- Inner model
In

mathematical logic , suppose "T" is a theory in the language:$L\; =\; langle\; in\; angle$

of set theory.

If "M" is a model of $L$ describing a

set theory and "N" is a class of "M" such that:$langle\; N,\; in\_M,\; cdots\; angle$

is a model of "T" containing all

ordinal s of "M" then we say that "N" is an**inner model**of "T" (in "M").This term

**"inner model**" is sometimes applied to models which are proper classes; the term**set model**is used for models which 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 atransitive 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 then the assumption that there exists a model. In fact, if there is a standard model, then there is a smallest standard model called the

contained in all standard models. It is countable and satisfiesminimal model V=L . The minimal model contains no standard model (as it is minimal) but (assuming the consistence of ZFC) it containssome model of ZFC by the Godel completeness theorem. This model is necessarily not well founded otherwise itsMostowski collapse would be a standard model. (It is not well founded as a relation in the universe, though itsatisfies 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.**Use**Usually when one talks about inner models of a theory, the theory one is discussing is

ZFC or some extension of ZFC (like ZFC+$exists$ ameasurable cardinal ). When no theory is mentioned, it is usually assumed that the model under discussion is an inner model of ZFC. However, it is not uncommon to talk about inner models ofsubtheories of ZFC (like ZF or KP) as well.**Related ideas**It was proved by Kurt Gödel that any model of ZF has a least inner model of ZF (which is also an inner model of ZFC + GCH), called the

constructible universe , or L.There is a branch of set theory called

inner model theory which studies ways of constructing least inner models of theories extending ZF. Inner model theory has led to the discovery of the exactconsistency strength of many important set theoretical properties.**ee also***Countable transitive models and generic filters

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