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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?

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?

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No, Gödel does

notshow that there are true claims S that are not provable.He shows rather that, given a consistent formal theory

Twhich contains enough arithmetic, then there will be a true arithmetical "Gödel sentence"Gwhich is not provable inT.But that Gödel sentenceG, though it can't be proved inT, can and will be provable in other formal theories (for example,Gis provable in the theory that you get by adding to the axioms ofTa new axiomCon(T)that encodes the claim thatTis consistent). So if we reflect on the axioms ofTand accept them as true, and so have good reason to think thatTis consistent, we'll have good reason to thinkT's Gödel sentenceG, which is provable inT + 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.