[FOM] Query on nonstandard models of the integers

[joeshipman@aol.com]

To clarify one further confusion that has been expressed: of course it 
is possible to express a version of the Fundamental Theorem of 
Arithmetic as a statement in Arithmetic. The problem is that the term 
"finite" has a nonstandard meaning in a nonstandard model, so that the 
"prime factorization" may have infinitely many (from the 
outside-the-model point of view) primes in it. This means that AS A 
COMMUTATIVE RING the nonstandard model does not possess the property of 
rings normally called "unique factorization" (which covers existence 
and uniqueness of finite factorizations into primes and does not 
contemplate infinite factorizations because in the theory of rings 
there is no infinitary multiplication, even though there is a "set 
product" defined for FINITE sets of elements).

In the nonstandard model one can indeed define an "infinitary 
multiplication", because it is a true theorem of arithmetic that for 
any function from {1,2,...n} to N there is a suitable product of the 
values of the function, and when n is infinite (as seen from outside 
the model) this is an infinitary operation. But that only makes true 
the pretend "Unique Factorization Theorem" that has been expressed as a 
sentence in the language of arithmetic, it does not mean that the model 
itself, viewed from outside, has the genuine Unique Factorization 
property as ring theorists normally understand it.

As I showed in my previous post, any nonstandard model of the theory of 
integers (or even of the subset of that theory that follows from 
Peano's axioms) has elements with infinitely many prime divisors and no 
expression as a finite (as seen from outside the model) product of 
primes.

-- JS
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