[FOM] Choice of new axioms 1 (reply to Friedman)
joeshipman@aol.com
joeshipman at aol.com
Mon Feb 13 11:27:50 EST 2006
Harvey makes some good points, and in some other cases misses my
points (which is my fault for not stating them clearly enough.)
Friedman:
2. RVM is neither an intuitively justifiable axiom nor is it a
convention.
Shipman:
It may not be intuitive to you, but it certainly represents an
intuition shared by many mathematicians prior to Banach-Tarski.
Otherwise, why would anyone have been SURPRISED by Banach-Tarski?
Friedman:
It also has the following major drawback: it asserts the existence of
a
function from sets of reals to reals without indicating an example.
Shipman:
That is a drawback relative to ZF, but not relative to ZFC. Is V=L
"fixing the problem with the Axiom of Choice" a good enough reason to
adopt it?
*****
Shipman, earlier:
>
> 2) RVM settles even more questions than V=L does, in particular it
> implies Con(ZFC) and lots of other new arithmetical statements while
> V=L proves no new arithmetical statements
Friedman:
This is incorrect as indicated above in 3. (3. RVM has only very few
interesting consequences for the projective
hierarchy, and almost no interesting consequences for arbitrary sets
of
reals.)
Shipman:
It is going too far to say that statement of mine is "incorrect". If
you are going to measure "more" in the crudest possible way, both RVM
and V=L settle aleph_zero new questions in the language of set theory,
so only in that sense is my statement 2) incorrect. But RVM certainly
settles infinitely many new ARITHMETICAL questions, questions of a type
V=L has nothing to say about, including practically all your recent
independent statements. As far as statements about projective sets,
etc., are concerned, I won't argue with you about whether the
aleph_zero new questions about projective sets that RVM settles are
"less interesting" than the aleph_zero new questions about projective
sets that V=L settles.
Friedman:
>
> 4) If V=L settles a lot of set-theoretic questions "the wrong way",
> then either RVM settles them "the right way" or else there is more
than
> one "wrong way" and Friedman shouldn't have used the word "the".
Whatever few statements in the projective hierarchy settled by RVM,
they are
the "right way" according to set theorists. If my memory serves me
right,
"most" of the copious statements in the projective hierarchy settled
by V =
L (and not ZFC), are settled the "wrong way" according to set
theorists, and
some of them are settled the "right way" according to set theorists.
But
this correctness for some statements in the projective hierarchy is,
according to set theorists, in the category of "a stopped clock is
correct
twice a day".
Shipman:
Can you be more specific about why the way V=L settles projective
statements is considered "wrong" by set theorists, and would you
venture an opinion on whether this attitude of set theorists should be
considered authoritative?
********
Friedman:
>
> This relates to the ongoing discussion of the real numbers as a
> foundation for physics. It is my belief that it is something of a
> historical accident that the axioms of set theory arose in their
> current form.
This I do not believe. It is "obvious" that ZFC is some sort of
"canonical
extrapolation of finite set theory to infinite set theory", in a sense
that
remains to be fully elucidated. I am sure that this has been at least
a
decisive factor in the ultimate source of its acceptance as the
current gold
standard. So I don't see any chance of a different historical
development.
Shipman:
I have said nothing against ZFC. I am sure that for exactly the
reasons you state, ZFC would have been developed and found of immense
value in any alternate history; but the existence of a real-valued
measure on [0,1] would have been considered an important "open
question" as CH was, and the eventual proof that RVM implied ZFC was
consistent would be taken as an argument IN FAVOR of RVM.
*********
Friedman:
I don't see any physical intuition that tells me how to measure an
arbitrary
set of real numbers. This is on several levels. I don't see the
physical
significance of an arbitrary set of reals, and I don't see the
physical
meaning of a mapping of all sets of reals into the reals, and I don't
see
the physical meaning of arbitrary infinite sequences of arbitrary sets
of
reals.
Shipman:
That is your own personal physical intuition, which is well-informed
by contemporary physics, and which is rightly shared by many of TODAY'S
physicists. But you are missing my point. I am not claiming that RVM
has any justification in physical intuition NOW. I am claiming that it
USED to have an intuitive justification back around 1900, when both
mathematical and physical intuition were more naive. That is why the
Banach-Tarski result was SURPRISING. My point is that if physics and
mathematics had developed certain results in different sequences than
they were actually developed, then at the time RVM was formulated
precisely it would STILL have had the backing of intuition and
therefore been taken much more seriously as a mathematical axiom, and
if it had been shown within a few years to establish Con(ZFC) than that
would have been regarded as even more reason to take it seriously.
