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 Lα, that is, [math]\displaystyle{ (X,\in)\prec (L_\alpha,\in) }[/math], then in fact there is some ordinal [math]\displaystyle{ \beta\leq\alpha }[/math] such that [math]\displaystyle{ X=L_\beta }[/math].
More can be said: If X is not transitive, then its transitive collapse is equal to some [math]\displaystyle{ L_\beta }[/math], and the hypothesis of elementarity can be weakened to elementarity only for formulas which are [math]\displaystyle{ \Sigma_1 }[/math] in the Lévy hierarchy. Also, the assumption that X be transitive automatically holds when [math]\displaystyle{ \alpha=\omega_1 }[/math].
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)
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