Inner model


In set theory, a branch of mathematical logic, an inner model for a theory T is a substructure of a model M of a set theory that is both a model for T and contains all the ordinals of M.

Definition

Let be the language of set theory. Let S be a particular set theory, for example the ZFC axioms and let T also be a theory in.
If M is a model for S, and N is a -structure such that
  1. N is a substructure of M, i.e. the interpretation of in N is
  2. N is a model for T
  3. the domain of N is a transitive class of M
  4. N contains all ordinals of M
then we say that N is an inner model of T. Usually T will equal S, so that N is a model for S 'inside' the model M of S.
If only conditions 1 and 2 hold, N is called a standard model of T, a standard submodel of T if S = T. A model N of T in M is called transitive when it is standard and condition 3 holds. If the axiom of foundation is not assumed all three of these concepts are given the additional condition that N be well-founded. Hence inner models are transitive, transitive models are standard, and standard models are well-founded.
The assumption that there exists a standard submodel of ZFC is stronger than the assumption that there exists a model. In fact, if there is a standard submodel, then there is a smallest standard submodel
called the minimal model contained in all standard submodels. The minimal submodel contains no standard submodel but 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 submodel.
In particular in the minimal submodel there is a model of ZFC but there is no standard submodel 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. 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 as well.

Related ideas

It was proved by Kurt Gödel that any model of ZF has a least inner model of ZF, called the constructible universe, or L.
There is a branch of set theory called inner model theory that 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.