# Set Theory – Basic Question in Axiomatic Set Theory

elementary-set-theoryset-theory

Given two classes $$x$$ and $$y$$, does this statement hold $$x \in y \lor x \not \in y$$? I know that if $$x$$ is a proper class, then it should hold since $$x$$ is a proper class if and only for every class $$z$$, then $$x \not \in z$$. However, if $$x$$ is a set, then intuitively it seems that this statement will hold, but I know that in mathematics there are some statements that cannot be proved using the current axioms. Sorry I am not too acquainted with ZFC or NBG.

I have made my comment into an answer since the OP seemed happy with it.

The Law of Excluded Middle: In classical predicate logic (and some other types of logic), $$\Psi \lor \lnot \Psi$$ for all statements $$\Psi$$.

$$x\in y$$ is a statement in the language of ZFC, and ZFC is a theory in predicate logic. Therefore $$(x\in y)\lor \lnot (x\in y)$$.

The intuitionist school of logic holds the following position:

"If we know that it is not the case that a statement $$\Psi$$ is provably false, we don't necessarily know that $$\Psi$$ is provably true either."

It follows that in intuitionist logic we cannot use the law of excluded middle to prove statements.