Inner model theory

• 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, ${\displaystyle \kappa }$-complete ultrafilter U over an ordinal ${\displaystyle \kappa }$ (see zero dagger).

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.

• 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

• Core model
• Inner model