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 universeL developed by
Kurt Gödel. Every model M of ZF has an inner model LM 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, LM 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.