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 L_α, that is, ( X , ∈ ) ≺ ( L α , ∈ ) , then in fact there is some ordinal β ≤ α such that X = L β .

More can be said: If X is not transitive, then its transitive collapse is equal to some L β , and the hypothesis of elementarity can be weakened to elementarity only for formulas which are Σ 1 in the Lévy hierarchy. Also, the assumption that X be transitive automatically holds when α = ω 1 .

The lemma was formulated and proved by Kurt Gödel in his proof that the axiom of constructibility implies GCH.