As far as I understand, the current solution is to have the axiom set ensure that the sentence F is meaningless. For example, if we consider the naive set theory, then the set reveals that the NST is contradictory, since any answer to the question “Is its own element?” implies Y. But, if we try to construct the set in the ZFC axioms, then we notice that the ZFC doesn’t actually allow us to create a set of all sets. What it does allow is to select a subset from any already-created set and to create new sets following strict rules…
As far as I understand, the current solution is to have the axiom set ensure that the sentence F is meaningless. For example, if we consider the naive set theory, then the set