Standard model (set theory)

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).

Often, when one talks about a model $M$ of set theory, it is assumed that $M$ is a set model, i.e. the domain of $M$ is a set in $V$ . If the domain of $M$ is a proper class, then $M$ is a class model. An inner model is necessarily a class model, because inner models are required to contain all the ordinals of $V$ .