No, not quite. The distinction is subtle, but I’ll try to lay it out.
there are statements that are true (semantically valid) but unprovable, and hence they provide a counterexample to “If X |=Y then X |- Y”
This is not the case. The problem is with the ‘and hence’: ‘X ⊨ Y’ should not be read as ‘Y is true in X’ but rather ‘Y is true in every model of X’. There are statements which are true in the standard model of Peano arithmetic which are not entailed by it—that is, ‘Y is true in the standard model of Peano arithmetic’ is not the same as ‘PA ⊨ Y’. So this is not a counterexample.
Rather, the notion of ‘complete’ in the incompleteness theorem is that a model is complete if for every statement, either the statement or its negation is entailed by the model (and hence provable by semantic completeness). Gödel’s incompleteness theorem shows that no theory T containing PA is complete in this sense; that is, there statements S which are true in T but not entailed by it (and so not provable). This is because there are models of T in which S is not true: that is, systems in which every statement in T holds, but S does not. (These systems have some additional structure beyond that required by T.)
Wikipedia may do a better job of explaining this than I can at the moment.
I did mean “semantically valid” to be “true in all models”, I was just thinking about second-order Peano arithmetic not first-order ;) I think it’s more natural precisely because then it exactly does show incompleteness.
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”.
No, not quite. The distinction is subtle, but I’ll try to lay it out.
This is not the case. The problem is with the ‘and hence’: ‘X ⊨ Y’ should not be read as ‘Y is true in X’ but rather ‘Y is true in every model of X’. There are statements which are true in the standard model of Peano arithmetic which are not entailed by it—that is, ‘Y is true in the standard model of Peano arithmetic’ is not the same as ‘PA ⊨ Y’. So this is not a counterexample.
Rather, the notion of ‘complete’ in the incompleteness theorem is that a model is complete if for every statement, either the statement or its negation is entailed by the model (and hence provable by semantic completeness). Gödel’s incompleteness theorem shows that no theory T containing PA is complete in this sense; that is, there statements S which are true in T but not entailed by it (and so not provable). This is because there are models of T in which S is not true: that is, systems in which every statement in T holds, but S does not. (These systems have some additional structure beyond that required by T.)
Wikipedia may do a better job of explaining this than I can at the moment.
I did mean “semantically valid” to be “true in all models”, I was just thinking about second-order Peano arithmetic not first-order ;) I think it’s more natural precisely because then it exactly does show incompleteness.
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”.