[FOM] Ruling-out Nonstandard Models of 1st-Order PA

rtragesser@mac.com rtragesser at mac.com
Mon Oct 23 16:00:39 EDT 2006

The Tennenbaum-Kreisel Theorem
and ruling out Non-Standard Models of First order Arithmetic  
informally and formally:

A natural question is: Formal First-Order Peano Arithmetic [abbrv. PA  
henceforth]  has non-standard models; what informal ideas are  
sufficiently clear and distinct, such that (formal) PA supplemented  
by these informal ideas with "informal rigor" rules out the  
nonstandards models ?

	There are two families of such ideas at hand:

(1) ideas of the natural numbers, and
(2) ideas of Archimedean-ness,---

[I]Ideas of the natural numbers.
  For most of us, we have some sufficiently clear and distinct idea  
such as such
	[1a.i] an idea of the well-ordering type omega, or,
	[1a.ii] less abstractly,--such an idea of "the natural number  
series" or,
	[1.a.iii] still less abstractly,--such an idea of the Hindu-Arabic  
decimal system of numerals 1, 2, ...., 9, 10, 11, ... (ideally  
continuing ad infinitum) [because of the way this numeral system is  
cyclically built and keeps tabs on itself,  the idea of it is more  
clear an distinct than, say, the unary system of numerals, 1,  11,  
111, ....]

[II]Ideas of Archimedean-ness.
	Nonstandard models are non-Archimedean, so some clear and distinct  
ideas along the lines,
             j<k,  then some j+j+...+j, that is, j added to itself  
some "finite number of times",  j+...+j > k ,  where, e.g., "finite  
number of times" is given an informally rigorous sense through  
[1.a.iii], viz., enumerated by those numerals up to some numeral.

     Are there other such informal ideas?
     There is a possible third tantalizingly suggested by,-

  The Tennenbaum-Kreisel Theorem: THERE IS NO NONSTANDARD MODEL OF PA  
RECURSIVE.  (From Boolos-Burgess-Jeffrey, 4th ed., p.306, without  
reference to the literature or folklore.)

	The tantalizing suggestion is that, formal PA supplemented by the  
informal but clear and distinct idea,

[III] the arithmetical "+" is reckonable, or calculable.

  This is tantalizing, but does the Tennenbaum-Kreisel Theorem really  
support it???
I can't answer this question -- it's proved difficult to tease out  
the issues. (So this is a query to FOM.)

Remark.  The arithmetical rules a+b=b+a, ab=ba, a+(b+c)=(a+b)+c, ...,  
a(b+c)=ab+ac, (for c<b) a(b-c)=ab-ac, (a+b)'=a+b'  have a two-fold  
sense.  As Felix Klein remarked, they are rules of reckoning.  But  
they are also theoretical rules (e.g.,  a(b+c)=ab+ac, (for c<b) a(b-c) 
=ab-ac entail that common factors, and in particular gcf, are  
preserved under addition and subtraction.)  Isn't a formal-logical  
codification of arithmetic that doesn't force this double-sense (and  
so in particularity, the reckonability of '+') on any interpretation  
or model of it be in some fundamental sense deficient?

robert tragesser

Robert Tragesser
email: rtragesser at mac.com
Ph: 845-358-4515, Cell: 860-227-7940
26 DePew Avenue #1
Nyack, NY 10960-3839

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