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 Gรถdel'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.)