Inner model

In mathematical logic, suppose T is a theory in the language

of set theory.

If M is a model of describing a set theory and N is a class of M such that

is a model of T containing all ordinals of M then we say that N is an inner model of T (in M). Ordinarily these models are transitive subsets or subclasses of the von Neumann universe V, or sometimes of a generic extension of V.

This term inner model is sometimes applied to models that are proper classes; the term set model is used for models that are sets.

A model of set theory is called standard if the element relation of the model is the actual element relation restricted to the model. A model is called transitive when it is standard and the base class is a transitive class of sets. A model of set theory is often assumed to be transitive unless it is explicitly stated that it is non-standard. Inner models are transitive, transitive models are standard, and standard models are well-founded.

The assumption that there exists a standard model of ZFC (in a given universe) is stronger than the assumption that there exists a model. In fact, if there is a standard model, then there is a smallest standard model called the minimal model contained in all standard models. The minimal model contains no standard model (as it is minimal) but (assuming the consistency of ZFC) it contains some model of ZFC by the Gödel completeness theorem. This model is necessarily not well founded otherwise its Mostowski collapse would be a standard model. (It is not well founded as a relation in the universe, though it satisfies the axiom of foundation so is "internally" well founded. Being well founded is not an absolute property.) In particular in the minimal model there is a model of ZFC but there is no standard model of ZFC.