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Logic, Mathematics
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?
May 7, 2009

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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 notS 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 notS  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!!