[FOM] Query on nonstandard models of the integers
joeshipman@aol.com
joeshipman at aol.com
Sun Nov 4 20:49:01 EST 2007
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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