Is the set theory (ZF) a structure? According to the definition, generally speaking, a structure $\langle A;R;F,C\rangle$ is such that $A$ is a non-empty set, $R$ is the set of relations, $F$ is the set of functions, and $C$ is a set of constants. For example $\langle\mathbb{R};; +,\cdot, ^{-1};0,1\rangle$ would be the field of the real numbers. Now, in the construction of the set theory (ZF), we need to make a structure. But then $A$ would be the set of sets, which is impossible... What am I missing?
 A: This is a very delicate point in set theory.
First of all we need to point out that the concept of "set" is not an absolute one. Different models of set theory, will think that different mathematical objects are indeed sets.
This is an external point of view. We consider the universes of set theory as sets, from the outside. The Russell paradox, and indeed all the classical paradoxes of set theory, state that the universe cannot be a set from its own point of view. This is where internal and external points of view are important.
Furthermore, if in a universe of set theory which satisfies the axioms of $\sf ZF$ and we can find a set $M$ and a relation $E$ such that $\langle M,E\rangle$ is a model of $\sf ZF$ then by the completeness theorem we have proved that $\sf ZF$ is consistent - in that particular universe. But from the incompleteness theorem we know that a theory like $\sf ZF$ cannot prove its own consistency, therefore we can never even prove that such set structure exists.
If $V$ is the universe of all sets that mathematics have, and suppose that it satisfies all the axioms of $\sf ZF$, then we know that (1) it is not a set itself; (2) we cannot prove that there is a set which is a structure satisfying the axioms of $\sf ZF$; (3) if there is such set, then we can talk about objects which are in that structure, which that particular structure thinks of as sets, and we can contrast them to actual sets.
This is close to the Skolem paradox, by the way.
In any case, as I said, this is a very delicate point and one has to study quite a lot of logic and set theory in order to understand it completely and become comfortable with it.
