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

More can be said: If X is not transitive, then its transitive collapse is equal to some ${\displaystyle L_{\beta }}$, and the hypothesis of elementarity can be weakened to elementarity only for formulas which are ${\displaystyle \Sigma _{1}}$ in the Lévy hierarchy.^[1] Also, Devlin showed the assumption that X is transitive automatically holds when ${\displaystyle \alpha =\omega _{1}}$.^[2]

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)