Condensation lemma
In set theory, a branch of mathematics, the condensation lemma is a result about sets in the
constructible universe.
It states that if X is a transitive set and is an elementary submodel of some level of the constructible hierarchy Lα, that is,, then in fact there is some ordinal such that.
More can be said: If X is not transitive, then its transitive collapse is equal to some, and the hypothesis of elementarity can be weakened to elementarity only for formulas which are in the Lévy hierarchy. Also, the assumption that X be transitive automatically holds when.
The lemma was formulated and proved by Kurt Gödel in his proof that the axiom of constructibility implies GCH.
