I could live with that (since I am more concerned with Godel’s 2nd theorem than with the 1st), were it not for what Boolos says:

if math is not a lot of bunk, then no claim of the form “claim X can’t be proved” can be proved.

Steve again: The Godel sentence [...] is also provably true, using the axioms of arithmetic plus the additional assumption that arithmetic is consistent.

This I can definitely accept; but since the assumption (that arithmetic is consistent) is a working hypothesis, that leads back to my taking the truth and unprovability of the “true but unprovable” statement as a working hypothesis.

* within axiomatic arithmetic

But why is it unprovable that 24537 is indeed a “bad” number if in fact it is a “bad” number? Does it follow from the fact that there are infinitely large quantities of prime numbers, hence one can’t rule out that the difference between two of them will generate the number 24537, thus making it unprovable?

No, it does not follow from this fact. If you were trying to prove that 24537 is bad, your first idea might be “let’s try every possible pair of primes”. That won’t work, because there are infinitely many prime pairs. This means that your FIRST IDEA for a proof won’t work, but that doesn’t rule out the possibility of some OTHER proof. There are plenty of cases in mathematics where the first obvious attempt at a proof doesn’t work, but some other proof works just fine.

To see that we’ve produced an unprovable statement, you really do need all of the steps in the argument I’ve given here.

I am feeling tired and must have missed something, but wouldn’t putting this statement after the number 2. make it both true (which it is) and provable (because it follows number 2, which is a “good number?”

Obviously I am missing why it is important that the number in the statement is put after its own number. If the statement states any truly bad number to be bad, why not pair that with a different “good” number and be done with it?

http://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf

Also, I’m sure you know about this but there’s an interesting paradox that follows if we assume some first order axiom system A for arithmetic is consistent, and we take another important theorem by Godel, The Completeness Theorem:
http://mathworld.wolfram.com/GoedelsCompletenessTheorem.html

Since A is consistent, then as a consequence of the second incompleteness theorem, we know that A’ = A + Provable_in_A(”1=2″) is consistent too… and then by the completeness theorem we know that there’s a model M of arithmetic that makes A’ true. Of course M would have to be nonstandard in this case and contain some “infinitary” numbers…