So, as far as I understand classical mathematics assumes the law of excluded middle. I wonder then, how is it compatible with the fact that we know some statements are undecidable? Such as the axiom of choice and the continuum hypothesis, both of which have been shown to be neither provable nor disprovable. Doesn't that already violate the law of excluded middle?
I understand that these statements are undecidable only in a specific axiomatic system. But let's consider this statement: "Assuming ZF and classical logic, the axiom of choice holds." This is neither true nor false. Doesn't that violate the law of excluded middle? Thank you