. The congress uses a quota system to
achieve its institution and geography diversity objectives."

One main goal of the congress is to assemble a spectrum of affiliated
research conferences, workshops, and symposiums into a coordinated
research meeting held in a common place at a common time. This model
facilitates communication among researchers in different fields of computer
science, computer engineering, and applied computing. The Congress also
encourages multi-disciplinary and inter-disciplinary research initiatives;
i.e., facilitating increased opportunities for cross-fertilization across
sub-disciplines. 60211


MEASURABLE SCIENTIFIC IMPACT OF CONGRESS:

As of December 14, 2014, proceedings of the federated congress that this
conference is part of, have received over 27,914 citations (includes 3,346
self-citations). Citation data is obtained from Microsoft Academic Search.
The citation data does not even include more than 15,000 other citations
to papers published by conferences whose first offerings were initiated
by the congress. Individual proceedings/books (2013 & 2014) of the
federated congress can be purchased from major science book distributors:
(such as: EBSCO and others):
http://www.worldacademyofscience.org/worldcomp14/ws/proceedings
http://www.worldacademyofscience.org/worldcomp13/ws/proceedings

MISCELLANEOUS:
The information that appears in this announcement is correct as of
January 1, 2015.

CONTACT:
Inquiries should be sent to: sc at world-comp.org

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

This email was sent to:  martin at eipye.com

To opt out of this email list: (Please allow up to 24 hours for any changes
to be reflected)
http://world-comp.org/cgi-bin/rm/full.cgi?60A52426-5F75-11DE-A965-BEC5C32F065D
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150311/6219234d/attachment-0001.html>

From jbell at uwo.ca  Wed Mar 11 19:31:51 2015
From: jbell at uwo.ca (John Bell)
Date: Wed, 11 Mar 2015 19:31:51 -0400
Subject: [FOM] Potential and Actual Infinity
In-Reply-To: <mailman.760.1426111024.28387.fom@cs.nyu.edu>
References: <mailman.760.1426111024.28387.fom@cs.nyu.edu>
Message-ID: <675571BC9D904082B1A736CC99DA7776@SandraVAIO>

A nice way of representing potential infinity is to allow sets to undergo explicit variation over time, as in the topos E of of sets varying over the natural numbers. The objects of this topos are all sequences of maps between sets

A0 ?-> A1 ?>  A2 ?> .....An ?> ....

Such an object may be thought of as a set A ?varying over discrete time: An is the ?state? of A at time n. 
Now consider the temporally varying set , call it K,


{0} ?-> {0, 1} ?> {0,1,2}  -->  ...  --> {0,1,2, ..., n}  --> ...

in which all the arrows are identity maps. In E, K ?grows? indefinitely and hence potentially infinite. On the other hand at each specific time K?s s state  is finite and so K is not actually infinite. In short, in E, K is potentially, but not actually infinite..

In the internal logic of E, K fails be finite in that it  is not equipollent with any natural number. On the other hand  K is not transfinite in that the set of natural numbers cannot be injected into it.

-- John Bell
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150311/a0693b12/attachment.html>

From di.gama at gmail.com  Wed Mar 11 23:56:18 2015
From: di.gama at gmail.com (Mario Carneiro)
Date: Wed, 11 Mar 2015 23:56:18 -0400
Subject: [FOM] Potential and Actual Infinity
In-Reply-To: <675571BC9D904082B1A736CC99DA7776@SandraVAIO>
References: <mailman.760.1426111024.28387.fom@cs.nyu.edu>
	<675571BC9D904082B1A736CC99DA7776@SandraVAIO>
Message-ID: <CAFXXJSvcB_i1C4aP7ThLLa1NFRX1oGNjXchx=b8tQ9_Bwa7FTQ@mail.gmail.com>

On Wed, Mar 11, 2015 at 7:31 PM, John Bell <jbell at uwo.ca> wrote:

>  In the internal logic of *E, K *fails be finite in that it  is not
> equipollent with any natural number. On the other hand  K is not
> transfinite in that the set of natural numbers cannot be injected into it.
>

It seems that this notion of "not finite" as not equipollent with a natural
number is quite weak in E, seeing as it would also label the set

W = {0} -> {0,1} -> {0} -> {0,1} -> ...

as "infinite" in this sense, even though W is also a subset of {0,1}.
(Assuming I have correctly captured the "internal logic of E"...)

Mario Carneiro
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150311/0fff7175/attachment.html>

From joeshipman at aol.com  Thu Mar 12 00:30:16 2015
From: joeshipman at aol.com (Joseph Shipman)
Date: Thu, 12 Mar 2015 00:30:16 -0400
Subject: [FOM] Potential and Actual Infinity
Message-ID: <DB9265C5-1E9C-4E36-B676-F9A14ECC9012@aol.com>

This discussion seems to be making too much of a simple point. In Peano Arithmetic, which can be equivalently formalized by taking ZF and replacing the axiom of Infinity with its negation, there is no actual infinity. An actually infinite set gives a logically stronger system and we understand nowadays when this is necessary, and exactly how it is stronger (quantifiers are switched so instead of forall n thereexists X (X has at least n elements), we have thereexists X forall n (X has at least n elements).)

Potential infinity gets you quite far, you need actual infinity to get further (and to get to some important things, like the Robertson-Seymour Graph Minor Theorem, you even need uncountable infinities).

The potential/actual distinction is just a type of set/class distinction where we understand that we don't freely quantify over classes. PA respects this because it never requires the domain of quantification to be a completed object--this is a misconception fueled by the translation of the universal quantifier as "for all" rather than "for any",

-- JS

From katzmik at macs.biu.ac.il  Thu Mar 12 04:55:21 2015
From: katzmik at macs.biu.ac.il (katzmik at macs.biu.ac.il)
Date: Thu, 12 Mar 2015 10:55:21 +0200 (IST)
Subject: [FOM] Potential and Actual Infinity
In-Reply-To: <675571BC9D904082B1A736CC99DA7776@SandraVAIO>
References: <mailman.760.1426111024.28387.fom@cs.nyu.edu>
	<675571BC9D904082B1A736CC99DA7776@SandraVAIO>
Message-ID: <64929.132.71.121.186.1426150521.squirrel@webmail.cs.biu.ac.il>

John,

Thanks for this interesting contribution.  Perhaps you could mention that the
arrows are inclusion maps rather than identity maps.  If "inclusion" does not
make sense in this topos, still a different term seems appropriate for the
arrows. MK

On Thu, March 12, 2015 01:31, John Bell wrote:
> A nice way of representing potential infinity is to allow sets to undergo
> explicit variation over time, as in the topos E of of sets varying over the
> natural numbers. The objects of this topos are all sequences of maps between
> sets
>
> A0 ?-> A1 ?>  A2 ?> .....An ?> ....
>
> Such an object may be thought of as a set A ?varying over discrete time: An is
> the ?state? of A at time n.
> Now consider the temporally varying set , call it K,
>
>
> {0} ?-> {0, 1} ?> {0,1,2}  -->  ...  --> {0,1,2, ..., n}  --> ...
>
> in which all the arrows are identity maps. In E, K ?grows? indefinitely and
> hence potentially infinite. On the other hand at each specific time K?s s
> state  is finite and so K is not actually infinite. In short, in E, K is
> potentially, but not actually infinite..
>
> In the internal logic of E, K fails be finite in that it  is not equipollent
> with any natural number. On the other hand  K is not transfinite in that the
> set of natural numbers cannot be injected into it.
>
> -- John Bell
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>



From cp at cpporter.com  Thu Mar 12 10:06:44 2015
From: cp at cpporter.com (Christopher Porter)
Date: Thu, 12 Mar 2015 15:06:44 +0100
Subject: [FOM] Varieties of Algorithmic Information (VAI) 2015
Message-ID: <61ece0392847f2dd6f310ccfd3990546@cpporter.com>


-----------------------------------------------------------------------
VAI 2015 - Call for Abstracts
-----------------------------------------------------------------------

Varieties of Algorithmic Information 2015
June 15-18, Heidelberg, Germany
http://vai2015.sciencesconf.org

-----------------------------------------------------------------------

Deadlines:

   Abstract submission: April 1, 2015
   Author notification: April 15, 2015
   Registration: May 15, 2015

-----------------------------------------------------------------------

The notion of algorithmic information is referred to throughout work in 
computability theory, algorithmic randomness, and related areas. 
However, in many cases, this notion of algorithmic information is used 
both informally and in a number of different senses. The goal of 
Varieties of Algorithmic Information is to clarify the various notions 
of algorithmic information as they appear in computability-theoretic 
investigations, to ascertain the similarities and differences between 
them, and to foster interaction between mathematicians, computer 
scientists, and philosophers with interest in the topic.

The conference will consist of four invited talks (by Vasco Brattka, 
Walter Dean, Jan Reimann and Nikolay Vereshchagin) as well as 
contributed talks. If you are interested in giving a talk on a topic 
related to the theme of the conference, please submit an abstract via 
the conference webpage (abstracts will be subject to a brief review 
process). If you simply wish to attend the conference, contact the 
organisers at vai at computability.fr.

This conference is made possible by the support of the John Templeton 
Foundation (grant #48003).

We look forward to seeing you in Heidelberg!

The organizing committee: Laurent Bienvenu, Christopher Porter and 
Wolfgang Merkle

From tchow at alum.mit.edu  Fri Mar 13 13:55:43 2015
From: tchow at alum.mit.edu (Timothy Y. Chow)
Date: Fri, 13 Mar 2015 13:55:43 -0400 (EDT)
Subject: [FOM] Potential and Actual Infinity
In-Reply-To: <mailman.23.1426262782.8783.fom@cs.nyu.edu>
References: <mailman.23.1426262782.8783.fom@cs.nyu.edu>
Message-ID: <alpine.LFD.2.11.1503131351160.16449@laurent.mit.edu>

Joseph Shipman wrote:

> This discussion seems to be making too much of a simple point. In Peano 
> Arithmetic, which can be equivalently formalized by taking ZF and 
> replacing the axiom of Infinity with its negation, there is no actual 
> infinity.

Yes, this is what I said in my initial post, and I think I was the first 
one to bring up the term "potential infinity" in the current discussion.

The question I raised in that same post, however, does not seem to have 
been answered yet.  Namely, is there a way to prove the consistency of PA 
assuming only "potential infinity"?  What we might call the classical 
approach to potential infinity turns this question into, can PA prove its 
own consistency?  So the classical form of the question has a negative 
answer.  However, McCall seems to want to claim some kind of positive 
answer.  Arnon Avron also claimed that potential infinity was all that was 
needed to prove the consistency of PA, but unless I missed something, has 
not responded to my request for a more formal justification of this claim.

Tim

From JoeShipman at aol.com  Fri Mar 13 20:32:50 2015
From: JoeShipman at aol.com (Joseph Shipman)
Date: Fri, 13 Mar 2015 20:32:50 -0400
Subject: [FOM] Potential and Actual Infinity
In-Reply-To: <alpine.LFD.2.11.1503131351160.16449@laurent.mit.edu>
References: <mailman.23.1426262782.8783.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503131351160.16449@laurent.mit.edu>
Message-ID: <DF62FCE4-B876-4630-815C-0323075D3758@aol.com>

It's not clear how to make sense of this question. I can prove Con(PA) from the axiom that epsilon_0 is well-ordered, or from Goodstein's Theorem, or from various pi^0_1 statements which do not themselves speak about or seem to entail an actually infinite set. However, if we want to find a proof of Con(PA) from within our official formal foundational system ZFC, it is absolutely necessary that somewhere in that proof, the Axiom of Infinity is used, and that states that an actually infinite set exists. So there is a strong technical sense in which the answer to your question is "Not if ZFC-derivable is the standard for provability".

-- JS

Sent from my iPhone

> On Mar 13, 2015, at 1:55 PM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:
> 
> Joseph Shipman wrote:
> 
>> This discussion seems to be making too much of a simple point. In Peano Arithmetic, which can be equivalently formalized by taking ZF and replacing the axiom of Infinity with its negation, there is no actual infinity.
> 
> Yes, this is what I said in my initial post, and I think I was the first one to bring up the term "potential infinity" in the current discussion.
> 
> The question I raised in that same post, however, does not seem to have been answered yet.  Namely, is there a way to prove the consistency of PA assuming only "potential infinity"?  What we might call the classical approach to potential infinity turns this question into, can PA prove its own consistency?  So the classical form of the question has a negative answer.  However, McCall seems to want to claim some kind of positive answer.  Arnon Avron also claimed that potential infinity was all that was needed to prove the consistency of PA, but unless I missed something, has not responded to my request for a more formal justification of this claim.
> 
> Tim
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom

From sambin at math.unipd.it  Fri Mar 13 20:14:16 2015
From: sambin at math.unipd.it (sambin at math.unipd.it)
Date: Sat, 14 Mar 2015 01:14:16 +0100
Subject: [FOM] Potential and Actual Infinity
In-Reply-To: <alpine.LFD.2.11.1503131351160.16449@laurent.mit.edu>
References: <mailman.23.1426262782.8783.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503131351160.16449@laurent.mit.edu>
Message-ID: <20150314011416.16661fre3wuu964g@webmail.math.unipd.it>

Quoting "Timothy Y. Chow" <tchow at alum.mit.edu>:

> Arnon Avron also claimed that potential infinity was all that was  
> needed to prove the consistency of PA, but unless I missed  
> something, has not responded to my request for a more formal  
> justification of this claim.
>
> Tim

Gerhard Gentzen in the 30s proved that PA is consistent using  
induction up to the famous ordinal epsilon-zero. To define such an  
ordinal, no actual infinity is necessary.

Giovanni Sambin

----------------------------------------------------------------
This message was sent using IMP, the Internet Messaging Program.



From aa at tau.ac.il  Sat Mar 14 02:36:00 2015
From: aa at tau.ac.il (Arnon Avron)
Date: Sat, 14 Mar 2015 08:36:00 +0200
Subject: [FOM] "Proof" of the consistency of PA published by Oxford UP
In-Reply-To: <alpine.LFD.2.11.1503091134520.14548@laurent.mit.edu>
References: <alpine.LFD.2.11.1503062159110.25430@laurent.mit.edu>
	<20150309085622.GC2866@localhost.localdomain>
	<alpine.LFD.2.11.1503091134520.14548@laurent.mit.edu>
Message-ID: <20150314063559.GA12088@localhost.localdomain>

Tim,

You recently raise more issues and questions than I can answer
immediately. So for the time being this is my reply only (or mainly) 
to the first of three postings of yours that were explicitly
connected to things I said. 

> Again to play devil's advocate, suppose that one accepts the
> induction schema outright only for quantifier-free formulas?

This is indeed the only "reasonable" choice for a devil's advocate,
because my argument is based on the principle that
if one adds one quantifier on N to a formula which 
expresses a meaningful property of tuples of natural
numbers, then one gets another formula with the same
property. Everyone who accepts this principle,
and accepts induction on quantifier-free formulas of PA
necessarily accepts  it for each formula of PA. She
might pretend to do it on a "case-by-case basis", refusing
to admit that she accepts it "for all", but both of us
would know that she would accept it for each specific
formula I bring her, and this is the same for me
(on the basis of seeing N as a potentially infinite set).
However, if one refuses to accept the above principle
then one is really stuck with induction on quantifier-free formulas.

Now I am unable to understand someone who says that A(x,y)
expresses a meaningful property which is  meaningful for every pair
of natural numbers (or for each pair of natural numbers: 
for me these are the same -  in Hebrew there is no such
distinction...), but maintains that the existence
of a natural number  y such that A(x,y)
is not a property of x. I have no way to convince
such a person that he is  wrong. I can only think
that he is deceiving himself, and wait till I catch him
in a case he says something which is in direct conflict
with his official beliefs (this is almost sure to happen...).

> Specific formulas with quantifiers could also be accepted on a
> case-by-case basis, but only after one has gone through the mental
> process of thinking about the formula and convincing oneself that
> induction for that property is "obvious."

I have already hinted above what I think about this scenario.
 
> >I do not see the connection between the two parts of this
> >paragraph (before and after the "Hence..."). Yes, I do know "many
> >ways of weakening the induction axiom to produce a system whose
> >consistency doesn't imply the consistency of PA". The simplest of
> >them is to throw induction away altogether (like in Q). Still, I
> >do not have any picture (let alone "fairly clear" one) what it
> >could mean for the *natural numbers* to exist, yet for PA to be
> >inconsistent.
> 
> "Fairly clear" might be an overstatement, but as an analogy, let's
> consider the power set axiom P of ZF.  Someone might feel that they
> can grasp the set-theoretic universe (including of course the set of
> natural numbers) clearly enough to be convinced that ZF - P is
> consistent, yet not grasp the concept of an "arbitrary subset" with
> enough clarity to rule out the possibility that ZF is inconsistent.

