Condensation lemma explained

In set theory, a branch of mathematics, the condensation lemma is a result about sets in theconstructible 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 be 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

• Book: Devlin , Keith . Keith Devlin

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

Notes and References

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.