Revision in response to early comments. Users of set theory need an implementation (in case "model" means something different) of the axioms. I would expect something like this:

An implementation consists of a "collection-of-elements" $X$, and a relation (logical pairing) $E:X\times X\to \{0,1\}$. A logical function $h:X\to\{0,1\}$ is a set if it is of the form $x\mapsto E[x,a]$ for some $a\in X$. Sets are required to satisfy the following axioms: ....

The background "collection-of-elements" needs some properties to even get started. For instance "of the form $x\mapsto E[x,a]$" needs first-order quantification. Mathematical standards of precision seem to require some discussion, but so far I haven't seen anything like this. The first version of this question got answers like $X$ is "the domain of discourse" (philosophy??), "everything" (naive set theory?) and "a set" (circular). Is this a missing definition? Taking it seriously seems to give a rather fruitful perspective.