Properly speaking, the signature of ZFC includes a binary relation *symbol* rather than a binary relation. In set theory this symbol is usually denoted $\in$ but it could be denoted equally well as $R$ or $\prec$. In an arbitrary model of set theory, the "sets" might actually be any objects: cats, books, chairs, etc. But if we're are only interested in the elements *qua* elements of that model, we would likely call them "sets" anyway, and call the relation "membership", in the context of that model.

It is very common, when talking about a first-order theory, to conflate the symbols in the theory with their intended interpretations. For example, when we define Peano arithmetic in the signature of ordered rings, we might say that the signature has a single binary addition function $+$. Of course we already know what the "addition" function is on natural numbers, but the interpretation of the $+$ function in an arbitrary model of PA may have very little to do with addition on natural numbers. Nevertheless we usually call the elements of an arbitrary model of PA the "numbers" of the model, and we call the interpretation of the $+$ symbol the "addition" on those numbers. It's simply too cumbersome to say "The objects in the model which are intended to be numbers" or "the function in the model which is intended to be addition".

Similarly, even though the elements of an arbitrary model of ZFC might not "really" be sets, or the interpretation of the $\in$ symbol may not really be set membership, we often speak as if they are. The key observation is that, if someone "lived inside" the model, and only had access to the $\in$ relation, that person would have no way to tell that the things they see are not sets. One way of making this observation precise is the following lemma, which is proved from "outside" a model $(X, R)$ of set theory.

**Mostowski Collapsing Lemma.** Suppose that $R$ is a binary relation in an arbitrary class $X$ (of arbitrary objects) such that:

- For each $y \in X$, the collection $\{ x \in X : xRy\}$ is a set
- The model $(X,R)$ is well founded – every subset of $X$ has an $R$-minimal element
- The model $(X,R)$ satisfies the axiom of extensionality

Then there is a transitive class $C$ (of sets) such that the structure $(C, \in)$ is isomorphic to $(X, R)$, and both $C$ and the isomorphism are uniquely determined by $X$ and $R$.

This lemma says that if we look from the outside at a model that looks even vaguely like a (well-founded) model of ZFC, we can replace it with an isomorphic model whose elements are actually sets and whose binary relation is actually set membership. This doesn't work formally for non-well-founded models, because the actual set membership relation is well founded. But "from the inside" we wont be able to tell that any model of ZFC is not well founded.