The difference is that induction is an essential part of our
mental construction of the natural numbers (1,2,3, *and so on*...).
Therefore the validity of induction is inherent in the very
concept of natural numbers - a concept which  is crystal clear to 
everyone. In contrast, the concept of a set has never been
so clear (originally it meant, I believe, only definable sets,
though "definable" is of course a fuzzy notion). Accordingly,
I find this last scenario you describe as not impossible, but the 
analogy is in my opinion is very weak.

> Do you agree with that?  If so, then returning to PA, someone might
> feel that they understand what is meant by "1, 2, 3, 4, and so on
> indefinitely" yet not understand what an "arbitrary first-order
> property" of the natural numbers is with enough clarity to rule out
> the possibility that PA is inconsistent.  

I am not assuming the notion of an "arbitrary first-order property
of the natural numbers". I am assuming the absolutely clear notion
of an "arbitrary first-order formula in the language of PA",
and the in-general fuzzy property (of formulas in some language)
of "expressing property of natural numbers".
I have explained what makes me absolutely certain that understanding 
what is meant by "1, 2, 3, 4, and so on indefinitely"
forces one to see with 100% certainty that N is a model of PA,
and so that PA is consistent. If you claim that there
is some circularity in my thinking I'll accept, and note
that we cannot even start to talk about such issues
(what is a formula of PA? What is a proof in PA? What does it
mean to say that PA is inconsistent) without being involved in
such circularity, i.e. understanding the things we pretend
not to understand.

> The induction that you
> describe for first-order properties is conceptually more complicated
> than the induction needed to grasp the successor operation.  And of
> course, we have technically precise ways of demonstrating that the
> greater conceptual complexity is real and not illusory.  Given this,
> I don't find it convincing to argue that induction for first-order
> formulas is part of the *fundamental meaning* of the natural
> numbers, any more than I find it convincing to argue that the power
> set axiom is part of the *fundamental meaning* of the set-theoretic
> universe.

 Really, Tim? (or are you still playing the devil's advocate?)

Arnon

From panu.raatikainen at helsinki.fi  Sat Mar 14 05:01:49 2015
From: panu.raatikainen at helsinki.fi (Panu Raatikainen)
Date: Sat, 14 Mar 2015 11:01:49 +0200
Subject: [FOM] Potential and Actual Infinity
In-Reply-To: <alpine.LFD.2.11.1503131351160.16449@laurent.mit.edu>
References: <mailman.23.1426262782.8783.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503131351160.16449@laurent.mit.edu>
Message-ID: <20150314110149.Horde.DgXHvelY3r5B0T23cUfZbg6@webmail.helsinki.fi>

Timothy Y. Chow" <tchow at alum.mit.edu>:

> Joseph Shipman wrote:
>
>> This discussion seems to be making too much of a simple point. In  
>> Peano Arithmetic, which can be equivalently formalized by taking ZF  
>> and replacing the axiom of Infinity with its negation, there is no  
>> actual infinity.
>
> Yes, this is what I said in my initial post, and I think I was the  
> first one to bring up the term "potential infinity" in the current  
> discussion.


This is certainly one possible way to interpret what a commitment to  
"actual infinity" is.

But surely it is not the common sense intended by, e.g., finitists and  
constructivists, who use the notion so much:
they suggest that an unlimited use of classical logic to more complex,  
quantified formulas in the context of arithmetic (where the domain of  
quantification is necessarily infinite) already commits one to "actual  
infinity".


I should perhaps also repeat my old point once again:

There are two different senses of "an axiom of infinity" in the logic  
literature.
On the one hand, in logic, it often means any sentence which forces  
the domain to be infinite (also the
standard axioms of successor are together an axiom of infinity in this sense).
In set theory, on the other hand, the axiom of infinity is the axiom  
which says that there is an infinite set. The axioms of ZFC without  
this axiom already make the domain infinite, but it is this axiom  
which gives ZFC its extreme power. It is much stronger assumption   
than an axiom of infinity in the first sense.


All the Best

Panu


-- 
Panu Raatikainen

Ph.D., Adjunct Professor in Theoretical Philosophy

Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies
P.O. Box 24  (Unioninkatu 38 A)
FIN-00014 University of Helsinki
Finland

E-mail: panu.raatikainen at helsinki.fi

http://www.mv.helsinki.fi/home/praatika/


From gergely.szekely at gmail.com  Sat Mar 14 05:35:24 2015
From: gergely.szekely at gmail.com (=?UTF-8?Q?Gergely_Sz=C3=A9kely?=)
Date: Sat, 14 Mar 2015 10:35:24 +0100
Subject: [FOM] FINAL CALL: Logic, Relativity and Beyond Conference (Budapest,
 9-13 Aug 2015)
Message-ID: <CAHE+U3+RohaXsWz2+oRED+jr_oGOsLZCG-67gX=RuH+Xa7moAQ@mail.gmail.com>

FINAL ANNOUNCEMENT:

Logic, Relativity and Beyond
2nd international conference
http://www.renyi.hu/conferences/lrb15/

August 9-13 2015, Budapest, Hungary

*********************************************
IMPORTANT DATES:

Deadline for abstract/paper submission: 20 March, 2015
Notifying the authors: 20 May, 2015
Early registration: 31 May, 2015
Conference: 9-13 August, 2015

There are several new and rapidly evolving research areas blossoming out
from the interaction of logic and relativity theory. The aim of this
conference series, which take place once every 2 or 3 years, is to attract
and bring together mathematicians, physicists, philosophers of science, and
logicians from all over the word interested in these and related areas to
exchange new ideas, problems and results.

Topics include (but are not restricted to):
*  Special and general relativity
*  Axiomatizing physical theories
*  Logical foundations of spacetime
*  Computability and physics
*  Logic of causality
*  Relativistic computation
*  Knowledge acquisition in science
*  Branching spacetime
*  Concept algebras and algebraic logic
*  Logic of time and space
*  Cylindric and relation algebras
*  Relativity theory and philosophy of science

*****************************************************
Invited speakers:
- S. Barry Cooper (University of Leeds)
- Alexander K. Guts (Omsk State University)
- Mark Hogarth (University of Cambridge)
- Thomas M?ller (Universit?t Konstanz)
- Istvan Racz (Wigner Institute)
- Laszlo E. Szabo (Eotvos University)

*****************************************************
Program Committee:
- Istvan Nemeti (Chair, Renyi Institute)
- Thomas Benda (Yang Ming University)
- Miklos Ferenczi (Budapest University of Technology)
- Michele Friend (The George Washington University)
- Judit X. Madarasz (Renyi Institute)
- John Byron Manchak (University of Washington)
- Tomasz Placek (Jagiellonian University)
- Ildiko Sain (Renyi Institute)
- Mike Stannett (University of Sheffield)
- Gergely Szekely (Renyi Institute)
- Christian Wuthrich (University of California)

*******************************************************
Organizing Committee:
- Gergely Szekely (Chair, Renyi Institute)
- Hajnal Andreka (Renyi Institute)
- Koen Lefever (Vesalius College)
- Attila Molnar (Eotvos University)
- Mike Stannett (University of Sheffield)

We invite you to submit your abstract via the following link:
http://www.easychair.org/conferences/?conf=lrb15

Looking forward to seeing you in Budapest.

If you know anyone who may be interested in this conference, please notify
them.

[Apologies for multiple postings]
Contact: Gergely Szekely - lrb15 at renyi.mta.hu

If you wish to be removed from this notification list, please send an email
to szekely.gergely at renyi.mta.hu with REMOVE LRB LIST in the subject line.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150314/0b33b3e8/attachment.html>

From williamtait at mac.com  Sun Mar 15 14:02:18 2015
From: williamtait at mac.com (WILLIAM TAIT)
Date: Sun, 15 Mar 2015 13:02:18 -0500
Subject: [FOM] Potential and Actual Infinity
In-Reply-To: <20150314110149.Horde.DgXHvelY3r5B0T23cUfZbg6@webmail.helsinki.fi>
References: <mailman.23.1426262782.8783.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503131351160.16449@laurent.mit.edu>
	<20150314110149.Horde.DgXHvelY3r5B0T23cUfZbg6@webmail.helsinki.fi>
Message-ID: <338B0253-FA7D-4515-A9A2-F2B1A505D226@mac.com>

Some remarks on the discussion of the distinction between the actual and potential infinite. 

An actual infinity is an infinite object, (representable by us as an infinite set, as e.g. Joe Shipman notes).

John Bell has shown how the potential infinite can be 'represented' by an object; but the historical notion of a potential infinity (i.e. Aristotle's) is not of an object, but of a species of objects together with an operation for constructing more and more objects of the species. (In John's example, the objects in the topos E are in fact actual infinities: \omega sequences of maps. It is only by restricting the language in which we speak of them to the internal language of the topos that they are merely 'potentially infinite'.)

Panu Raatikainen (3/4) has raised a point that has always troubled me. Both Hilbert and the early intuitionists have associated commitment to the actual infinite with the use of classical logic, so that, for example, the use of quantification over the integers combined with classical logic commits one to the the set of integers as an actual infinity. I would like someone to explain why this is the same notion of actual infinity as Aristotle's. (One might ask, too, whether quantification over the integers using intuitionistic logic commits one to the actual infinite---and why.)

Giovanni Sambin (\pi) refers to Gentzen's consistency proof for PA using induction up to \epsilon_0 as an example of a consistency proof for that system that does not employ the actual infinite. Certainly Gentzen was also of that view and indeed argued for it. But does it really avoid the actual infinite? In the coding of the ordinals up to \epsilon_0 by natural numbers, let m be the code for \omega. As a natural number, m is certainly finite; but in its role as representative of \omega, it has infinitely many predecessors. (Of course, the form of induction on \epsilon_0 used in Gentzen's proof is the weak free-variable form: having established for a primitive recursive A

A(0) and (A(f(x))--> A(x) (where f satisfies the condition f(x) =0 or f(x) < x, < being the \epsilon_0 ordering)

we can conclude A(x). But it is still not clear to me how the argument avoids the actual infinite. (One needs this form of induction for a wide class of functions f.) Or better: I am unsure what the claim means that Gentzen's argument avoids the actual infinite.

Best wishes to all,
Bill



From joeshipman at aol.com  Sat Mar 14 18:09:06 2015
From: joeshipman at aol.com (Joseph Shipman)
Date: Sat, 14 Mar 2015 18:09:06 -0400
Subject: [FOM] Potential and Actual Infinity
In-Reply-To: <20150314011416.16661fre3wuu964g@webmail.math.unipd.it>
References: <mailman.23.1426262782.8783.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503131351160.16449@laurent.mit.edu>
	<20150314011416.16661fre3wuu964g@webmail.math.unipd.it>
Message-ID: <14D5DCEA-71E9-42F7-B5D1-424444DA3557@aol.com>

To define epsilon-zero, no infinity is necessary, but try proving as rigorously as you can that it is in fact well-ordered! Specify clearly the axiomatic system in which you expect to be able to prove this.

-- JS

Sent from my iPhone

> On Mar 13, 2015, at 8:14 PM, sambin at math.unipd.it wrote:
> 
> Quoting "Timothy Y. Chow" <tchow at alum.mit.edu>:
> 
>> Arnon Avron also claimed that potential infinity was all that was needed to prove the consistency of PA, but unless I missed something, has not responded to my request for a more formal justification of this claim.
>> 
>> Tim
> 
> Gerhard Gentzen in the 30s proved that PA is consistent using induction up to the famous ordinal epsilon-zero. To define such an ordinal, no actual infinity is necessary.
> 
> Giovanni Sambin
> 
> ----------------------------------------------------------------
> This message was sent using IMP, the Internet Messaging Program.
> 
> 
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom

From frode.bjordal at ifikk.uio.no  Tue Mar 17 11:34:02 2015
From: frode.bjordal at ifikk.uio.no (=?UTF-8?Q?Frode_Bj=C3=B8rdal?=)
Date: Tue, 17 Mar 2015 12:34:02 -0300
Subject: [FOM] A query on explicit definitions of inacessible cardinals.
Message-ID: <CAGarh+b0qOX58WQRcbhxTO-M6xfYfz8+X7rmyENbN274hZC_Zg@mail.gmail.com>

I am skeptical about the existence of relative inaccessible cardinals for
which one cannot give a precise explicit definition in terms of its set
condition in the language of set theory, or a slight extension thereof in
librationism ? as I prefer (see http://arxiv.org/abs/1407.3877 and
http://apcz.pl/czasopisma/index.php/LLP/article/view/LLP.2012.016). In a
question at Mathoverflow (
http://mathoverflow.net/questions/199850/a-query-on-how-to-climb-inaccessibles-in-%C2%A3
<http://mathoverflow.net/questions/199850/a-query-on-how-to-climb-inaccessibles-in-?>)
I give such an explicit definition for Mahlo-cardinals by means of a fixed
point construction I call manifestation point (it goes back to Cantini and
Visser and is related to work by Kleene and, ultimately, G?del) in the
librationist framework. With additional assumptions giving us ?M this means
that we can have an interpretation of ZFC+Mahlo (in the sense isolated) in
?M; we then follow the strategy I use in http://arxiv.org/abs/1407.3877 to
isolate an interpretation of ZFC in ? + the Skolem-Fraenkel Postulation.

 It seems to me, and also from the literature, that it is dubious that we
can press such a justification as this of inaccessible cardinals much
higher. Can a limit be pinpointed?

?Best regards

Frode?


..........................................
Professor Dr. Frode Bj?rdal
Universitetet i Oslo Universidade Federal do Rio Grande do Norte
quicumque vult hinc potest accedere ad paginam virtualem meam
<http://www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150317/876439f0/attachment.html>

From martin at eipye.com  Tue Mar 17 11:50:35 2015
From: martin at eipye.com (Martin Davis)
Date: Tue, 17 Mar 2015 08:50:35 -0700
Subject: [FOM] MASTER'S DEGREE IN LOGIC AND PHILOSOPHY OF SCIENCE IN SPAIN
Message-ID: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>

NINE EDITION OF THE MASTER IN LOGIC AND PHILOSOPHY OF SCIENCE (2015-16)

This is a JOINT MASTER, with the participation of the following Spanish
institutions: Universidad de Salamanca, Universidad de Valladolid,
Universidad de Santiago de Compostela, Universidad de La Laguna,
Universidad de Granada, Universidad de A Coru?a, Universidad de Valencia
and CSIC (Spanish National Research Council).



It is programmed as BLENDED LEARNING (B-Learning). The on-campus sessions
are provided during two weeks at the beginning of the
first term, and twoweeks at the beginning of the second term. The rest of
the Master is taught on-line.


Students can complete the program in one year (full time) or two (pat
time). It is structured in four non exclusive branches: a) argumentation,
b) logic, c) philosophy and history of science and technology, and d) mind
and language.

Most of the courses are taught in Spanish, some of them being offered in
English. Students whose speaking and/or writing language is English,
Portuguese, Italian or French are also welcome.

We welcome applications from a wide range of backgrounds: philosophy,
science, technology or humanities.

More information: http:// <http://logicae.usal.es/drupal/?q=en/pagina/home>
epimenides.usal.es





The Master provides direct access to the PhD PROGRAM IN LOGIC AND
PHILOSOPHY OF SCIENCE: http://doctoradologifici.usal.es





The pre-enrollment can be done at any of the first four universities
mentioned above. In the case of the Universidad de Salamanca (the guest
university during this edition), the deadline for pre-enrollment ends on July
10  this year.



