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, $(X,\in )\prec (L_{\alpha },\in )$, then in fact there is some ordinal $\beta \le \alpha$ 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 ${\mathrm{\Sigma }}_1$ in the Lévy hierarchy. Also, 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.

References

• Devlin, Keith (1984). Constructibility. Springer. ISBN 3-540-13258-9.  (theorem II.5.2 and lemma II.5.10)