I’ll attempt to clarify a little, if that’s alright. Given a particular well-behaved theory T, Gödel’s (first!) incompleteness theorem exhibits a statement G that is neither provable nor disprovable in T—that is, neither G nor ¬G is syntactically entailed by T. It follows by the completeness theorem that there are models of T in which G is true, and models of T in which ¬G is true.
Now G is often interpreted as meaning “G is not provable in T”, which is obviously true. However, this interpretation is an artifact of the way we’ve carefully constructed G, using a system of relations on Gödel numbers designed to carefully reflect the provability relations on statements in the language of T. But these Gödel numbers are elements of whatever model of T we’re using, and our assumption that the relations used in the construction of G have anything to do with provability in T only apply if the Gödel numbers are from the usual, non-crazy model N of the natural numbers. There are unusual, very-much-crazy models of the natural numbers which are not N, however, and if we’re using one of those then our relations are unintuitive and wacky and have nothing at all to do with provability in T, and in these G can be true or false as it pleases. So when we say “G is true”, we actually mean “G is true if we’re using the standard model of the natural numbers, which may or may not even be a valid model of T in the first place”.
I’ll attempt to clarify a little, if that’s alright. Given a particular well-behaved theory T, Gödel’s (first!) incompleteness theorem exhibits a statement G that is neither provable nor disprovable in T—that is, neither G nor ¬G is syntactically entailed by T. It follows by the completeness theorem that there are models of T in which G is true, and models of T in which ¬G is true.
Now G is often interpreted as meaning “G is not provable in T”, which is obviously true. However, this interpretation is an artifact of the way we’ve carefully constructed G, using a system of relations on Gödel numbers designed to carefully reflect the provability relations on statements in the language of T. But these Gödel numbers are elements of whatever model of T we’re using, and our assumption that the relations used in the construction of G have anything to do with provability in T only apply if the Gödel numbers are from the usual, non-crazy model N of the natural numbers. There are unusual, very-much-crazy models of the natural numbers which are not N, however, and if we’re using one of those then our relations are unintuitive and wacky and have nothing at all to do with provability in T, and in these G can be true or false as it pleases. So when we say “G is true”, we actually mean “G is true if we’re using the standard model of the natural numbers, which may or may not even be a valid model of T in the first place”.