All information about pre-enrollment can be found at*:*



http://www.usal.es/webusal/en/node/42002


For any query: omtorres at usal.es
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150317/bfa83415/attachment.html>

From maa4200 at gmail.com  Tue Mar 17 13:56:46 2015
From: maa4200 at gmail.com (Fred Richman)
Date: Tue, 17 Mar 2015 13:56:46 -0400
Subject: [FOM] Potential and Actual Infinity
In-Reply-To: <338B0253-FA7D-4515-A9A2-F2B1A505D226@mac.com>
References: <mailman.23.1426262782.8783.fom@cs.nyu.edu>
	<alpine.LFD.2.11.1503131351160.16449@laurent.mit.edu>
	<20150314110149.Horde.DgXHvelY3r5B0T23cUfZbg6@webmail.helsinki.fi>
	<338B0253-FA7D-4515-A9A2-F2B1A505D226@mac.com>
Message-ID: <CAEMYdipm3o-TfcwthCvJqavrbykhO78mKga5WyYmhJkXAnZonQ@mail.gmail.com>

What has always bothered me is sort of a converse to what Bill alluded to.
Namely, why should commitment to actual infinity entail classical logic? A
number of people seem to think that it does

--Fred
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150317/d1b51c5e/attachment.html>

From paul at mtnmath.com  Tue Mar 17 13:42:59 2015
From: paul at mtnmath.com (Paul Budnik)
Date: Tue, 17 Mar 2015 10:42:59 -0700
Subject: [FOM] Potential and Actual Infinity
Message-ID: <550867A3.9050907@mtnmath.com>

Potential infinity is a philosophical concept which is usually defined 
in a way that requires time as a fundamental assumption. For example the 
Wikipedia entry on Actual Infinity says "This is contrasted with 
*potential infinity*, in which a non-terminating process (such as "add 1 
to the previous number") produces an unending "infinite" sequence of 
results, but each individual result is finite and is achieved in a 
finite number of steps." Or consider Aristotle's definition from the 
same article "/Potential infinity/ is something that is never complete: 
more and more elements can be always added, but never infinitely many."

Quantifying over the integers and a recursive relations (PI_1) is 
equivalent to asking if an easily defined Turing Machine (TM) has an 
unbounded number of outputs. Generalizations of this question can be 
defined for Pi_n. With the aid of recursive ordinal notations it can be 
generalized further.

For more about this approach see "Objective mathematics in a finite 
unbounded universe" (www.mtnmath.com/math/objMath.pdf) and "Generalizing 
Kleene?s O ..." (www.mtnmath.com/ord/kleeneo.pdf).

Paul Budnik
www.mtnmath.com

-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150317/bb2cfc7e/attachment-0001.html>

From pmt6sbc at maths.leeds.ac.uk  Wed Mar 18 10:48:17 2015
From: pmt6sbc at maths.leeds.ac.uk (S B Cooper)
Date: Wed, 18 Mar 2015 14:48:17 GMT
Subject: [FOM] Computability in Europe 2015: Call for Informal Presentations
Message-ID: <201503181448.t2IEmH3X029363@maths.leeds.ac.uk>

-------------------------------------------------------------------
  COMPUTABILITY IN EUROPE 2015: Evolving Computability
                    Bucharest, Romania
                      June 29 - July 3
               http://fmi.unibuc.ro/CiE2015/
-------------------------------------------------------------------

FUNDING DEADLINE APPROACHING 
 - ASL STUDENT TRAVEL GRANTS:  March 28, 2015

SUBMISSION DEADLINE FOR INFORMAL PRESENTATIONS:  APRIL 24, 2015
-------------------------------------------------------------------
 
CALL FOR INFORMAL PRESENTATIONS

There is a remarkable difference in conference style between computer 
science and mathematics conferences. Mathematics conferences allow 
for informal presentations that are prepared very shortly before the 
conference and inform the participants about current research and 
work in progress. The format of computer science conferences with 
pre-conference proceedings is not able to accommodate this form of 
scientific communication.

Continuing the tradition of past CiE conferences, also this year's 
CiE conference endeavours to get the best of both worlds. In addition 
to the formal presentations based on our LNCS proceedings volume, we 
invite researchers to present informal presentations. For this, please 
send us a brief description of your talk (between one paragraph and 
one page) by:

APRIL 24, 2015

Please submit your abstract electronically, via EasyChair 
<https://www.easychair.org/login.cgi?conf=cie2015>, 
selecting the category "Informal Presentation".

You will be notified whether your talk has been accepted for informal 
presentation usually within a week or two after your submission.

If you intend to apply for the ASL Student Travel Award, you might 
need us to confirm that your are going to give a presentation at 
CiE 2015 (applications of students who are presenting get higher 
priority). Therefore, we would like to ask you to submit your informal 
presentations by March 25 so that we can send you the notification 
before the ASL deadline of March 28.

FUNDING OPPORTUNITIES:

CiE 2015 has received funding from the ASL (Association for Symbolic 
Logic) and EATCS (European Association for Theoretical Computer 
Science) that allows students who are members of ASL or EATCS and 
want to attend CiE 2015 to apply for travel funds or a reduction of 
the early registration fee.

Preference will be given to presenters of accepted papers.

Applications for ASL travel grants have to be addressed directly to 
ASL, with a strict deadline of March 28, 2015.

Applications for EATCS travel grants have to be sent to 
cie2015 at fmi.unibuc.ro 
prior to the early registration deadline.

___________________________________________________________________
CiE 2015                               http://fmi.unibuc.ro/CiE2015
ASSOCIATION COMPUTABILITY IN EUROPE http://www.computability.org.uk
CiE Conference Series                    http://www.illc.uva.nl/CiE
CiE Membership Application Form http://www.lix.polytechnique.fr/CIE
Computability (Journal of CiE)  http://www.computability.de/journal
CiE on FaceBook                    https://www.facebook.com/AssnCiE
Association CiE on Twitter       https://twitter.com/AssociationCiE
___________________________________________________________________



From georg.moser at exchange.uibk.ac.at  Fri Mar 20 02:40:52 2015
From: georg.moser at exchange.uibk.ac.at (Georg Moser)
Date: Fri, 20 Mar 2015 07:40:52 +0100
Subject: [FOM] =?utf-8?q?Call_for_Papers=3A_Hilbert=E2=80=99s_Epsilon_and_?=
 =?utf-8?q?Tau_in_Logic=2C_Informatics_and_Linguistics?=
In-Reply-To: <201503172333.t2HNXVYA027405@easychair.org>
References: <201503172333.t2HNXVYA027405@easychair.org>
Message-ID: <550BC0F4.3030600@exchange.uibk.ac.at>

Call for Papers

Hilbert?s Epsilon and Tau in Logic, Informatics and Linguistics

Dates: June 10-12, 2015
Location: Montpellier, France

Workshop Webpage: https://sites.google.com/site/epsilon2015workshop/
Contact email: Epsilon2015 at easychair.org

Submission deadline: April 1st, 2015
Submission webpage: https://easychair.org/conferences/?conf=epsilon2015

Organizers / workshop co-chairs:
Stergios Chatzikyriakidis, LIRMM-CNRS, University of Montpellier
Fabio Pasquali, University of Marseille
Christian Retor?, University of Montpellier & LIRMM-CNRS
Host: I2M-CNRS and University of Montpellier

Workshop information:
This workshop aims at promoting work on Hilbert?s epsilon calculus in a 
number of relevant fields ranging from Philosophy and Mathematics to 
Linguistics and Informatics. The Epsilon and Tau operators were 
introduced by David Hilbert, inspired by Russell's Iota operator for 
definite descriptions, as binding operators that form terms from 
formulae. One of their main features is that substitution with Epsilon 
and Tau terms expresses quantification. This leads to a calculus which 
is a strict and conservative extension of First Order Predicate Logic. 
The calculus was developed for studying first order logic in view of the 
program of providing a rigorous foundation of mathematics via syntactic 
consistency proofs. The first relevant outcomes that certainly deserve a 
mention are the two "Epsilon Theorems" (similar to quantifiers 
elimination), the first correct proof of Herbrand?s theorem or the use 
of the Epsilon operator in Bourbaki?s ?l?ments de Math?matique. Nowadays 
the!
   interest in the Epsilon substitution method has spread in a variety 
of fields: Mathematics, Logic, Philosophy, History of Mathematics, 
Linguistic, Type Theory, Computer science, Category Theory and others.

Submission
The workshop welcomes submissions of up to 4 (but not less than 2) 
pages. Usual spacing, font and margin should be used (single-spaced, 
11pt or larger, and 1 inch margin on A4 or letter size paper). Abstracts 
should be submitted by April 1st, 2015 as pdf files through the 
EasyChair conference system ( 
https://easychair.org/conferences/?conf=epsilon2015). An indicative list 
of themes that are of particular interest to the conference are 
(non-exhaustive):

- History of Logic
- Philosophy
- Proof theory
- Model theory
- Category theory
- Type theory
- Quantification in Natural language
- Noun-Phrases Semantics
- Proof Assistants (e.g. Coq, Isabelle, ... )
- Other subnectors (e.g. Russell's iota, ?-operator, ... )

Reviewing:
Abstracts will be reviewed by members of the program committee, and, 
where appropriate, outside reviewers. The organizers will be responsible 
for making decisions partly in consultation with the program committee. 
Notifications will be made by May 1st, 2015.

Post-Proceedings:
Selected papers from the workshop will appear as a special volume in 
Journal of Logics and their Applications

Important dates:
April 1st, 2015: Submission deadline
May 1st,2015: Notification of acceptance
June 10-12, 2015: Workshop

Invited speakers:

Claus-Peter Wirth (University of Saarland): The descriptive operators 
iota, tau and epsilon - on their origin, partial and complete 
specification, model-theoretic semantics, practical applicability (with 
the support of the Hilbert Bernays Project (sponsored by IFCoLog)).

Vito Michele Abrusci (University of Roma Tre): Hilbert's tau and epsilon 
in proof theory.

Hartley Slater (University of Western Australia): Linguistic and 
philosophical ramifications of the epsilon calculus.


Program Committee:

Daisuke Bekki (Ochanomizu University)

Stergios Chatzikyriakidis (LIRMM-CNRS & University of Montpellier)

Francis Corblin  (University of Paris-Sorbonne & Institut Jean Nicod CNRS)

Michael Gabbay (University of Cambridge)

Makoto Kanazawa (National Institute of Informatics of Tokyo)

Ruth Kempson (King's College, London)

Alda Mari (CNRS Institut Jean Nicod & ENS & EHESS)

Richard Moot (CNRS LaBRI & Bordeaux University)

Georg Moser (University of Innsbruck)

Bruno Woltzenlogel Paleo (Vienna University of Technology)

Michel Parigot (CNRS-PPS & University of Paris Diderot 7)

Fabio Pasquali (University of Aix-Marseille & I2M CNRS)

Christian Retor? (University of Montpellier & LIRMM-CNRS)

Mark Steedman (University of Edimburgh)



From panu.raatikainen at helsinki.fi  Fri Mar 20 12:04:37 2015
From: panu.raatikainen at helsinki.fi (Panu Raatikainen)
Date: Fri, 20 Mar 2015 18:04:37 +0200
Subject: [FOM] Infinitary languages, consistency and models
In-Reply-To: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>
References: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>
Message-ID: <20150320180437.Horde.IELTzNaMjUPwZYtTGLHVHQ1@webmail.helsinki.fi>

According to Scott (1965), Ryll-Nardzewski (unpublished) has  
constructed a syntactically consistent and complete set of sentences  
of the infinitary language L_omega1 omega which has no model.

(i) Does anyone know what is the general idea of the construction?

(ii) Does anyone have any idea whether something analogous holds for  
infinitary languages which allow uncountable conjunctions/disjunctions?


Best,

Panu
-- 
Panu Raatikainen

Ph.D., Adjunct Professor in Theoretical Philosophy

Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies
P.O. Box 24  (Unioninkatu 38 A)
FIN-00014 University of Helsinki
Finland

E-mail: panu.raatikainen at helsinki.fi

http://www.mv.helsinki.fi/home/praatika/


From gergely.szekely at gmail.com  Sat Mar 21 05:45:25 2015
From: gergely.szekely at gmail.com (=?UTF-8?Q?Gergely_Sz=C3=A9kely?=)
Date: Sat, 21 Mar 2015 10:45:25 +0100
Subject: [FOM] Deadline Extension: Logic,
 Relativity and Beyond Conference (Budapest, 9-13 Aug 2015)
Message-ID: <CAHE+U3L+key+7EHi8_aC-8qoSM1D8ws=nTWoBf+W+pem1eN_XQ@mail.gmail.com>

Because several people have asked it, the deadline will be extended:

*New deadline* for abstract or paper submission: *5 April*, 2015

You can submit your abstract or paper (or extended abstract) via the
following link:
http://www.easychair.org/conferences/?conf=lrb15

Papers and extended abstracts should be no more than 12 pages (excluding
bibliography), and should be submitted in pdf formatted for A4 paper.
Extended abstracts and papers will be published in electronic form as a
conference proceedings.

See below, for more details of the conference.

==============================

Logic, Relativity and Beyond
2nd international conference
http://www.renyi.hu/conferences/lrb15/

August 9-13 2015, Budapest, Hungary

*********************************************
IMPORTANT DATES:

*New deadline* for abstract or paper submission: *5 April*, 2015
Notifying the authors: 20 May, 2015
Early registration: 31 May, 2015
Conference: 9-13 August, 2015

There are several new and rapidly evolving research areas blossoming out
from the interaction of logic and relativity theory. The aim of this
conference series, which take place once every 2 or 3 years, is to attract
and bring together mathematicians, physicists, philosophers of science, and
logicians from all over the word interested in these and related areas to
exchange new ideas, problems and results.

Topics include (but are not restricted to):
*  Special and general relativity
*  Axiomatizing physical theories
*  Logical foundations of spacetime
*  Computability and physics
*  Logic of causality
*  Relativistic computation
*  Knowledge acquisition in science
*  Branching spacetime
*  Concept algebras and algebraic logic
*  Logic of time and space
*  Cylindric and relation algebras
*  Relativity theory and philosophy of science

*****************************************************
Invited speakers:
- S. Barry Cooper (University of Leeds)
- Alexander K. Guts (Omsk State University)
- Mark Hogarth (University of Cambridge)
- Thomas M?ller (Universit?t Konstanz)
- Istvan Racz (Wigner Institute)
- Laszlo E. Szabo (Eotvos University)

*****************************************************
Program Committee:
- Istvan Nemeti (Chair, Renyi Institute)
- Thomas Benda (Yang Ming University)
- Miklos Ferenczi (Budapest University of Technology)
- Michele Friend (The George Washington University)
- Judit X. Madarasz (Renyi Institute)
- John Byron Manchak (University of Washington)
- Tomasz Placek (Jagiellonian University)
- Ildiko Sain (Renyi Institute)
- Mike Stannett (University of Sheffield)
- Gergely Szekely (Renyi Institute)
- Christian Wuthrich (University of California)

*******************************************************
Organizing Committee:
- Gergely Szekely (Chair, Renyi Institute)
- Hajnal Andreka (Renyi Institute)
- Koen Lefever (Vesalius College)
- Attila Molnar (Eotvos University)
- Mike Stannett (University of Sheffield)

We invite you to submit your abstract or paper (or extended abstract) via
the following link:
http://www.easychair.org/conferences/?conf=lrb15

Looking forward to seeing you in Budapest.

If you know anyone who may be interested in this conference, please notify
them.

Contact: Gergely Szekely - lrb15 at renyi.mta.hu

[Apologies for multiple postings.]

If you wish to be removed from this notification list, please send an email
to szekely.gergely at renyi.mta.hu with UNSUBSCRIBE LRB LIST in the subject
line.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150321/4a1b0d4a/attachment.html>

From ali.enayat at gmail.com  Sat Mar 21 19:01:04 2015
From: ali.enayat at gmail.com (Ali Enayat)
Date: Sun, 22 Mar 2015 00:01:04 +0100
Subject: [FOM] Petr Vopenka
Message-ID: <CAPzKPNv9F67Qjxckz2SDBW8T6NWs+W3-d=Bon5teb6svUgNFkw@mail.gmail.com>

Dear FOMers,

Sad news: Petr Vopenka passed away on March 20, 2015.

Some of you might be interested to watch a (free) 22-min Cech
documentary on Vopenka (with English subtitles) made in 2006; there is
a surprise half-way through, about the origins of Vopenka cardinals.

http://docalliancefilms.com/film/891

Best regards,

Ali Enayat

From panu.raatikainen at helsinki.fi  Sun Mar 22 03:15:36 2015
From: panu.raatikainen at helsinki.fi (Panu Raatikainen)
Date: Sun, 22 Mar 2015 09:15:36 +0200
Subject: [FOM] Second-order logic and neo-logicism
In-Reply-To: <20150320180437.Horde.IELTzNaMjUPwZYtTGLHVHQ1@webmail.helsinki.fi>
References: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>
	<20150320180437.Horde.IELTzNaMjUPwZYtTGLHVHQ1@webmail.helsinki.fi>
Message-ID: <20150322091536.Horde._kC78UE9TIT2HypdbiIC9w1@webmail.helsinki.fi>


The following new paper might interest some here:

Panu Raatikainen: "Neo-logicism and its logic", History and Philosophy  
of Logic (forthcoming)

http://philpapers.org/rec/RAANAI


It has greatly benefited from certain old discussions here in FOM on  
the second-order logic; special thanks to Martin Davis!


