Standard model (set theory)

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

Usually, when one talks about a model ${\displaystyle M}$ of set theory, it is assumed that ${\displaystyle M}$ is a set model, i.e. the domain of ${\displaystyle M}$ is a set in ${\displaystyle V}$. If the domain of ${\displaystyle M}$ is a proper class, then ${\displaystyle M}$ is a class model. An inner model is necessarily a class model.