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&alpha;, that is, $$(X,\in)\prec (L_\alpha,\in)$$, then in fact there is some ordinal $$\beta\leq\alpha$$ such that $$X=L_\beta$$.

More can be said: If X is not transitive, then its transitive collapse is equal to some $$L_\beta$$, and the hypothesis of elementarity can be weakened to elementarity only for formulas which are $$\Sigma_1$$ in the Lévy hierarchy. Also, Devlin showed the assumption that X be transitive automatically holds when $$\alpha=\omega_1$$.

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