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 $\beta \leq \alpha$ 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.^[1] Also, Devlin showed the assumption that X is transitive automatically holds when $\alpha =\omega _{1}$ .^[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)

Inline citations

^ R. B. Jensen, The Fine Structure of the Constructible Hierarchy (1972), p.246. Accessed 13 January 2023.
^ W. Marek, M. Srebrny, "Gaps in the Constructible Universe" (1973), p.364.

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[1] R. B. Jensen, The Fine Structure of the Constructible Hierarchy (1972), p.246. Accessed 13 January 2023.

[2] W. Marek, M. Srebrny, "Gaps in the Constructible Universe" (1973), p.364.

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