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, Devlin showed the assumption that X is 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

• Template:Cite book (theorem II.5.2 and lemma II.5.10)

Inline citations

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