Standard model (set theory)

Not to be confused with standard model (particle physics).

In set theory, a standard model for a theory T is a model M for T where the membership relation ∈_M is the same as the membership relation ∈ 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 ∈ y ∈ M implies x ∈ M is a standard transitive model (or simply a transitive model).

Usually, 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.