Set Theory ZF Axioms Doubt I have a pretty basic question about the symbolic representation of the axiom of extensionality for set theory, which states that 
$$
\forall A \forall B [ \forall x (x\in A \iff x\in B)] \iff A = B
$$
Now, I have been reading the Enderton's Elements of Set Theory, and it states that the notation $\forall x$ denotes "for every set x". However, my question is: x is not really a set, is it? It should be an object, right? For example, if I have the set {1,2,3}, wouldn't it be correct if I wrote $ 1 \in \{ 1,2,3 \} $, and here $1$ is not a set, right?
 A: In axiomatic set theory, we usually take the position that everything is a set, and in particular everything we need to care about as elements of a set are themselves sets.
The expectation is then that things like "$1$" are to be interpreted as abbreviations for particular sets that represent the numbers. A common choice of representative for the natural number $1$ is the set $\{\{\}\}$, as you will no doubt learn when you read further on in the book.
The reason why this is done is that it turns out it can be made to work (if you're willing not to worry too much about not being able put the Platonic integers into sets, but only certain other sets that represent them), and it simplifies the axioms that they don't have to deal with things that are not set. This is technically convenient.
It is possible to do axiomatic set theory in a way such that sets can contain things that are not themselves set -- usually those things are called urelements, and are assumed to have no properties (as relevant for set theory) except that they can be distinguished from each other somehow. This is not what is usually done (and is not what "ZF" or "ZFC" usually refers to), but, again, you can do it if you want to.