All the Best

Panu



Abstract:
The rather unrestrained use of second-order logic in the neo-logicist  
program is critically examined. It is argued in some detail that it  
brings with it genuine set-theoretical existence assumptions, and that  
the mathematical power that Hume?s Principle seems to provide, in the  
derivation of Frege?s Theorem, comes largely from the ?logic? assumed  
rather than from Hume?s principle. It is shown that Hume?s principle  
is in reality not stronger than the very weak Robinson Arithmetic Q.  
Consequently, only few rudimentary facts of arithmetic are logically  
derivable from Hume?s principle. And that hardly counts as a  
vindication of logicism.
-- 
Panu Raatikainen

Ph.D., Adjunct Professor in Theoretical Philosophy

Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies
P.O. Box 24  (Unioninkatu 38 A)
FIN-00014 University of Helsinki
Finland

E-mail: panu.raatikainen at helsinki.fi

http://www.mv.helsinki.fi/home/praatika/


From josef.urban at gmail.com  Sun Mar 22 07:04:51 2015
From: josef.urban at gmail.com (Josef Urban)
Date: Sun, 22 Mar 2015 12:04:51 +0100
Subject: [FOM] Petr Vopenka
In-Reply-To: <CAPzKPNv9F67Qjxckz2SDBW8T6NWs+W3-d=Bon5teb6svUgNFkw@mail.gmail.com>
References: <CAPzKPNv9F67Qjxckz2SDBW8T6NWs+W3-d=Bon5teb6svUgNFkw@mail.gmail.com>
Message-ID: <CAFP4q16YEtT7ryO1OCumxS2zs=pRxhFmxmEMDXOJ=LM9-dPa=Q@mail.gmail.com>

He recently claimed that actual infinity is really in some sense
contradictory (e.g. in this talk in Czech:
https://www.youtube.com/watch?v=zcaZJXO-p4g ). More explanations are in his
recent book, which is hopefully being translated, but I am not sure there
is a "formal proof" there.

I found his investigations and seminars on "psychoanalysis" of science and
mathematics extremely inspiring.  Counterweight to the standard bulk of
overspecialized and unmotivated definition/theorem/proof lectures.

Josef


On Sun, Mar 22, 2015 at 12:01 AM, Ali Enayat <ali.enayat at gmail.com> wrote:

> Dear FOMers,
>
> Sad news: Petr Vopenka passed away on March 20, 2015.
>
> Some of you might be interested to watch a (free) 22-min Cech
> documentary on Vopenka (with English subtitles) made in 2006; there is
> a surprise half-way through, about the origins of Vopenka cardinals.
>
> http://docalliancefilms.com/film/891
>
> Best regards,
>
> Ali Enayat
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150322/0db84ac8/attachment.html>

From marek at cs.uky.edu  Sun Mar 22 19:51:29 2015
From: marek at cs.uky.edu (Victor Marek)
Date: Sun, 22 Mar 2015 19:51:29 -0400
Subject: [FOM] More on Vopenka movie
Message-ID: <20150322235129.GA5762@cs.uky.edu>

The place where the movie about Petr Vopenka is:
http://dafilms.com
That site has search window. One can put VOPENKA in the search window.
It is better to do such search, as the URL contains reference in Czech.

But a movie itself does not really provide the true context of dramatic life
of Vopenka, and in particular persecution of Vopenka for over 20 years
(1968-1989). Surprisingly for me, his collaborators (Petr Hajek, Tomas Jech,
Antonin Sochor) are not called by the director to give a testimonial. A pity,
for Petr Vopenka lived the life of truth and was willing to pay for it.

Victor Marek

Victor W. Marek                                 Department of Computer Science
marek at cs.uky.edu                                        University of Kentucky
marek at cs.engr.uky.edu                                 Lexington, KY 40506-0633
859-257-3496 (office)                                     859-257-3961 (Dept)
http://www.cs.uky.edu/~marek                              859-257-1505 (FAX)

From mara at usal.es  Sun Mar 22 12:58:50 2015
From: mara at usal.es (Maria Manzano)
Date: Sun, 22 Mar 2015 17:58:50 +0100
Subject: [FOM] The Life and Work of Leon Henkin. Essays on His Contributions
Message-ID: <3C6170FD-E5E6-4606-8878-C5873A411970@usal.es>

We are glad to announce the launching of the book

The Life and Work of Leon Henkin
Essays on His Contributions
Manzano, Mar?a, Sain, Ildik?, Alonso, Enrique (Eds.)
http://www.springer.com/birkhauser/mathematics/book/978-3-319-09718-3
The papers are individually available here:
http://link.springer.com/book/10.1007/978-3-319-09719-0

Mar?a Manzano
Enrique Alonso
Ildik? Sain
Editors
**********************************************************************************************

From the preface:

Leon Henkin (1921-2006) was an extraordinary logician and an excellent teacher. His writings became influential from the very start of his career with his doctoral thesis, The completeness of formal systems, defended in 1947 under the direction of Alonzo Church.

Answering an invitation from Alfred Tarski, Henkin joined the Mathematics Department in Berkeley (University of California) in 1953. When Tarski and Henkin were able to assemble a number of logicians from the Department of Mathematics and Philosophy, they created an interdepartmental agency, the very famous Group in Logic and the Methodology of Science. He stayed with the Department until 1991, when he retired and became an Emeritus Professor.

Henkin was often described as a social activist, he labored much of his career to boost the number of women and underrepresented minorities in the upper echelons of mathematics. He was also very aware that we are beings immersed in the crucible of history from which we find it hard to escape.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150322/db4c92dc/attachment.html>

From schweber at berkeley.edu  Mon Mar 23 04:32:46 2015
From: schweber at berkeley.edu (Noah David Schweber)
Date: Mon, 23 Mar 2015 01:32:46 -0700
Subject: [FOM] Workshop on Vaught's Conjecture
Message-ID: <CAFoZ5beE3x2H5ZvCe3KFLn9Xk5PyS+KPT74wXsYDvvamRH_Y=A@mail.gmail.com>

First Announcement

2nd Workshop on Vaught's Conjecture

June 1-5, 2015 at the University of California - Berkeley

https://math.berkeley.edu/~schweber/vcc15/


Organized by:

 - Julia Knight (University of Notre Dame, Julia.F.Knight.1 at nd.edu
<knight.1 at nd.edu>)

 - Antonio Montalban (UC Berkeley, antonio at math.berkeley.edu)

 - Thomas Scanlon (UC Berkeley, scanlon at math.berkeley.edu)

 - Noah Schweber (UC Berkeley, schweber at math.berkeley.edu)


********


A workshop on the mathematics surrounding Vaught's Conjecture (on the
number of isomorphism types of countable models of a countable complete
theory elementary first order theory) will be held at the University of
California at Berkeley from June 1 to June 6, 2015. The first workshop on
Vaught's Conjecture was held at the University of Notre Dame, in May of
2005.  This workshop resulted in a number of new ideas and collaborations,
some of which were published in a special issue of the Notre Dame Journal
of Formal Logic. We hope that this second workshop will build on the
success of the first.


There will be tutorials by Uri Andrews, Su Gao, and Chris Laskowski; the
invited speakers currently include:


Nate Ackerman

John Baldwin

Howard Becker

Samuel Coskey

Cameron Freer

Sy Friedman

Robin Knight

Paul Larson

David Marker

Ludomir Newelski

Richard Rast

Gerald Sacks

Slawomir Solecki

Ioannis Souldatos
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150323/5a522e22/attachment.html>

From lanzetr at gmail.com  Mon Mar 23 12:25:51 2015
From: lanzetr at gmail.com (Ran Lanzet)
Date: Mon, 23 Mar 2015 18:25:51 +0200
Subject: [FOM] Second-order logic and neo-logicism
References: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>	<20150320180437.Horde.IELTzNaMjUPwZYtTGLHVHQ1@webmail.helsinki.fi>
	<20150322091536.Horde._kC78UE9TIT2HypdbiIC9w1@webmail.helsinki.fi>
Message-ID: <021601d06586$09589e10$1c09da30$@gmail.com>

I am probably missing something here, and will be glad if you could clarify.

As far as I understand, your main argument against neo-logicism is roughly this:
1. The rules of 2nd-order logic (SOL) employed by the neo-logicist are very strong, in the sense of entailing some serious mathematical content. In particular:
	a. They are provably equivalent to the "basic rules" of SOL plus the unrestricted impredicative comprehension scheme.
	b. Once we accept those rules as the background logic, we get immediately from the very weak Q+ to the very strong PA2.
2. Hence, it does not seem reasonable to accept the neo-logicist's version of SOL as logic.

Now I believe the neo-logicist would happily accept (1): after all, her basic claim is that, essentially, all of ordinary mathematics is derivable from logic (more precisely: from her favorite version of SOL plus Hume's principle (HP); and I'm sure she will happily accept that SOL and not HP does the majority of work here). She will, though, undoubtedly object to your step from (1) to (2). She might argue as follows: the move from (1) to (2) is unwarranted, unless we accept the following principle: 
(*) 	if a set of rules entails substantial mathematical theorems, then it is unreasonable to regard that set of rules as part of logic. 
But accepting this principle -- so she might argue -- is to beg the question against logicism.

Question: what did I miss here? Or, more specifically: why is the suggested reply ineffective against your argument?

Best,
Ran        

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Panu Raatikainen
Sent: Sunday, March 22, 2015 09:16
To: Foundations of Mathematics
Subject: [FOM] Second-order logic and neo-logicism


The following new paper might interest some here:

Panu Raatikainen: "Neo-logicism and its logic", History and Philosophy of Logic (forthcoming)

http://philpapers.org/rec/RAANAI


It has greatly benefited from certain old discussions here in FOM on the second-order logic; special thanks to Martin Davis!


All the Best

Panu



Abstract:
The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions, and that the mathematical power that Hume?s Principle seems to provide, in the derivation of Frege?s Theorem, comes largely from the ?logic? assumed rather than from Hume?s principle. It is shown that Hume?s principle is in reality not stronger than the very weak Robinson Arithmetic Q.  
Consequently, only few rudimentary facts of arithmetic are logically derivable from Hume?s principle. And that hardly counts as a vindication of logicism.
--
Panu Raatikainen

Ph.D., Adjunct Professor in Theoretical Philosophy

Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies P.O. Box 24  (Unioninkatu 38 A)
FIN-00014 University of Helsinki
Finland

E-mail: panu.raatikainen at helsinki.fi

http://www.mv.helsinki.fi/home/praatika/

_______________________________________________
FOM mailing list
FOM at cs.nyu.edu
http://www.cs.nyu.edu/mailman/listinfo/fom

-----
No virus found in this message.
Checked by AVG - www.avg.com
Version: 2015.0.5856 / Virus Database: 4311/9342 - Release Date: 03/20/15


From gprimiero at libero.it  Mon Mar 23 13:23:09 2015
From: gprimiero at libero.it (gprimiero at libero.it)
Date: Mon, 23 Mar 2015 18:23:09 +0100 (CET)
Subject: [FOM] CfP: HAPOC3 -- Third International Conference for the History
 and Philosophy of Computing
Message-ID: <412187662.6815901427131389974.JavaMail.httpd@webmail-20.iol.local>

---------------------------------------------------------------------------------------------------------------------------
Call For Papers
HaPoC 3: Third International Conference for the History and Philosophy 
of Computing
8 -- 11 October, 2015, Pisa
hapoc2015.di.unipi.it
---------------------------------------------------------------------------------------------------------------------------

The DHST commission for the history and philosophy of computing 
(www.hapoc.org) is happy to announce the third HAPOC conference. The 
series aims at creating an interdisciplinary focus on computing, 
stimulating a dialogue between the historical and philosophical 
viewpoints. To this end, the conference hopes to bring together 
researchers interested in the historical developments of computing, as 
well as those reflecting on the sociological and philosophical issues 
springing from the rise and ubiquity of computing machines in the 
contemporary landscape. In the past editions, the conference has 
successfully presented a variety of voices, contributing to the creation 
of a fruitful dialogue between researchers with different backgrounds 
and sensibilities.

For HaPoC 2015 we welcome contributions from historians and philosophers 
of computing as well as from philosophically aware computer scientists 
and mathematicians. Topics include but are not limited to
? History and Philosophy of Computation (interpretation of the 
Church-Turing thesis; models of computation; logical/mathematical 
foundations of computer science; information theory...)
? History and Philosophy of Programming (classes of programming 
languages; philosophical status of programming...)
? History and Philosophy of the Computer (from calculating machines to 
the future of the computer; user interfaces; abstract architectures...)
? History and Epistemology of the use of Computing in the sciences 
(simulation vs. modelisation; computer-assisted proofs; linguistics...)
? Computing and the Arts: historical and conceptual issues (temporality 
in digital art; narration in interactive art work...)
? Social, ethical and pedagogical aspects of Computing (pedagogy of 
computer science; algorithms and copyright; Internet, culture, society...)

Our invited speakers are Nicola Angius (Universit? di Sassari, IT), 
Lenore Blum (Carnagie Mellon University, USA),David Allan Grier (IEEE &amp; George Washington University, USA),Furio Honsell (Universit? di Udine, IT),Pierre Mounier-Kuhn (CNRS &amp; Universit? Paris-Sorbonne, F),Franck Varenne (Universit? de Rouen, F).

We cordially invite researchers working in a field relevant to the 
topics of the conference to submit a short abstract of approximately 200 
words and an extended abstract of at most a 1000 words (references 
included) to www.easychair.org/conferences/?conf=hapoc2015

Abstracts must be written in English and anonymised. Please note that 
the format of uploaded files must be either .pdf or .doc. In order to 
access the submission page, an EasyChair account will be required. 
Please notice that what is called ?abstract? in the EasyChair ?Title, 
Abstract and Other Information? section corresponds to the short 
abstract of this call, and what is called ?paper? in the EasyChair 
?Upload Paper? section corresponds to the extended abstract of this call.

Please check out the website of HaPoC 2015 for more information on the 
conference at http://hapoc2015.sciencesconf.org

A post-proceedings volume is going to appear in the IFIP Advances in 
Information and Communication Technology series, published by Springer.