*********
Shipman, earlier:
>and it would eventually have been found to be a
> proper extension to ZFC (in this alternate universe Solovay would
have
> shared the Fields Medal with Cohen, for the epochal verification
that
> RVM was indeed more powerful).
Friedman:
A serious misreading of mathematical politics, if only for the fact
that
Solovay's work appears in the proceedings of a 1967 conference, with a
footnote "the main results of this paper were proved in the spring of
1966",
and Cohen was awarded the Fields medal in summer of 1966. Also, this
result
of Solovay occupies only a small fraction of that paper, in which much
more
original deep results appear. Exact quote from Solovay at the
beginning of
the relevant 3 page section: "Our method is the method of inner models
of
Godel [2]. We use in an essential way some recent work of Jack
Silver".
These comments have nothing to do with any kind of evaluation of the
total
of Solovay's impressive (early, middle, late) work in set theory, and
qualifications for Fields Medals.
Shipman:
I have the highest regard for Solovay's work, but I wrote imprecisely
and did not mean to imply that Solovay had shown RVM was independent.
In the alternate history I am proposing, his work would have
established all the arithmetical consequences of MC, which would
previously have been open questions, as fully proven theorems.
**********
Shipman, earlier:
> The eventual discovery that matter was not infinitely divisible
would
> not have threatened the use of RVM, because once it had become
accepted
> and found widely useful, it would have attained a valued
MATHEMATICAL
> status.
Friedman:
RVM is not widely useful in mathematics or physics. What wide
applications
do you have in mind?
Shipman:
In addition to Con(ZFC), how about the "Strong Fubini Theorems" I deal
with in my thesis (see the October 1990 Transactions of the AMS)? I
show that RVM imples that Fubini's theorem applies to all
(non-negative) mutlivariate real functions, not just the measurable
ones, in the sense that iterated integrals may not always exist but
WHENEVER they exist they are equal. Furthermore, this has applications
to physics because it allows one to prove a "no hidden variables"
theorem that rules out the hidden-variables theories that had been
proposed by the physicist Itamar Pitowsky and the mathematician Stanley
Gudder in a series of papers.
Of course, I am not sure RVM would have become "widely useful", but
I'll bet it would have become useful enough to withstand challenges
from people who say "that really shouldn't be an axiom, it shouldn't be
regarded as true".
********
Friedman:
I agree that the foundations of physics is in an extremely problematic
state, but this does not have any apparent connection with
nonmeasureable
sets and set theory.
In fact, the only substantial subject I know of for which the
foundations is
not in an extremely problematic state is mathematics.
Shipman:
I agree that nonmeasurable sets are probably not the problem, but set
theory is part of the problem. Specifically, it seems to be remarkably
difficult for physicists to formulate the current crop of fundamental
theories in a way that is compatible with the current foundation of
mathematics in Set Theory, but I am not yet prepared to conclude that
this is entirely the fault of the physicists (though I'm open to
arguments that it is).
*********
Friedman:
> although MATTER is not infinitely divisible, SPACE still is,
Why do you think that space is? I consider it a coin toss whether
physicists
at the end of the century will be adopting physical theories with
space
infinitely divisible, or space not infinitely divisible. I would not
predict
one way or the other.
Shipman:
Because the fundamental theories are still officially formulated in
terms of real or complex manifolds.
Friedman:
>and the
> full apparatus of analysis is necessary to mathematize our
fundamental
> physical theories,
I don't know what you mean by "full apparatus".
Shipman:
I mean that no reformulation of fundamental physics has been plausibly
proposed that avoids using the real numbers and only talks about finite
objects (or even countable objects; all the fundamental theories
involve ontologies going way beyond second-order arithmetic).
Friedman:
Also, there seems to be doubt that there are any truly fundamental
physical
theories left that have been mathematized. I have heard the view that
any
truly fundamental theory must incorporate quantum mechanics (quantum
field
theory), which has not been properly mathematized.
Shipman:
General Relativity has certainly been adequately mathematized. Quantum
Field Theory has been pseudo-mathematized in the sense that certain
simpler subtheories (for example Quantum Electrodynamics) can be
formulated in a logically impeccable way EXCEPT that there is no proof
that the procedure for generating experimental predictions, as an
algorithm, is convergent.
Friedman:
>even though the "physical meaning" of the real
> numbers is completely opaque.
>
I don't know quite what this means. At least at some naïve intuitive
level,
the physical meaning of the real numbers is apparent. Arbitrary sets
of real
numbers is a different matter.
Shipman:
Earlier you doubted that space is infinitely divisible, but this is
exactly the issue here. If space is NOT infinitely divisible, and there
is not an infinite amount of information contained in a finite region
of spacetime, it's not clear how any "real number", in its entirety,
can faithfully represent something physical.
-- JS
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