Inner 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, the smallest inner model containing all real numbers and all ordinals.
• L, the class constructed relative to a normal, non-principal, -complete ultrafilter U over an ordinal .

  Consistency results