I am exploring Axiomatic set theory out of curiosity, and was referring to https://youtu.be/AAJB9l-HAZs as an introduction to the subject. My question is here (time in video 10:47) :-
The first axiom of the theory is stated as this: Axiom on $\in$-relation: $x \ \in\ y$ is a proposition if and only if x and y are sets.
This does not seem like an axiom that is in an axiomatic system. An axiom in an axiomatic system is a formula that is not a tautology in that logic system, but is assumed to be a universal truth (Please correct me if I am incorrect). To me, this "axiom" seems to just be saying this - $\in$ is a (binary) predicate. The variables on which the truth value of $\in$ depends are called sets.
If I remember correctly, there exist some formulas in predicate logic which can neither be proved/disproved in the logic. Perhaps the professor wants to say that the only if the formulas can be proved/disproved, the variables are sets? He goes on to 'provide an application of this axiom' by showing how Russel's paradox is avoided. To me, it simply seems like he demonstrates how the formula: $\exists u ((\forall z(z \in u \iff \lnot z \in z)))$ can neither be proved nor disproved using predicate logic. (I may be extremely wrong here). Since it can neither be proved nor disproved, $u$ is not a set?
Another thing is the formal formula of the axiom which is stated: $\forall x \forall y \ ((x \in y) \oplus \lnot (x \in y))$ - I think that this is simply a tautology? Why is this an axiom?