Ruling-out Nonstandard Models of 1st-Order PA

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 WITH DOMAIN THE NATURAL NUMBERS IN WHICH THE ADDITION FUNCTION IS 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-ee4meeAH724@xxxxxxxxxxxxxxxx

Ph: 845-358-4515, Cell: 860-227-7940

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