Actually, you can’t quite escape the problem of the excluded middle by asserting that “This sentence is false” is not well-formed, or meaningful; because Gödel’s sentence G is a perfectly well-formed (albeit horrifically complicated) statement about the properties of natural numbers which is undecidable in exactly the same way as Epimenides’ paradox.
Mathematicians who prefer to use the law of excluded middle (i.e. most of us, including me) have to affirm that (G or ~G) is indeed a theorem, although neither G nor ~G are theorems! (This doesn’t lead to a contradiction within the system, fortunately, because it’s also impossible to formally prove that neither G nor ~G are theorems.)
Neither G nor -G are theorems, but (G or -G) is a theorem.
As an analogy (whose deepness I don’t quite know), this is like if the logical system was a person (who I will name Peano) who I determine will never figure out whether G is true or false. Peano, however, knows that G must be either true or false. He can know (G or -G) even if he can never know G nor know -G.
The tricky thing, I think, is to not accidentally mix the meta lels.
Actually, you can’t quite escape the problem of the excluded middle by asserting that “This sentence is false” is not well-formed, or meaningful; because Gödel’s sentence G is a perfectly well-formed (albeit horrifically complicated) statement about the properties of natural numbers which is undecidable in exactly the same way as Epimenides’ paradox.
Mathematicians who prefer to use the law of excluded middle (i.e. most of us, including me) have to affirm that (G or ~G) is indeed a theorem, although neither G nor ~G are theorems! (This doesn’t lead to a contradiction within the system, fortunately, because it’s also impossible to formally prove that neither G nor ~G are theorems.)
Neither G nor -G are theorems, but (G or -G) is a theorem.
As an analogy (whose deepness I don’t quite know), this is like if the logical system was a person (who I will name Peano) who I determine will never figure out whether G is true or false. Peano, however, knows that G must be either true or false. He can know (G or -G) even if he can never know G nor know -G.
The tricky thing, I think, is to not accidentally mix the meta lels.
No, it’s not the same. Gödel sentences can be resolved by adding axioms. You can’t add axioms to resolve ‘This sentence is false’.