In set theory, **inner model theory** is the study of certain models of ZFC or some fragment or strengthening thereof. Ordinarily these models are transitive subsets or subclasses of the von Neumann universe *V*, or sometimes of a generic extension of *V*. Inner model theory studies the relationships of these models to determinacy, large cardinals, and descriptive set theory. Despite the name, it is considered more a branch of set theory than of model theory.

## Examples

- The class of all sets is an inner model containing all other inner models.
- The first non-trivial example of an inner model was the constructible universe
*L*developed by Kurt Gödel. Every model*M*of ZF has an inner model*L*^{M}satisfying the axiom of constructibility, and this will be the smallest inner model of*M*containing all the ordinals of*M*. Regardless of the properties of the original model,*L*^{M}will satisfy the generalized continuum hypothesis and combinatorial axioms such as the diamond principle ◊. - HOD, the class of sets that are hereditarily ordinal definable, form an inner model, which satisfies ZFC.
- The sets that are hereditarily definable over a countable sequence of ordinals form an inner model, used in Solovay's theorem.
- L(R), the smallest inner model containing all real numbers and all ordinals.
- L[U], the class constructed relative to a normal, non-principal, -complete ultrafilter U over an ordinal (see zero dagger).

## Consistency results

One important use of inner models is the proof of consistency results. If it can be shown that every model of an axiom *A* has an inner model satisfying axiom *B*, then if *A* is consistent, *B* must also be consistent. This analysis is most useful when *A* is an axiom independent of ZFC, for example a large cardinal axiom; it is one of the tools used to rank axioms by consistency strength.

## References

- Jech, Thomas (2003),
*Set Theory*, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag - Kanamori, Akihiro (2003),
*The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings*(2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-00384-7