IMPORTANT DATES:
Submission deadline: June 19, 2015
Notification of acceptance: July 19, 2015

The 2015 conference is located in Pisa, the cradle of Italian computer 
science: here the first Italian computers were designed in the 
mid-Fifties and the first Master course in informatics was established 
in 1969. The Museum of Computing Machinery, part of the University of 
Pisa, shows some artefacts from the early days of Italian CS, as well a 
selection of personal computing machines. Besides its artistic 
attractions, among them the world-famous leaning tower, during the days 
of the conference Pisa will host the Internet Festival, devoted to all 
the aspects of the net (www.internetfestival.it)

-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150323/8e5d3013/attachment.html>

From williamtait at mac.com  Tue Mar 24 12:51:52 2015
From: williamtait at mac.com (WILLIAM TAIT)
Date: Tue, 24 Mar 2015 11:51:52 -0500
Subject: [FOM] Second-order logic and neo-logicism
In-Reply-To: <021601d06586$09589e10$1c09da30$@gmail.com>
References: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>
	<20150320180437.Horde.IELTzNaMjUPwZYtTGLHVHQ1@webmail.helsinki.fi>
	<20150322091536.Horde._kC78UE9TIT2HypdbiIC9w1@webmail.helsinki.fi>
	<021601d06586$09589e10$1c09da30$@gmail.com>
Message-ID: <523DC5F6-3908-4B71-9510-11B8A5EEC8E9@mac.com>

Could you say more, at least briefly, about what counts as 'ordinary mathematics' in

> her basic claim is that, essentially, all of ordinary mathematics is derivable from logic (more precisely: from her favorite version of SOL plus Hume's principle (HP

Thanks,
Bill

> On Mar 23, 2015, at 11:25 AM, Ran Lanzet <lanzetr at gmail.com> wrote:
> 
> I am probably missing something here, and will be glad if you could clarify.
> 
> As far as I understand, your main argument against neo-logicism is roughly this:
> 1. The rules of 2nd-order logic (SOL) employed by the neo-logicist are very strong, in the sense of entailing some serious mathematical content. In particular:
> 	a. They are provably equivalent to the "basic rules" of SOL plus the unrestricted impredicative comprehension scheme.
> 	b. Once we accept those rules as the background logic, we get immediately from the very weak Q+ to the very strong PA2.
> 2. Hence, it does not seem reasonable to accept the neo-logicist's version of SOL as logic.
> 
> Now I believe the neo-logicist would happily accept (1): after all, her basic claim is that, essentially, all of ordinary mathematics is derivable from logic (more precisely: from her favorite version of SOL plus Hume's principle (HP); and I'm sure she will happily accept that SOL and not HP does the majority of work here). She will, though, undoubtedly object to your step from (1) to (2). She might argue as follows: the move from (1) to (2) is unwarranted, unless we accept the following principle: 
> (*) 	if a set of rules entails substantial mathematical theorems, then it is unreasonable to regard that set of rules as part of logic. 
> But accepting this principle -- so she might argue -- is to beg the question against logicism.
> 
> Question: what did I miss here? Or, more specifically: why is the suggested reply ineffective against your argument?
> 
> Best,
> Ran        
> 
> -----Original Message-----
> From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Panu Raatikainen
> Sent: Sunday, March 22, 2015 09:16
> To: Foundations of Mathematics
> Subject: [FOM] Second-order logic and neo-logicism
> 
> 
> The following new paper might interest some here:
> 
> Panu Raatikainen: "Neo-logicism and its logic", History and Philosophy of Logic (forthcoming)
> 
> http://philpapers.org/rec/RAANAI
> 
> 
> It has greatly benefited from certain old discussions here in FOM on the second-order logic; special thanks to Martin Davis!
> 
> 
> All the Best
> 
> Panu
> 
> 
> 
> Abstract:
> The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions, and that the mathematical power that Hume?s Principle seems to provide, in the derivation of Frege?s Theorem, comes largely from the ?logic? assumed rather than from Hume?s principle. It is shown that Hume?s principle is in reality not stronger than the very weak Robinson Arithmetic Q.  
> Consequently, only few rudimentary facts of arithmetic are logically derivable from Hume?s principle. And that hardly counts as a vindication of logicism.
> --
> Panu Raatikainen
> 
> Ph.D., Adjunct Professor in Theoretical Philosophy
> 
> Theoretical Philosophy
> Department of Philosophy, History, Culture and Art Studies P.O. Box 24  (Unioninkatu 38 A)
> FIN-00014 University of Helsinki
> Finland
> 
> E-mail: panu.raatikainen at helsinki.fi
> 
> http://www.mv.helsinki.fi/home/praatika/
> 
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
> 
> -----
> No virus found in this message.
> Checked by AVG - www.avg.com
> Version: 2015.0.5856 / Virus Database: 4311/9342 - Release Date: 03/20/15
> 
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom


From joeshipman at aol.com  Tue Mar 24 12:54:01 2015
From: joeshipman at aol.com (Joe Shipman)
Date: Tue, 24 Mar 2015 12:54:01 -0400
Subject: [FOM] Second-order logic and neo-logicism
In-Reply-To: <021601d06586$09589e10$1c09da30$@gmail.com>
References: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>
	<20150320180437.Horde.IELTzNaMjUPwZYtTGLHVHQ1@webmail.helsinki.fi>
	<20150322091536.Horde._kC78UE9TIT2HypdbiIC9w1@webmail.helsinki.fi>
	<021601d06586$09589e10$1c09da30$@gmail.com>
Message-ID: <24B7149A-C780-4F41-8E0C-9FD15DC9DA8C@aol.com>

That is one of the two points I was going to make. My other point is that there is a more cogent argument against "second-order logicism" than that SOL entails strong mathematical theorems, namely:

"Accepting your full semantics for SOL doesn't get us new mathematics without a stronger deductive calculus. The most you can do in general is to say that a for a strong mathematical statement X, either X or ~X is a logical truth, without specifying which is the case. Please give me some axioms and inference rules to go along with your SOL semantics, which are themselves justifiable as logical principles rather than mathematical ones."

Personally, I think this is an objection which can be well met up to a point--probably a system as strong as Maclane Set Theory can be justified as purely "logical". 

I agree with Ran that the objections as previously stated seemed to count mathematical power as a strike against a logicist development in an unfairly question-begging way, but perhaps my way of framing this will lead to more fruitful and technically interesting discussion.

-- JS

Sent from my iPhone

> On Mar 23, 2015, at 12:25 PM, "Ran Lanzet" <lanzetr at gmail.com> wrote:
> 
> I am probably missing something here, and will be glad if you could clarify.
> 
> As far as I understand, your main argument against neo-logicism is roughly this:
> 1. The rules of 2nd-order logic (SOL) employed by the neo-logicist are very strong, in the sense of entailing some serious mathematical content. In particular:
>    a. They are provably equivalent to the "basic rules" of SOL plus the unrestricted impredicative comprehension scheme.
>    b. Once we accept those rules as the background logic, we get immediately from the very weak Q+ to the very strong PA2.
> 2. Hence, it does not seem reasonable to accept the neo-logicist's version of SOL as logic.
> 
> Now I believe the neo-logicist would happily accept (1): after all, her basic claim is that, essentially, all of ordinary mathematics is derivable from logic (more precisely: from her favorite version of SOL plus Hume's principle (HP); and I'm sure she will happily accept that SOL and not HP does the majority of work here). She will, though, undoubtedly object to your step from (1) to (2). She might argue as follows: the move from (1) to (2) is unwarranted, unless we accept the following principle: 
> (*)    if a set of rules entails substantial mathematical theorems, then it is unreasonable to regard that set of rules as part of logic. 
> But accepting this principle -- so she might argue -- is to beg the question against logicism.
> 
> Question: what did I miss here? Or, more specifically: why is the suggested reply ineffective against your argument?
> 
> Best,
> Ran        
> 
> -----Original Message-----
> From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Panu Raatikainen
> Sent: Sunday, March 22, 2015 09:16
> To: Foundations of Mathematics
> Subject: [FOM] Second-order logic and neo-logicism
> 
> 
> The following new paper might interest some here:
> 
> Panu Raatikainen: "Neo-logicism and its logic", History and Philosophy of Logic (forthcoming)
> 
> http://philpapers.org/rec/RAANAI
> 
> 
> It has greatly benefited from certain old discussions here in FOM on the second-order logic; special thanks to Martin Davis!
> 
> 
> All the Best
> 
> Panu
> 
> 
> 
> Abstract:
> The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions, and that the mathematical power that Hume?s Principle seems to provide, in the derivation of Frege?s Theorem, comes largely from the ?logic? assumed rather than from Hume?s principle. It is shown that Hume?s principle is in reality not stronger than the very weak Robinson Arithmetic Q.  
> Consequently, only few rudimentary facts of arithmetic are logically derivable from Hume?s principle. And that hardly counts as a vindication of logicism.
> --
> Panu Raatikainen
> 
> Ph.D., Adjunct Professor in Theoretical Philosophy
> 
> Theoretical Philosophy
> Department of Philosophy, History, Culture and Art Studies P.O. Box 24  (Unioninkatu 38 A)
> FIN-00014 University of Helsinki
> Finland
> 
> E-mail: panu.raatikainen at helsinki.fi
> 
> http://www.mv.helsinki.fi/home/praatika/
> 
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
> 
> -----
> No virus found in this message.
> Checked by AVG - www.avg.com
> Version: 2015.0.5856 / Virus Database: 4311/9342 - Release Date: 03/20/15
> 
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom

From panu.raatikainen at helsinki.fi  Tue Mar 24 17:17:15 2015
From: panu.raatikainen at helsinki.fi (Panu Raatikainen)
Date: Tue, 24 Mar 2015 23:17:15 +0200
Subject: [FOM] Second-order logic and neo-logicism
In-Reply-To: <021601d06586$09589e10$1c09da30$@gmail.com>
References: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>
	<20150320180437.Horde.IELTzNaMjUPwZYtTGLHVHQ1@webmail.helsinki.fi>
	<20150322091536.Horde._kC78UE9TIT2HypdbiIC9w1@webmail.helsinki.fi>
	<021601d06586$09589e10$1c09da30$@gmail.com>
Message-ID: <20150324231715.Horde.fVefkc5AELp_MtrDM1A4oA2@webmail.helsinki.fi>

Dear Ran,

What you're missing is that impredicative SOL does not only have some  
mathematical content, but in particular substantial *set-theoretical  
power* - which both Wright and Rossberg, for example, grant is  
problematic in this context.  (see the quotes in the paper)

Best, Panu




Lainaus Ran Lanzet <lanzetr at gmail.com>:

> I am probably missing something here, and will be glad if you could clarify.
>
> As far as I understand, your main argument against neo-logicism is  
> roughly this:
> 1. The rules of 2nd-order logic (SOL) employed by the neo-logicist  
> are very strong, in the sense of entailing some serious mathematical  
> content. In particular:
> 	a. They are provably equivalent to the "basic rules" of SOL plus  
> the unrestricted impredicative comprehension scheme.
> 	b. Once we accept those rules as the background logic, we get  
> immediately from the very weak Q+ to the very strong PA2.
> 2. Hence, it does not seem reasonable to accept the neo-logicist's  
> version of SOL as logic.
>
> Now I believe the neo-logicist would happily accept (1): after all,  
> her basic claim is that, essentially, all of ordinary mathematics is  
> derivable from logic (more precisely: from her favorite version of  
> SOL plus Hume's principle (HP); and I'm sure she will happily accept  
> that SOL and not HP does the majority of work here). She will,  
> though, undoubtedly object to your step from (1) to (2). She might  
> argue as follows: the move from (1) to (2) is unwarranted, unless we  
> accept the following principle:
> (*) 	if a set of rules entails substantial mathematical theorems,  
> then it is unreasonable to regard that set of rules as part of logic.
> But accepting this principle -- so she might argue -- is to beg the  
> question against logicism.
>
> Question: what did I miss here? Or, more specifically: why is the  
> suggested reply ineffective against your argument?
>
> Best,
> Ran
>
> -----Original Message-----
> From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On  
> Behalf Of Panu Raatikainen
> Sent: Sunday, March 22, 2015 09:16
> To: Foundations of Mathematics
> Subject: [FOM] Second-order logic and neo-logicism
>
>
> The following new paper might interest some here:
>
> Panu Raatikainen: "Neo-logicism and its logic", History and  
> Philosophy of Logic (forthcoming)
>
> http://philpapers.org/rec/RAANAI
>
>
> It has greatly benefited from certain old discussions here in FOM on  
> the second-order logic; special thanks to Martin Davis!
>
>
> All the Best
>
> Panu
>
>
>
> Abstract:
> The rather unrestrained use of second-order logic in the  
> neo-logicist program is critically examined. It is argued in some  
> detail that it brings with it genuine set-theoretical existence  
> assumptions, and that the mathematical power that Hume?s Principle  
> seems to provide, in the derivation of Frege?s Theorem, comes  
> largely from the ?logic? assumed rather than from Hume?s principle.  
> It is shown that Hume?s principle is in reality not stronger than  
> the very weak Robinson Arithmetic Q.
> Consequently, only few rudimentary facts of arithmetic are logically  
> derivable from Hume?s principle. And that hardly counts as a  
> vindication of logicism.
> --
> Panu Raatikainen
>
> Ph.D., Adjunct Professor in Theoretical Philosophy
>
> Theoretical Philosophy
> Department of Philosophy, History, Culture and Art Studies P.O. Box  
> 24  (Unioninkatu 38 A)
> FIN-00014 University of Helsinki
> Finland
>
> E-mail: panu.raatikainen at helsinki.fi
>
> http://www.mv.helsinki.fi/home/praatika/
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>
> -----
> No virus found in this message.
> Checked by AVG - www.avg.com
> Version: 2015.0.5856 / Virus Database: 4311/9342 - Release Date: 03/20/15
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom


-- 
Panu Raatikainen

Ph.D., Adjunct Professor in Theoretical Philosophy

Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies
P.O. Box 24  (Unioninkatu 38 A)
FIN-00014 University of Helsinki
Finland

E-mail: panu.raatikainen at helsinki.fi

http://www.mv.helsinki.fi/home/praatika/


From lanzetr at gmail.com  Wed Mar 25 01:18:42 2015
From: lanzetr at gmail.com (Ran Lanzet)
Date: Wed, 25 Mar 2015 07:18:42 +0200
Subject: [FOM] Second-order logic and neo-logicism
In-Reply-To: <523DC5F6-3908-4B71-9510-11B8A5EEC8E9@mac.com>
References: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>	<20150320180437.Horde.IELTzNaMjUPwZYtTGLHVHQ1@webmail.helsinki.fi>	<20150322091536.Horde._kC78UE9TIT2HypdbiIC9w1@webmail.helsinki.fi>	<021601d06586$09589e10$1c09da30$@gmail.com>
	<523DC5F6-3908-4B71-9510-11B8A5EEC8E9@mac.com>
Message-ID: <026c01d066bb$29dd4970$7d97dc50$@gmail.com>

I guess I could have written "PA2" instead of "ordinary mathematics".
Best,
Ran

-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of WILLIAM TAIT
Sent: Tuesday, March 24, 2015 18:52
To: Foundations of Mathematics
Cc: William Tait
Subject: Re: [FOM] Second-order logic and neo-logicism

Could you say more, at least briefly, about what counts as 'ordinary mathematics' in

> her basic claim is that, essentially, all of ordinary mathematics is derivable from logic (more precisely: from her favorite version of SOL plus Hume's principle (HP

Thanks,
Bill

> On Mar 23, 2015, at 11:25 AM, Ran Lanzet <lanzetr at gmail.com> wrote:
> 
> I am probably missing something here, and will be glad if you could clarify.
> 
> As far as I understand, your main argument against neo-logicism is roughly this:
> 1. The rules of 2nd-order logic (SOL) employed by the neo-logicist are very strong, in the sense of entailing some serious mathematical content. In particular:
> 	a. They are provably equivalent to the "basic rules" of SOL plus the unrestricted impredicative comprehension scheme.
> 	b. Once we accept those rules as the background logic, we get immediately from the very weak Q+ to the very strong PA2.
> 2. Hence, it does not seem reasonable to accept the neo-logicist's version of SOL as logic.
> 
> Now I believe the neo-logicist would happily accept (1): after all, her basic claim is that, essentially, all of ordinary mathematics is derivable from logic (more precisely: from her favorite version of SOL plus Hume's principle (HP); and I'm sure she will happily accept that SOL and not HP does the majority of work here). She will, though, undoubtedly object to your step from (1) to (2). She might argue as follows: the move from (1) to (2) is unwarranted, unless we accept the following principle: 
> (*) 	if a set of rules entails substantial mathematical theorems, then it is unreasonable to regard that set of rules as part of logic. 
> But accepting this principle -- so she might argue -- is to beg the question against logicism.
> 
> Question: what did I miss here? Or, more specifically: why is the suggested reply ineffective against your argument?
> 
> Best,
> Ran        
> 
> -----Original Message-----
> From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Panu Raatikainen
> Sent: Sunday, March 22, 2015 09:16
> To: Foundations of Mathematics
> Subject: [FOM] Second-order logic and neo-logicism
> 
> 
> The following new paper might interest some here:
> 
> Panu Raatikainen: "Neo-logicism and its logic", History and Philosophy of Logic (forthcoming)
> 
> http://philpapers.org/rec/RAANAI
> 
> 
> It has greatly benefited from certain old discussions here in FOM on the second-order logic; special thanks to Martin Davis!
> 
> 
> All the Best
> 
> Panu
> 
> 
> 
> Abstract:
> The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions, and that the mathematical power that Hume?s Principle seems to provide, in the derivation of Frege?s Theorem, comes largely from the ?logic? assumed rather than from Hume?s principle. It is shown that Hume?s principle is in reality not stronger than the very weak Robinson Arithmetic Q.  
> Consequently, only few rudimentary facts of arithmetic are logically derivable from Hume?s principle. And that hardly counts as a vindication of logicism.
> --
> Panu Raatikainen
> 
> Ph.D., Adjunct Professor in Theoretical Philosophy
> 
> Theoretical Philosophy
> Department of Philosophy, History, Culture and Art Studies P.O. Box 24  (Unioninkatu 38 A)
> FIN-00014 University of Helsinki
> Finland
> 
> E-mail: panu.raatikainen at helsinki.fi
> 
> http://www.mv.helsinki.fi/home/praatika/
> 
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
> 
> -----
> No virus found in this message.
> Checked by AVG - www.avg.com
> Version: 2015.0.5856 / Virus Database: 4311/9342 - Release Date: 03/20/15
> 
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom

