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.
References
(theorem II.5.2 and lemma II.5.10)