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Condensation lemma

From Wikipedia, the free encyclopedia

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, , then in fact there is some ordinal such that .

More can be said: If X is not transitive, then its transitive collapse is equal to some , and the hypothesis of elementarity can be weakened to elementarity only for formulas which are in the Lévy hierarchy.[1] Also, Devlin showed the assumption that X be transitive automatically holds when .[2]

The lemma was formulated and proved by Kurt Gödel in his proof that the axiom of constructibility implies GCH.

References

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  • Devlin, Keith (1984). Constructibility. Springer. ISBN 3-540-13258-9. (theorem II.5.2 and lemma II.5.10)

Inline citations

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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.