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, ${\displaystyle (X,\in )\prec (L_{\alpha },\in )}$, then in fact there is some ordinal ${\displaystyle \beta \leq \alpha }$ such that ${\displaystyle X=L_{\beta }}$.

More can be said: If X is not transitive, then its transitive collapse is equal to some ${\displaystyle L_{\beta }}$, and the hypothesis of elementarity can be weakened to elementarity only for formulas which are ${\displaystyle \Sigma _{1}}$ in the Lévy hierarchy. ^[1] Also, Devlin showed the assumption that X is transitive automatically holds when ${\displaystyle \alpha =\omega _{1}}$. ^[2]

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

Condensation Arguments

A condensation argument is an argument which uses, in some way, shape, or form, the condensation lemma. Typically, this is done by means of reducing ${\displaystyle \Sigma _{n}}$ predicates over a specific ${\displaystyle J_{\alpha }}$ to ${\displaystyle \Sigma _{1}}$ predicates over an amenable structure, and then working in that said amenable structure instead of ${\displaystyle J_{\alpha }}$ itself.

References

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

Inline citations

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.