_______________________________________________
FOM mailing list
FOM at cs.nyu.edu
http://www.cs.nyu.edu/mailman/listinfo/fom

-----
No virus found in this message.
Checked by AVG - www.avg.com
Version: 2015.0.5856 / Virus Database: 4311/9342 - Release Date: 03/20/15


From lanzetr at gmail.com  Wed Mar 25 02:40:27 2015
From: lanzetr at gmail.com (Ran Lanzet)
Date: Wed, 25 Mar 2015 08:40:27 +0200
Subject: [FOM] Second-order logic and neo-logicism
In-Reply-To: <20150324231715.Horde.fVefkc5AELp_MtrDM1A4oA2@webmail.helsinki.fi>
References: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>	<20150320180437.Horde.IELTzNaMjUPwZYtTGLHVHQ1@webmail.helsinki.fi>	<20150322091536.Horde._kC78UE9TIT2HypdbiIC9w1@webmail.helsinki.fi>	<021601d06586$09589e10$1c09da30$@gmail.com>
	<20150324231715.Horde.fVefkc5AELp_MtrDM1A4oA2@webmail.helsinki.fi>
Message-ID: <027301d066c6$959ff3f0$c0dfdbd0$@gmail.com>

Thank you, Panu!
Rossberg and Wright's point is (if I'm looking at the right quotes, and reading them correctly) that set theory, being the powerful mathematical theory that it is, should not be assumed in the neo-logicist project, and that consequently (since SOL is assumed by neo-logicism) if 
	1. 	SOL is in fact set theory, 
then this would be a serious problem for neo-logicism.
You explain that 
	2. 	substantial set theoretical principles are derivable from SOL. 
But does (2) entail (1)? Does the neo-logicist have to consider (2) problematic for her project if she considers (1) problematic?
The neo-logicist would surely say that PA2 should not be assumed in her project (as this would be question begging). Yet she does not consider the fact that 
	3.	PA2 is "derivable from" (can be interpreted in) SOL+HP
problematic. In particular, she does not consider (3) to entail that SOL+HP is a mathematical (rather than logical) theory. So I ask:
(i) Why should the neo-logicist be bothered by (2) anymore than she is bothered by (3)?
(ii) If (3) is a problem for the neo-logicist to the same extent that (2) is, then can't you simplify your argument by relying on (3) instead of on (2)? If not, why?
Best,
Ran   


-----Original Message-----
From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf Of Panu Raatikainen
Sent: Tuesday, March 24, 2015 23:17
To: Foundations of Mathematics
Subject: Re: [FOM] Second-order logic and neo-logicism

Dear Ran,

What you're missing is that impredicative SOL does not only have some mathematical content, but in particular substantial *set-theoretical
power* - which both Wright and Rossberg, for example, grant is problematic in this context.  (see the quotes in the paper)

Best, Panu




Lainaus Ran Lanzet <lanzetr at gmail.com>:

> I am probably missing something here, and will be glad if you could clarify.
>
> As far as I understand, your main argument against neo-logicism is 
> roughly this:
> 1. The rules of 2nd-order logic (SOL) employed by the neo-logicist are 
> very strong, in the sense of entailing some serious mathematical 
> content. In particular:
> 	a. They are provably equivalent to the "basic rules" of SOL plus the 
> unrestricted impredicative comprehension scheme.
> 	b. Once we accept those rules as the background logic, we get 
> immediately from the very weak Q+ to the very strong PA2.
> 2. Hence, it does not seem reasonable to accept the neo-logicist's 
> version of SOL as logic.
>
> Now I believe the neo-logicist would happily accept (1): after all, 
> her basic claim is that, essentially, all of ordinary mathematics is 
> derivable from logic (more precisely: from her favorite version of SOL 
> plus Hume's principle (HP); and I'm sure she will happily accept that 
> SOL and not HP does the majority of work here). She will, though, 
> undoubtedly object to your step from (1) to (2). She might argue as 
> follows: the move from (1) to (2) is unwarranted, unless we accept the 
> following principle:
> (*) 	if a set of rules entails substantial mathematical theorems,  
> then it is unreasonable to regard that set of rules as part of logic.
> But accepting this principle -- so she might argue -- is to beg the 
> question against logicism.
>
> Question: what did I miss here? Or, more specifically: why is the 
> suggested reply ineffective against your argument?
>
> Best,
> Ran
>
> -----Original Message-----
> From: fom-bounces at cs.nyu.edu [mailto:fom-bounces at cs.nyu.edu] On Behalf 
> Of Panu Raatikainen
> Sent: Sunday, March 22, 2015 09:16
> To: Foundations of Mathematics
> Subject: [FOM] Second-order logic and neo-logicism
>
>
> The following new paper might interest some here:
>
> Panu Raatikainen: "Neo-logicism and its logic", History and Philosophy 
> of Logic (forthcoming)
>
> http://philpapers.org/rec/RAANAI
>
>
> It has greatly benefited from certain old discussions here in FOM on 
> the second-order logic; special thanks to Martin Davis!
>
>
> All the Best
>
> Panu
>
>
>
> Abstract:
> The rather unrestrained use of second-order logic in the neo-logicist 
> program is critically examined. It is argued in some detail that it 
> brings with it genuine set-theoretical existence assumptions, and that 
> the mathematical power that Hume?s Principle seems to provide, in the 
> derivation of Frege?s Theorem, comes largely from the ?logic? assumed 
> rather than from Hume?s principle.
> It is shown that Hume?s principle is in reality not stronger than the 
> very weak Robinson Arithmetic Q.
> Consequently, only few rudimentary facts of arithmetic are logically 
> derivable from Hume?s principle. And that hardly counts as a 
> vindication of logicism.
> --
> Panu Raatikainen
>
> Ph.D., Adjunct Professor in Theoretical Philosophy
>
> Theoretical Philosophy
> Department of Philosophy, History, Culture and Art Studies P.O. Box
> 24  (Unioninkatu 38 A)
> FIN-00014 University of Helsinki
> Finland
>
> E-mail: panu.raatikainen at helsinki.fi
>
> http://www.mv.helsinki.fi/home/praatika/
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom
>
> -----
> No virus found in this message.
> Checked by AVG - www.avg.com
> Version: 2015.0.5856 / Virus Database: 4311/9342 - Release Date: 
> 03/20/15
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom


--
Panu Raatikainen

Ph.D., Adjunct Professor in Theoretical Philosophy

Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies P.O. Box 24  (Unioninkatu 38 A)
FIN-00014 University of Helsinki
Finland

E-mail: panu.raatikainen at helsinki.fi

http://www.mv.helsinki.fi/home/praatika/

_______________________________________________
FOM mailing list
FOM at cs.nyu.edu
http://www.cs.nyu.edu/mailman/listinfo/fom

-----
No virus found in this message.
Checked by AVG - www.avg.com
Version: 2015.0.5856 / Virus Database: 4311/9342 - Release Date: 03/20/15


From vladik at utep.edu  Wed Mar 25 09:38:07 2015
From: vladik at utep.edu (Kreinovich, Vladik)
Date: Wed, 25 Mar 2015 13:38:07 +0000
Subject: [FOM] FW: conference in memory of Grigory "Grisha" Mints
Message-ID: <49A1861CB3A6E84A95F43B695D2B5C381B4573A7@ITDSRVMAIL011.utep.edu>

FYI
**************************************

                           International meeting
                        Third St.Petersburg Days of
                          LOGIC and COMPUTABILITY
                           devoted to memory of
                        Grigory MINTS (1939--2014)
                August 24-26, 2015    St.Petersburg, Russia
                             Call for papers

MAIN TOPICS
     This third meeting in the series of St.Petersburg Days of Logic and
     Computability is devoted to the memory of Grigory Mints. The main
     themes of the meeting are those related to his mathematical interests:
         * Proof theory
         * Intuitionistic logic
         * Modal logic
         * Non-classical logics
         * Automated deduction
         * Constructive mathematics
         * Applications of proof theory to category theory

PROGRAM COMMITTEE
         * Vladimir OREVKOV (St.Petersburg), chairperson
         * Solomon FEFERMAN (Stanford)
         * Boris KONEV (Liverpool)
         * Yuri MATIYASEVICH (St.Petersburg)
         * Anatol SLISSENKO (Paris)
         * Enn Tyugu (Tallinn)
         * Yuri MANIN (Germany)

ORGANIZING COMMITTEE
         * Boris KONEV (Liverpool)
         * Nikolai KOSSOVSKI (St.Petersburg)
         * Vladimir OREVKOV (St.Petersburg)
         * Alexei PASTOR (St.Petersburg)
         * Maxim VSEMIRNOV (St.Petersburg)

Working LANGUAGE: English

SUBMISSION of papers:
     If you wish to present a paper, please send an (extended) abstract
     in LaTeX on the conference e-mail: LogicDays at logic.pdmi.ras.ru
     You can find the template for your abstract on our website:
     http://www.pdmi.ras.ru/EIMI/2015/LC/abstract_guide.tex
     The deadline for submission is May 15, 2015.
     Notifications are due before June 15.

PUBLICATIONS
     Abstracts of talks will be available on WWW and will be delivered to
     the participants in printed form.

REGISTRATION
     Please register via http://www.pdmi.ras.ru/EIMI/2015/LC/app.html

Participation FEE:
     The fee will cover common meals and coffee breaks. The fee equivalent
     to 120 euro can be paid on arrival.  We hope to obtain the support of
     Russian Foundation of Basic Research to cover in part the expenses of
     Russian participants but unfortunately we cannot provide any financial
     support to foreign participants.

Location of the meeting:
     The "3rd DAYS" will take place at the Euler International Mathematical
     Institute (which is now a part of St.Petersburg Department of Steklov
     Institute of Mathematics of the Russian Academy of Sciences). The
     building of the Euler Institute is located at 10, Pesochnaya
     embankment, St.Petersburg.

ACCOMMODATION
     The Euler IMI has a preliminary reservation at the Nauka budget hotel
     and the Andersen hotel. More information can be found on our website:
     http://www.pdmi.ras.ru/EIMI/2015/LC/hotel.html

VISAS
     Please take into account that participants from most countries need a
     visa to enter Russia. After receiving your registration form the
     Organizing Committee will start preparing a formal invitation, which
     you will have to present together with your visa application form at
     a nearby Russian Consulate. It may take a long time before you have
     your visa, so we strongly recommend that you register at least
     three months in advance.

CONTACTS
         * Website of the meeting: http://www.pdmi.ras.ru/EIMI/2015/LC/index.html
         * E-mail: LogicDays at logic.pdmi.ras.ru
         * FAX:
              o 7 (812) 310 53 77 (Program Committee)
              o 7 (812) 234 58 19 (Organizing Committee)

Useful LINKS:
         * The First Days of Logic and Computability:
           http://logic.pdmi.ras.ru/LogicDays
         * The Second Days of Logic and Computability:
           http://logic.pdmi.ras.ru/2ndDays
         * Euler International Mathematical Institute:
           http://www.pdmi.ras.ru/EIMI
         * St.Petersburg Department of Steklov Institute of Mathematics:
           http://www.pdmi.ras.ru

More links can be found on the website of the meeting.

From richard_heck at brown.edu  Wed Mar 25 12:44:15 2015
From: richard_heck at brown.edu (Richard Heck)
Date: Wed, 25 Mar 2015 12:44:15 -0400
Subject: [FOM] Second-order logic and neo-logicism
In-Reply-To: <24B7149A-C780-4F41-8E0C-9FD15DC9DA8C@aol.com>
References: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>	<20150320180437.Horde.IELTzNaMjUPwZYtTGLHVHQ1@webmail.helsinki.fi>	<20150322091536.Horde._kC78UE9TIT2HypdbiIC9w1@webmail.helsinki.fi>	<021601d06586$09589e10$1c09da30$@gmail.com>
	<24B7149A-C780-4F41-8E0C-9FD15DC9DA8C@aol.com>
Message-ID: <5512E5DF.906@brown.edu>

On 03/24/2015 12:54 PM, Joe Shipman wrote:
> That is one of the two points I was going to make. My other point is that there is a more cogent argument against "second-order logicism" than that SOL entails strong mathematical theorems, namely:
>
> "Accepting your full semantics for SOL doesn't get us new mathematics without a stronger deductive calculus. The most you can do in general is to say that a for a strong mathematical statement X, either X or ~X is a logical truth, without specifying which is the case. Please give me some axioms and inference rules to go along with your SOL semantics, which are themselves justifiable as logical principles rather than mathematical ones."
>
> Personally, I think this is an objection which can be well met up to a point--probably a system as strong as Maclane Set Theory can be justified as purely "logical".

In fact, people interested in the so-called neo-Fregean program almost 
always think about the logic axiomatically, not semantically, for much 
this sort of reason. The interest is in certain axiomatizable fragments 
of SOL, and then of course in what justifies thinking of those fragments 
as logical (which is itself an epistemological notion in this context). 
It's dialectically important that that *not* depend upon assumption of 
the standard semantics but have some other basis.

As it happens, there has been a great deal of discussion of this sort of 
issue, as well as of technical issues connected to the question just how 
much deductive power one actually needs here, and then of further 
philosophical questions that emerge once that question has been 
answered. A great deal is also known about exactly what role HP plays in 
the proof of Frege's Theorem and what role it does not play. To point 
out that, if you start with Q, HP doesn't do any work in getting you the 
rest of PA is to point out something that has been known for *a very 
long time*. In particular, it has been known since the beginning (and is 
heavily emphasized in work by Boolos) that HP plays *no role whatsoever* 
in the proof of induction, and the exact relation between how much 
induction you get and how much comprehension you assume has been known 
since work by Linnebo published in 2004.

In fact, something even stronger---also pointed out by Boolos---is true. 
There is a sense in which HP plays no role even in the proof of the 
"crown jewel" of neo-Fregean logicism: the proof of the existence of 
successors. Given Frege's definitions of 0, predecession, and natural 
number, the existence of successors follows from the fact that 
succession is a function, i.e., from:

(*)    Pxy & Pxz --> y = z

where "Pxy" means: x immediately precedes y in the number-series. HP 
does play a role in the proof of (*), for which only predicative 
comprehension is needed. But the standard proof of the existence of 
successors from (*) requires impredicative comprehension, in particular, 
\Pi_1^1 comprehension, though a different (if similar) proof can be 
given in ramified predicative SOL. Thus, the real work HP does is in 
establishing the basic facts about predecession. And, as I said, this 
has been known for 20 years or so.

It's only more recently (starting with the same paper by Linnebo in 
2004) that attention has been paid to the role HP plays in 
characterizing addition and multiplication, but much the same turns out 
to be true in those cases. As emphasized by Burgess, it's important here 
that we think in terms of the cardinal definitions of these operations, 
not the ordinal ones.

I don't know whether, as you say, "a system as strong as Maclane Set 
Theory can be justified as purely 'logical'". What I do know is that 
such a claim is *extremely controversial* and would be *philosophically 
significant* if it could be established. I think the same is true of the 
claim that Q is "logical", and even of the claim that R is "logical".

So I disagree rather strongly with the claim to the contrary that is 
made in the paper. It's a common mistake to think logicism must be the 
view that *all* of (ordinary?) mathematics is "logical". But even Frege 
did not hold that view.

> I agree with Ran that the objections as previously stated seemed to count mathematical power as a strike against a logicist development in an unfairly question-begging way, but perhaps my way of framing this will lead to more fruitful and technically interesting discussion.

I for one find it hard to see what is new in this objection, as compared 
to a similar objection due to Boolos (yet again) that is 25 years old or 
so now.

