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I know that Gödel shows that there are true claims S that are not provable. The epistemic question is "How do we know S is true". Is it "true" in the same way that axioms of Euclid's geometry are true?

April 22, 2009

Response from Peter Smith on April 22, 2009

No, Gödel does not show that there are true claims S that are not provable.

He shows rather that, given a consistent formal theory T which contains enough arithmetic, then there will be a true arithmetical "Gödel sentence" G which is not provable in T. But that Gödel sentence G, though it can't be proved in T, can and will be provable in other formal theories (for example, G is provable in the theory that you get by adding to the axioms of T a new axiom Con(T) that encodes the claim that T is consistent). So if we reflect on the axioms of T and accept them as true, and so have good reason to think that T is consistent, we'll have good reason to think T's Gödel sentence G, which is provable in T + Con(T), is true. (And there's nothing especially mysterious about the notion of truth here: it is the common-or-garden notion of arithmetic truth that is invoked when we say of even the simplest sentence of formal arithmetic, as it might be "1 + 0 = 1", that it is true.)

For more -- a great deal more! -- on this, see for example my Introduction to Gödel's Theorems.


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