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 ordinals 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 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 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 minimal model contained in all standard models. It is countable and satisfies 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 its Mostowski 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+$exists$ a measurable 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 of subtheories 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 exact consistency strength of many important set theoretical properties.

ee also

*Countable transitive models and generic filters