Richard Heck


-----------------------
Richard G Heck Jr
Professor of Philosophy
Brown University

Website:   http://rgheck.frege.org/
Blog:      http://rgheck.blogspot.com/
Amazon:    http://amazon.com/author/richardgheckjr
Google+:   https://plus.google.com/108873188908195388170
Facebook:  https://www.facebook.com/rgheck

Check out my books "Reading Frege's Grundgesetze"
   http://tinyurl.com/ReadingFregesGrundgesetze
and "Frege's Theorem":
   http://tinyurl.com/FregesTheorem
or my Amazon author page:
   amazon.com/author/richardgheckjr



From Alan.Weir at glasgow.ac.uk  Wed Mar 25 06:58:05 2015
From: Alan.Weir at glasgow.ac.uk (Alan Weir)
Date: Wed, 25 Mar 2015 10:58:05 +0000
Subject: [FOM] Second-order logic and neo-logicism
Message-ID: <9113A29CF1E7784CA537AD964F9AC12708ADE48F@CMS10-01.campus.gla.ac.uk>

I don't think Panu Raatikainen's objections to neo-logicism (FOM Vol 147 Issue 22) beg the question again the neo-logicist as suggested by Ran Lanzet (Issue 23) by simply assuming that the strong second order logic required to get Frege's theorem is part of mathematics. The key issue, it seems to me, is that the logic which the neo-logicist assumes entails strong existence claims (the instances of the comprehension axioms for example). Neo-logicists often agree that in order that  they themselves not beg the question, they should work with a logic whose first-order component is free. But when you make the same adjustment at second order level then (at least depending on how you finesse the 'freeing up' of second order logic) the  logic becomes too weak to generate infinity from Hume's Principle. See Stewart Shapiro and Alan Weir ' "Neo-logicist" Logic is not Epistemically Innocent', Philosophia Mathematica, (2000), pp. 160-189.

Best

Alan

Professor Alan Weir
Philosophy,
Sgoil nan Daonnachdan,
Oilthigh Ghlaschu/University of Glasgow
GLASGOW G12 8QQ

-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150325/dcc9522a/attachment.html>

From bundala at berkeley.edu  Thu Mar 26 03:41:42 2015
From: bundala at berkeley.edu (Daniel Bundala)
Date: Thu, 26 Mar 2015 00:41:42 -0700
Subject: [FOM] VSTTE 15, Second Call for Papers
Message-ID: <CAGLq_MknT1xgsxtUor1042uuH=XDw3qzvS5065aAj=MQAXHFhA@mail.gmail.com>

**********************************************************************
7th Working Conference on Verified Software: Theories, Tools, and
Experiments
July 18 - 19, 2015
San Francisco, California, USA
http://www.eecs.berkeley.edu/vstte15

Co-located with 25th Conference on Computer Aided Verification
(http://i-cav.org/2015)
**********************************************************************

Full Paper Submission Deadline: April 27, 2015


SCOPE:

The Seventh Working Conference on Verified Software: Theories, Tools,
and Experiments follows a successful inaugural working conference at
Zurich in 2005 followed by conferences in Toronto (2008), Edinburgh
(2010), Philadelphia (2012), Atherton (2013), and Vienna (2014). The
goal of this conference is to advance the state of the art in the
science and technology of software verification, through the
interaction of theory development, tool evolution, and experimental
validation.

We welcome submissions describing significant advances in the
production of verified software, i.e., software that has been
proved to meet its functional specifications.  We are
especially interested in submissions describing large-scale
verification efforts that involve collaboration, theory unification,
tool integration, and formalized domain knowledge.  We welcome papers
describing novel experiments and case studies evaluating verification
techniques and technologies.  Topics of interest include education,
requirements modeling, specification languages,
specification/verification case-studies, formal calculi, software
design methods, automatic code generation, refinement methodologies,
compositional analysis, verification tools (e.g., static analysis, dynamic
analysis, model checking, theorem proving, satisfiability), tool
integration, benchmarks, challenge problems, and integrated
verification environments.


PAPER SUBMISSION

Papers will be evaluated by at least three members of the
Program Committee. We are accepting both long (limited to 16 pages)
and short (limited to 10 pages) paper submissions, written in English.
Short submissions also cover Verification Pearls describing an
elegant proof or proof technique.  Submitted research papers and
system descriptions must be original and not submitted for publication
elsewhere.

Research paper submissions must be in LNCS format and must
include a cogent and self-contained description of the ideas, methods,
results, and comparison to existing work.  Submissions of theoretical,
practical, and experimental contributions are equally encouraged,
including those that focus on specific problems or problem domains.

Papers should be submitted through:

https://www.easychair.org/conferences/?conf=vstte2015.

Submissions that arrive late, are not in the proper format, or are too
long will not be considered.  The post-conference proceedings of VSTTE
2015 will be published by Springer-Verlag in the LNCS series.  Authors
of accepted papers will be requested to sign a form transferring
copyright of their contribution to Springer-Verlag.  The use of LaTeX
and the Springer LNCS class files, obtainable
fromhttp://www.springer.de/comp/lncs/authors.html, is strongly
encouraged.

PUBLICATION

Accepted  papers will be  published as post-Proceedings, to
appear in Springer's Lectures Notes in Computer Science.

IMPORTANT DATES:

Abstract submission: April 20, 2015
Full paper submission: April 27, 2015
Notification: June 8, 2015


ORGANIZATION:
General Chair:
Martin Schaef (SRI International)

Program Chairs:
Arie Gurfinkel (Software Engineering Institute, Carnegie Mellon University)
Sanjit A. Seshia (University of California, Berkeley)

Publicity Chair:
Daniel Bundala (UC Berkeley)

PROGRAM COMMITTEE:
Elvira Albert (Complutense University of Madrid)
Nikolaj Bjorner (Microsoft Research)
Evan Chang (University of Colorado, Boulder)
Ernie Cohen (University of Pennsylvania)
Jyotirmoy Deshmukh (Toyota)
Jin Song Dong (National University of Singapore)
Vijay D'Silva (Google)
Vijay Ganesh (University of Waterloo)
Alex Groce (Oregon State)
Arie Gurfinkel (Software Engineering Institute, Carnegie Mellon University)
(co-chair)
Bill Harris (Georgia Institute of Technology)
Chris Hawblitzel (Microsoft Research)
Bart Jacobs (Katholieke Universiteit Leuven, Belgium)
Susmit Jha (United Technologies)
Rajeev Joshi (Laboratory for Reliable Software, Jet Propulsion Laboratory)
Vladimir Klebanov, Karlsruhe Institute of Technology, DE
Akash Lal (Microsoft Research India)
Ruzica Piskac (Yale)
Zvonimir Rakamaric (University of Utah)
Kristin Yvonne Rozier (University of Cincinnati)
Sanjit A. Seshia (UC Berkeley) (co-chair)
Natarajan Shankar (SRI)
Carsten Sinz (KIT)
Nishant Sinha (IBM Research Labs)
Alexander Summers (ETH Zurich)
Zachary Tatlock (University of Washington)
Sergey Tverdyshev (Sysgo AG)
Arnaud Venet (CMU / NASA Ames Research Center)
Karen Yorav (IBM Haifa Research Lab)

**********************************************************************
Please contact vstte2015 at easychair.org  for further information
**********************************************************************
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150326/285aa86d/attachment-0001.html>

From joeshipman at aol.com  Thu Mar 26 06:30:45 2015
From: joeshipman at aol.com (Joseph Shipman)
Date: Thu, 26 Mar 2015 06:30:45 -0400
Subject: [FOM] Second-order logic and neo-logicism
In-Reply-To: <5512E5DF.906@brown.edu>
References: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>
	<20150320180437.Horde.IELTzNaMjUPwZYtTGLHVHQ1@webmail.helsinki.fi>
	<20150322091536.Horde._kC78UE9TIT2HypdbiIC9w1@webmail.helsinki.fi>
	<021601d06586$09589e10$1c09da30$@gmail.com>
	<24B7149A-C780-4F41-8E0C-9FD15DC9DA8C@aol.com>
	<5512E5DF.906@brown.edu>
Message-ID: <34C034CC-203F-4EE1-848C-8E3042C21C35@aol.com>

I'm not claiming my objection is new. I am merely asking the neo-logicists to specify here, in reply to my post, the technical results that I assume they already know about:

1) what is the strongest deductive calculus for SOL that you accept as "logical" and therefore justified as part of the logicist project?

2) how much set theory do you get from this calculus?

A very strong deductive calculus for SOL is "any statement that ZFC proves is a second-order validity with standard semantics is valid". I assume they have something less strong in mind.

As for my statement about Maclane Set Theory, isn't this the same strength as a simple theory of types? Type theory can be developed logically in a less cumbersome way than Russell did.

-- JS

Sent from my iPhone

> On Mar 25, 2015, at 12:44 PM, Richard Heck <richard_heck at brown.edu> wrote:
> 
>> On 03/24/2015 12:54 PM, Joe Shipman wrote:
>> That is one of the two points I was going to make. My other point is that there is a more cogent argument against "second-order logicism" than that SOL entails strong mathematical theorems, namely:
>> 
>> "Accepting your full semantics for SOL doesn't get us new mathematics without a stronger deductive calculus. The most you can do in general is to say that a for a strong mathematical statement X, either X or ~X is a logical truth, without specifying which is the case. Please give me some axioms and inference rules to go along with your SOL semantics, which are themselves justifiable as logical principles rather than mathematical ones."
>> 
>> Personally, I think this is an objection which can be well met up to a point--probably a system as strong as Maclane Set Theory can be justified as purely "logical".
> 
> In fact, people interested in the so-called neo-Fregean program almost always think about the logic axiomatically, not semantically, for much this sort of reason. The interest is in certain axiomatizable fragments of SOL, and then of course in what justifies thinking of those fragments as logical (which is itself an epistemological notion in this context). It's dialectically important that that *not* depend upon assumption of the standard semantics but have some other basis.
> 
> As it happens, there has been a great deal of discussion of this sort of issue, as well as of technical issues connected to the question just how much deductive power one actually needs here, and then of further philosophical questions that emerge once that question has been answered. A great deal is also known about exactly what role HP plays in the proof of Frege's Theorem and what role it does not play. To point out that, if you start with Q, HP doesn't do any work in getting you the rest of PA is to point out something that has been known for *a very long time*. In particular, it has been known since the beginning (and is heavily emphasized in work by Boolos) that HP plays *no role whatsoever* in the proof of induction, and the exact relation between how much induction you get and how much comprehension you assume has been known since work by Linnebo published in 2004.
> 
> In fact, something even stronger---also pointed out by Boolos---is true. There is a sense in which HP plays no role even in the proof of the "crown jewel" of neo-Fregean logicism: the proof of the existence of successors. Given Frege's definitions of 0, predecession, and natural number, the existence of successors follows from the fact that succession is a function, i.e., from:
> 
> (*)    Pxy & Pxz --> y = z
> 
> where "Pxy" means: x immediately precedes y in the number-series. HP does play a role in the proof of (*), for which only predicative comprehension is needed. But the standard proof of the existence of successors from (*) requires impredicative comprehension, in particular, \Pi_1^1 comprehension, though a different (if similar) proof can be given in ramified predicative SOL. Thus, the real work HP does is in establishing the basic facts about predecession. And, as I said, this has been known for 20 years or so.
> 
> It's only more recently (starting with the same paper by Linnebo in 2004) that attention has been paid to the role HP plays in characterizing addition and multiplication, but much the same turns out to be true in those cases. As emphasized by Burgess, it's important here that we think in terms of the cardinal definitions of these operations, not the ordinal ones.
> 
> I don't know whether, as you say, "a system as strong as Maclane Set Theory can be justified as purely 'logical'". What I do know is that such a claim is *extremely controversial* and would be *philosophically significant* if it could be established. I think the same is true of the claim that Q is "logical", and even of the claim that R is "logical".
> 
> So I disagree rather strongly with the claim to the contrary that is made in the paper. It's a common mistake to think logicism must be the view that *all* of (ordinary?) mathematics is "logical". But even Frege did not hold that view.
> 
>> I agree with Ran that the objections as previously stated seemed to count mathematical power as a strike against a logicist development in an unfairly question-begging way, but perhaps my way of framing this will lead to more fruitful and technically interesting discussion.
> 
> I for one find it hard to see what is new in this objection, as compared to a similar objection due to Boolos (yet again) that is 25 years old or so now.
> 
> Richard Heck
> 
> 
> -----------------------
> Richard G Heck Jr
> Professor of Philosophy
> Brown University
> 
> Website:   http://rgheck.frege.org/
> Blog:      http://rgheck.blogspot.com/
> Amazon:    http://amazon.com/author/richardgheckjr
> Google+:   https://plus.google.com/108873188908195388170
> Facebook:  https://www.facebook.com/rgheck
> 
> Check out my books "Reading Frege's Grundgesetze"
>  http://tinyurl.com/ReadingFregesGrundgesetze
> and "Frege's Theorem":
>  http://tinyurl.com/FregesTheorem
> or my Amazon author page:
>  amazon.com/author/richardgheckjr
> 
> 
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> http://www.cs.nyu.edu/mailman/listinfo/fom

From julian.rohrhuber at musikundmedien.net  Thu Mar 26 07:40:26 2015
From: julian.rohrhuber at musikundmedien.net (Julian Rohrhuber)
Date: Thu, 26 Mar 2015 12:40:26 +0100
Subject: [FOM] Second-order logic and neo-logicism
In-Reply-To: <5512E5DF.906@brown.edu>
References: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>
	<20150320180437.Horde.IELTzNaMjUPwZYtTGLHVHQ1@webmail.helsinki.fi>
	<20150322091536.Horde._kC78UE9TIT2HypdbiIC9w1@webmail.helsinki.fi>
	<021601d06586$09589e10$1c09da30$@gmail.com>
	<24B7149A-C780-4F41-8E0C-9FD15DC9DA8C@aol.com> <5512E5DF.906@brown.edu>
Message-ID: <15EA2C67-93E9-40FD-A406-42B64AEBF137@musikundmedien.net>


> On 25.03.2015, at 17:44, Richard Heck <richard_heck at brown.edu> wrote:
> 
> It's a common mistake to think logicism must be the view that *all* of (ordinary?) mathematics is "logical". But even Frege did not hold that view.

Yes, I also often wonder why the difference between arithmetic and geometry that Frege made in this respect is omitted completely [1]. Logicism describes also a research programme, and orientation more than a conviction. What would be your definition?



[1] Many dictionary definitions seem to contribute to this omission, e.g. the Oxford Dictionary of English: "the theory that all mathematics can ultimately be deduced from purely formal logical axioms, introduced by Frege and developed by Bertrand Russell."

From vladik at utep.edu  Thu Mar 26 14:48:35 2015
From: vladik at utep.edu (Kreinovich, Vladik)
Date: Thu, 26 Mar 2015 18:48:35 +0000
Subject: [FOM] special interest group in logic and computations
Message-ID: <49A1861CB3A6E84A95F43B695D2B5C381B459349@ITDSRVMAIL011.utep.edu>

Dear Friends, 

Since last year, ACM, the main Computer Science professional organization, has a Special Interest Group in Logic and Computations SIGLOG, see http://siglog.hosting.acm.org/ it publishes a very interesting newsletter. Info about joining is available on group website. This may be of interest to many of the folks on this mailing list. 

Vladik

From richard_heck at brown.edu  Thu Mar 26 21:43:17 2015
From: richard_heck at brown.edu (Richard Heck)
Date: Thu, 26 Mar 2015 21:43:17 -0400
Subject: [FOM] Second-order logic and neo-logicism
In-Reply-To: <9113A29CF1E7784CA537AD964F9AC12708ADE48F@CMS10-01.campus.gla.ac.uk>
References: <9113A29CF1E7784CA537AD964F9AC12708ADE48F@CMS10-01.campus.gla.ac.uk>
Message-ID: <5514B5B5.50702@brown.edu>

On 03/25/2015 06:58 AM, Alan Weir wrote:
>
> I don?t think Panu Raatikainen?s objections to neo-logicism (FOM Vol 
> 147 Issue 22) beg the question again the neo-logicist as suggested by 
> Ran Lanzet (Issue 23) by simply assuming that the strong second order 
> logic required to get Frege?s theorem is part of mathematics. The key 
> issue, it seems to me, is that the logic which the neo-logicist 
> assumes entails strong existence claims (the instances of the 
> comprehension axioms for example). Neo-logicists often agree that in 
> order that they themselves not beg the question, they should work with 
> a logic whose first-order component is free. But when you make the 
> same adjustment at second order level then (at least depending on how 
> you finesse the ?freeing up? of second order logic) the  logic becomes 
> too weak to generate infinity from Hume?s Principle. See Stewart 
> Shapiro and Alan Weir ? ?Neo-logicist? Logic is not Epistemically 
> Innocent?, Philosophia Mathematica, (2000), pp. 160-189.
>

Hi, Alan,

Most of what I'm about to say is stuff you know, but I'll speak as if to 
you since I'm replying to you....

