This question is closely related to this one: https://mathoverflow.net/questions/27755/knuths-intuition-that-goldbach-might-be-unprovable. It stems from my ignorance about non-standard models of arithmetic. In a comment on the other question, Chandan Singh Dalawat reproduced the following interesting quotation:

> There are very many old problems in
> arithmetic whose interest is
> practically nil, i.e. the existence of
> odd perfect numbers, the iteration of
> numerical functions, the existence of
> infinitely many Fermat primes $2^{2^n}+1$,
> etc. Some of these questions may well
> be undecidable in arithmetic; the
> construction of arithmetical models in
> which questions of this type have
> different answers would be of great
> importance." (Bombieri, 1976)

What I'd like to know is, what might such a model look like, even roughly? For example, suppose one wanted to construct a model in which there were only finitely many Fermat primes. Would one do something like adjoin a nonstandard integer N, add the statement that all Fermat primes were less than N, and somehow demonstrate that that did not lead to a contradiction? (For all I know, that is an obviously flawed or even ridiculous suggestion.) A slightly more general question is this: is it conceivable that a number-theoretic independence proof might be achieved without one going too deeply into the number theory? (I ask that in the expectation, but not strong expectation, that the answer is no: that you can't get something for nothing.)