Editing Set theory (section)

=== Inner model theory ===
{{Main|Inner model theory}}

An ''inner model'' of Zermelo–Fraenkel set theory (ZF) is a transitive [[proper class|class]] that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the [[constructible universe]] ''L'' developed by Gödel.
One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether a model ''V'' of ZF satisfies the [[continuum hypothesis]] or the [[axiom of choice]], the inner model ''L'' constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent.

The study of inner models is common in the study of [[axiom of determinacy|determinacy]] and [[large cardinal]]s, especially when considering axioms such as the axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice).<ref>{{citation | last1=Jech | first1=Thomas | author1-link=Thomas Jech | title=Set Theory | edition= Third Millennium | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-44085-7 | year=2003 | zbl=1007.03002 | page=642 | url=https://books.google.com/books?id=CZb-CAAAQBAJ&pg=PA642 }}</ref>