[FOM] Ruling-out Nonstandard Models of 1st-Order PA
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
[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,
[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
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?
email: rtragesser at mac.com
Ph: 845-358-4515, Cell: 860-227-7940
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Nyack, NY 10960-3839
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