Standard model (set theory)

Summary

In set theory, a standard model for a theory $T$ (in the language of set theory) is a model $M$ for $T$ where the membership relation $\in_{M}$ is the same as the membership relation $\in$ of a set theoretical universe $V$ (restricted to the domain of $M$). In other words, $M$ is a substructure of $V$. A standard model $M$ that satisfies the additional transitivity condition that $x \in y \in M$ implies $x \in M$ is a standard transitive model (or simply a transitive model).