For what it's worth, I think that the objection you and Stewart 
raise---whether it ultimately can be resisted or not---is importantly 
different from "second-order logic has significant mathematical power" 
(or "set-theoretic strength", if there's meant to be a difference 
between those). It's not just the complaint that the logic has 
(second-order) existential implications (Quine et alia) nor that the 
logic is strong (already in Boolos, with maybe the most sophisticated 
discussion in Koellner [1]). The way you present it here, the objection 
turns in part on the idea that there are dialectical reasons that the 
neo-Fregean needs the *first*-order logic to be free, and that then has 
implications for what the *second*-order logic ought to be like. That is 
a *much* more subtle complaint. And, as the title of your paper makes 
clear, it's ultimately an *epistemological* complaint, which is 
essential, since the issue in which neo-Fregean logicists are interested 
is primarily epistemological (though of course it has ontological aspects).

One way to put my point is to say that your complaint actually has 
nothing to do with the question whether 'neo-logicist logic' is really 
*logic* rather than mathematics. Which is all to the good, since the 
question what "logic" really is isn't exactly uncontroversial enough 
that we can just work with the intuitive notion. Most discussions framed 
in those terms therefore end up leaving me feeling like I've got no idea 
what we're supposed to be talking about.

The excessive focus on whether second-order logic is really "logic" 
simply obscures what *I* take the key issue here to be, namely: Does the 
sort of reasoning required for the proof of Frege's Theorem (whatever 
sort that might be) preserve whatever interesting epistemic property one 
might think HP itself has (whatever that might be)? Of course, if one 
doesn't think HP *has* any interesting epistemic property, then one 
might not find this issue terribly gripping. But I am convinced that it 
is the right way to frame this *kind* of issue, anyway, which arises in 
plenty of other contexts.

And one can frame the issue in a more neutral way: Given some 
potentially interesting epistemic property, what sorts of reasoning 
preserve that property? Boolos's complaints about Frege's use of 
second-order logic in "Reading the Begriffsschrift" (as opposed to those 
in some other places) are largely driven by this kind of question, 
specialized to Frege's notion of analyticity---even though he was deeply 
skeptical about that notion (which is half of the point of his discussion).

 From a FOM point of view, the place where there's space for technical 
exploration is around the question: What sort of reasoning *actually is* 
required for the proof of Frege's Theorem? Yes, you can use second-order 
logic, but there are other sorts of systems one could use instead, such 
as the one explored in my paper "A Logic for Frege's Theorem" [2]. One 
might also lower one's ambitions from PA to weaker systems, and then 
various predicative logics will do, as discussed in my paper 
"Predicative Frege Arithmetic and 'Everyday Mathematics'" [3]. Other 
sorts of systems have been explored by Aldo Antonelli [4], Francesca 
Boccuni [5], and Fernando Ferreira [6,7] (and of course there are 
defenders of non-Fregean forms of logicism, too, e.g., Tennant and 
Zalta). So there's lot of room for technical gymnastics here.

Richard Heck


[1] http://philpapers.org/rec/KOESLO
[2] http://philpapers.org/rec/HECTLO
[3] http://philpapers.org/rec/HECPFA
[4] http://philpapers.org/rec/ANTNAV
[5] https://francescaboccuni.wordpress.com/publications/
[6] http://philpapers.org/rec/FERAFG
[7] "Impredicativity and Fregean arithmetic", UConn Abstractionism 
Workshop, 2014

-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150326/9fe967d1/attachment.html>

From beziau100 at gmail.com  Fri Mar 27 05:46:49 2015
From: beziau100 at gmail.com (jean-yves beziau)
Date: Fri, 27 Mar 2015 10:46:49 +0100
Subject: [FOM] a mytho-logical journey from Samos to Istanbul
Message-ID: <CAMrL2xsjEXr5Y3rgfkKVGksJgZUcXGz1ApW1fiz05u5y1VzUXQ@mail.gmail.com>

Take part to two important events of logic and enrich your culturalbackground
visiting pivotal sites in the history of humanity:
- Miletus, the city of Anaximander and Thales, a starting point for
philosophy and mathematics
- Ephesus with Celsius library and the nearby house of the virgin Mary
- Pergamon with  Asclepion Sanctuary and Trajan Temple
- Troy: considered during many centuries a fictitious city
We are organizing a journey
>From the 10th Panhellenic Logic Symposium
Samos Island, Greece, June 11-15, 2015
to the
5th World Congress and School on Universal Logic
Istanbul, Turkey, June 20-30, 2015http://www.uni-log.org/si
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150327/dac213a6/attachment.html>

From panu.raatikainen at helsinki.fi  Fri Mar 27 12:50:45 2015
From: panu.raatikainen at helsinki.fi (Panu Raatikainen)
Date: Fri, 27 Mar 2015 18:50:45 +0200
Subject: [FOM] Second-order logic and neo-logicism
In-Reply-To: <34C034CC-203F-4EE1-848C-8E3042C21C35@aol.com>
References: <CA+cpueJS+_QMjNXWX=o0_p--vO7e4P0+uVnMVi+OnJ2Fx=4Krg@mail.gmail.com>
	<20150320180437.Horde.IELTzNaMjUPwZYtTGLHVHQ1@webmail.helsinki.fi>
	<20150322091536.Horde._kC78UE9TIT2HypdbiIC9w1@webmail.helsinki.fi>
	<021601d06586$09589e10$1c09da30$@gmail.com>
	<24B7149A-C780-4F41-8E0C-9FD15DC9DA8C@aol.com> <5512E5DF.906@brown.edu>
	<34C034CC-203F-4EE1-848C-8E3042C21C35@aol.com>
Message-ID: <20150327185045.Horde.opXxWJ5dlpc_NANhkwfAVw1@webmail.helsinki.fi>

Just a brief comment:

I always thought these issues must be more or less clear for competent  
people like Burgess, Heck and Linnebo (though I don't think they  
express the critical points very clearly in their publications (at  
least the ones I know)).

However, I've had quite a lot of transaction with, e.g., British  
philosophers and, believe me, these issues are not at all clear to  
many of them - on the contrary, what I say seems to be almost a  
scandal for many. My paper is directed to them, and is a response to,  
e.g., Wright 2007.

I was simply trying to spell out as clearly as possible what I think  
are the relevant logical facts. I don't want to pretend I have  
anything really new to say for the real experts.

All the Best

Panu




-- 
Panu Raatikainen

Ph.D., Adjunct Professor in Theoretical Philosophy

Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies
P.O. Box 24  (Unioninkatu 38 A)
FIN-00014 University of Helsinki
Finland

E-mail: panu.raatikainen at helsinki.fi

http://www.mv.helsinki.fi/home/praatika/


From Alan.Weir at glasgow.ac.uk  Sat Mar 28 18:40:56 2015
From: Alan.Weir at glasgow.ac.uk (Alan Weir)
Date: Sat, 28 Mar 2015 22:40:56 +0000
Subject: [FOM] Re Second-Order  Logic and Logicism.
Message-ID: <9113A29CF1E7784CA537AD964F9AC12708AE02FF@CMS10-01.campus.gla.ac.uk>

"The excessive focus on whether second-order logic is really "logic" simply obscures what *I* take the key issue here to be, namely: Does the sort of reasoning required for the proof of Frege's Theorem (whatever 
sort that might be) preserve whatever interesting epistemic property one might think HP itself has (whatever that might be)?" Richard Heck, FOM 147.28.

I entirely agree Richard. That is exactly the sort of point Stewart and I were making. So perhaps I ought to have said that objections to the neo-logicist use of second-order logic needn't beg the question against them, even if some objections do. My own position is, indeed,  more radical than Panu's in his paper: I think that Hume's Principle can't even give us Q or the theorem of infinity without invoking  principles, whether you want to call them logical or not is irrelevant, which pose the same epistemological worries as standard mathematical theories construed platonistically. That is not just to repeat Quine's scepticism about second-order logic- as you put it: 

"the complaint that the logic has (second-order) existential implications (Quine et alia)"

Quine, remember, was very far from an ardent proponent of free logic at first-order level, his worries about second-order logic were more about the sort of existential commitments he thought it had. I have no hang-ups about the free segment of second order logic and believe in the existence of mind-independent properties, even more so than Duns Scotus. But this combination does not, I maintain, entail the epistemic innocence of the principles needed to get the infinity of the natural numbers from HP. 

I say that as a Scottish philosopher who is 100% not British, who rejects Britishness and all its works and pomps, even though, alas, I remain after last year's referendum, a UK citizen. Thus I am not one who naturally jumps in to defend 'Britain', but I do find Panu's remarks (same issue of FOM)

'However, I've had quite a lot of transaction with, e.g., British  philosophers and, believe me, these issues are not at all clear to many of them - on the contrary, what I say seems to be almost a scandal for many.'

and the contrast with 'competent people' like Burgess, Heck and Linnebo, puzzling. UK universities such as Birkbeck and St. Andrews and others have had many logicians interested in neo-logicism, (of whatever nationality) working at them, including Linnebo, also others such as Shapiro, Cook, logicians  perfectly cognizant with the complexities of second-order logic(s). And the final section criticises Hale, Wright, Boolos and Burgess for passages of which Panu says ''Whatever their actual intent, it is very easy to read such brief statements as suggesting that the full second-order arithmetic PA2 can be derived from HP alone, without any other substantial assumptions." Well, all these philosophers, not just the anti-logicists Boolos and Burgess, are well aware, even if those passages don't make it clear, that there are epistemological worries about the logic used to derive substantive mathematical results from HP.

So  whilst I am on the side of those who don't think neo-logicism yields any epistemological gains over a straightforward platonism, even leaving worries about abstraction principles aside,  I don't think only ignorance of the logical technicalities can explain taking the other view. If, for example, one had an anti-realist view of properties, but not objects, one might try to defend from the 'no better off than platonism' criticisms, the use, in proving e.g. theorems of infinity,  of predicative comprehension.

Alan Weir

Philosophy,
Sgoil nan Daonnachdan,
Oilthigh Ghlaschu/University of Glasgow
GLASGOW G12 8QQ

From bennett.mcnulty at gmail.com  Mon Mar 30 06:36:33 2015
From: bennett.mcnulty at gmail.com (Michael Bennett McNulty)
Date: Mon, 30 Mar 2015 12:36:33 +0200
Subject: [FOM] Philosophia Mathematica Virtual Special Issue on Mathematical
	Depth
Message-ID: <6C871F27-5FD0-402B-9816-80BD47E2A0F5@gmail.com>

Philosophia Mathematica has just released a virtual special issue on Mathematical Depth, based on the workshop held at UC Irvine in April of 2014.  Until November 2015, it will be available free of charge at this link:

http://www.oxfordjournals.org/our_journals/philmat/mathematical_depth_papers.html

The hard copy of the issue will be published in June.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150330/f3970bdb/attachment.html>

From beziau100 at gmail.com  Mon Mar 30 10:23:23 2015
From: beziau100 at gmail.com (jean-yves beziau)
Date: Mon, 30 Mar 2015 16:23:23 +0200
Subject: [FOM] First World Congress on Logic and Religion in Joao Pessoa,
	Brazil
Message-ID: <CAMrL2xtxFF3e00QmQAeq5mTOehEeL=oDLir_tk-ceQXb+cN8yw@mail.gmail.com>

In a fews days will start the
First World Congress on Logic and Religion
in Joao Pessoa, Brazil
http://www.uni-log.org/logos

This will be an excpetional event
with the participation of people like

Daniel von Wachter
of the International Academy of Philosophy, Liechtenstein
who will give the talk
"Is there room for miracles in the causal order of the world?"

Christoph Benzm?ller
Department of Mathematics and Computer Science, Free University of Berlin,
Germany
who will give the talk
"G?del's proof of existence of God revisited - Findings from a
Computer-supported Analysis"

Snezana Lawrence
Bath Spa University, UK
who will give the talk
"Mathematicians and their Gods"

Jean-Michel Kantor
University Denis Diderot- Paris7, France
who will give the talk
"God and Infinity"

Razvan Diaconescu
Simion Stoilow Institute of Mathematics of the Romanian Academy (IMAR)
who will give the talk
"The logical nature of the Nalanda tradition of Buddhism"

Benedikt Paul G?cke
Bochum University, Germany and Oxford University, UK
who will give the talk
"Did God know it? God's relation to a world of chance and randomness"

Purushottama Bilimoria
Denkin University and Melbourne University, Australia,
Editor-in-Chief Sophia
who will give the talk
"Thinking negation and nothingness in early Hinduism and classical Indian
philosophy"

Musa Akrami
Islamic Azad University, Science and
Research Branch of Tehran, Iran
who will give the talk
"Logic in Islamic and Medieval Islamic World"
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20150330/761b4a97/attachment.html>

From panu.raatikainen at helsinki.fi  Mon Mar 30 11:56:25 2015
From: panu.raatikainen at helsinki.fi (Panu Raatikainen)
Date: Mon, 30 Mar 2015 18:56:25 +0300
Subject: [FOM] Re Second-Order Logic and Logicism.
In-Reply-To: <9113A29CF1E7784CA537AD964F9AC12708AE02FF@CMS10-01.campus.gla.ac.uk>
References: <9113A29CF1E7784CA537AD964F9AC12708AE02FF@CMS10-01.campus.gla.ac.uk>
Message-ID: <20150330185625.Horde.3WQXMvZoJDIjlCbCBgrVsg1@webmail.helsinki.fi>

Alan Weir <Alan.Weir at glasgow.ac.uk>:

> but I do find Panu's  remarks ... [about British philosophers] ...
> and the contrast with 'competent people' like Burgess, Heck and  
> Linnebo, puzzling.


I am very sorry if my message suggested such a contrast. I certainly  
intended no such thing. These were two separate thoughts.

I was simply recording my subjective experience; that there exist such  
philosophers: with somewhat one-sided strong enthusiastic opinions  
about SOL and perhaps neo-logicism, but not aware about many the  
issues I discuss in my paper (which is hardly surprising; I guess some  
of them are not taught even in the advanced mathematical logic  
courses, but you'll have to learn them from various sources). As I  
haven't really spent much time elsewhere than in UK, that happens to  
be my sample. But certainly there are lots of "competent people" (in  
whatever sense) in UK.

Then again, there are some people such as Burgess, Heck and Linnebo  
(and few others) who have worked with these issues intensively for a  
long time; they hardly represent the average philosopher, and what may  
well be clear to them is hardly clear for everyone, and perhaps still  
worth spelling out explicitly.

BTW, I am inclined to agree that whether or not SOL is a "logic" is  
not the key issue (largely verbal). But whether or not SOL (even if  
given only in the form of the Gentzen-style introduction and  
elimination rules; cf. Wright 2007) brings with it, over HP or Q (or  
such), some additional set-theoretical power, is a different and  
substantial question. In my paper, I am trying to argue, and make the  
case as clear as I can, that it really does.


All the Best

Panu


-- 
Panu Raatikainen

Ph.D., Adjunct Professor in Theoretical Philosophy

Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies
P.O. Box 24  (Unioninkatu 38 A)
FIN-00014 University of Helsinki
Finland

E-mail: panu.raatikainen at helsinki.fi

http://www.mv.helsinki.fi/home/praatika/


From Vasco.Brattka at cca-net.de  Tue Mar 31 08:47:26 2015
From: Vasco.Brattka at cca-net.de (Vasco Brattka)
Date: Tue, 31 Mar 2015 14:47:26 +0200
Subject: [FOM] COMPUTABILITY - The Journal of the Association CiE - New
	Issue!
Message-ID: <551A975E.1090006@cca-net.de>

__________________________________________________________

Volume 4, Number 1, 2015 of

    COMPUTABILITY
    The Journal of the Association CiE

is now available via the journal web page

    http://www.computability.de/journal/
__________________________________________________________

The current issue contains the following articles

- Hyperprojective hierarchy of qcb0-spaces	p. 1
   Matthias Schr?der, Victor Selivanov	

- Mortality of iterated piecewise affine functions over
   the integers: Decidability and complexity	p. 19
   Amir M. Ben-Amram	

- On maximum conservative extensions	p. 57
   Henry Towsner		

- The complexity of satisfaction problems in reverse
   mathematics	p. 69
   Ludovic Patey	
__________________________________________________________

The journal Computability is a peer reviewed international
journal that is devoted to publishing original research of
highest quality, which is centered around the topic of
computability.

The subject is understood from a multidisciplinary perspective,
recapturing the spirit of Alan Turing (1912-1954) by linking
theoretical and real-world concerns from computer science,
mathematics, biology, physics, computational neuroscience,
history and the philosophy of computing.
__________________________________________________________

The journal is indexed in the following data bases:

- DBLP Bibliography Server
- Google Scholar
- Mathematical Reviews (MathSciNet)
- Zentralblatt MATH (zbMATH)
- Computability TUG bibliography archive

Further information on submissions can be found on the journal
web page mentioned above.