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This question is directed (mainly) to Peter Smith. I've read you "Introduction to Gödel's Theorems" (that's how I ended up here) and found it fascinating. At a certain point it the book, it is asserted that G (that is, a Gödel Sentence) is Goldbach type. My question is the following, what are the odds (I don't mean statistically, just your opinion) that the Goldbach conjecture is in some manner an example of a Gödel Sentence naturally (?) arising?
I am aware that most mathematicians believe the Goldbach Conjecture to be true, even if all attempts to prove it have failed so far. So, could it be that it actually is true, but to be proven, additional axioms would have to be added to regular arithmetic, or the former would have to be modified in some fashion? Has anyone tried to prove this? Have they succeeded?

Sorry for the messy English, I hope my question can be understood, and thanks for writing such an interesting book.

May 7, 2009

 Response from Peter Smith on May 9, 2009

I'm really glad you enjoyed the Gödel book!

Suppose that S is Goldbach's conjecture. And suppose theory T is your favourite arithmetic (which includes Robinson Arithmetic). Then Theorem 9.3 applies to S. So if not-S is not logically deducible from T, then S must be true.

So if we had a proof that S is a "naturally" arising Gödel sentence -- i.e. a demonstration that T proves neither S nor not-S -- we'd ipso facto have a proof that S is true.

That means that establishing that that S is a "naturally" arising Gödel sentence for T -- if that's what it is -- is at least as hard as proving Goldbach's Conjecture itself. Which, the evidence suggests, is very hard!

As to the "odds": my hunch is that GC is true, and can be proved in PA -- but I wouldn't bet even a decent meal out on it!!

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