TITLE: How would one even begin to try to prove that a simple number-theoretic statement is undecidable?
QUESTION [42 upvotes]: This question is closely related to this one: Knuth's 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.)

REPLY [2 votes]: I would like to extend the above question: It seems that most previous comments are dealing with independence of number theoretic statements from PA. I would like to know your input regarding the possible independence of number theoretic statements from ZFC. Unlike the Paris-Harrington theorem, in which we got the unprovability of a true statement, I would like to know whether some work has been done on proving the (symmetric/CH-like) independence of number theoretic statements from ZFC.
It seems that forcing is not very useful in this situation. Is it likely that such independence results may be somehow connected to possible (non-trivial) independence results from ZFC+V=L?