Return to Revisions

Just one clarification to Guillaume's answer: Yes, a model of ZFC is a set $E$ together with a binary relation $R$ on $E$ such that $(E,R)$ satisfies ZFC. The relation $R$ is extensional, that is every $x\in E$ is determined by the collection of $y\in E$ with $yRx$, because $(E,R)$ satisfies the axiom of extensionality.
However, even though $(E,R)$ satisfies the axiom of regularity, the relation $R$ need not be well-founded. $(E,R)$ thinks that $R$ is well-founded, but the real world knows an infinite $R$-decreasing chain in $E$. In this situation, $(E,R)$ is a non-standard model of ZFC which is clearly not isomorphic to a transitive model with the $\in$-relation.