Jech's text on Set Theory states the following:
If X and Y have the same elements, then X = Y :
∀u(u ∈ X ↔ u ∈ Y ) → X = Y.
The converse, namely, if X = Y then u ∈ X ↔ u ∈ Y, is an axiom of predicate calculus. Thus we have
X = Y if and only if ∀u(u ∈ X ↔ u ∈ Y).
The axiom expresses the basic idea of a set: A set is determined by its elements.
To check my understanding, are the following true?
(1) The axiom of extension states only that if two sets have the same members, then those two sets equal.
(2) It is, on the other hand, an axiom of the language of set theory (i.e., predicate calculus) that if two sets equal, then they have the same members.
A third question:
(3) It seems that the axioms of ZFC are very much independent of the language being worked with. That is, the axioms of ZFC don't state we must be working in the language predicate calculus. Then I take it there exist languages we could be working with that don't have the property that if two sets are equal, they have the same members (so that we could only conclude if two sets have the same members, they are the same and not the reverse). Is this true? Do mathematicians ever work in languages besides predicate calculus?