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This question is closely related to this one: Knuth's intuition that Goldbach might be unprovableKnuth'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.)

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.)

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.)

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gowers
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How would one even begin to try to prove that a simple number-theoretic statement is undecidable?

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.)