Condensation lemma

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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 Lα, 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, 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.

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