Set Theory – Basic Question in Axiomatic Set 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.

Best Answer

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.

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