set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

, a branch of mathematics, the condensation lemma is a result about sets in the

constructible universe In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L, is a particular Class (set theory), class of Set (mathematics), sets that can be described entirely in terms of simpler sets. L is the un ...

. It states that if ''X'' is a

transitive set In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions holds: * whenever x \in A, and y \in x, then y \in A. * whenever x \in A, and x is not an urelement, then x is a subset of A. S ...

and is an

elementary submodel In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences. If ''N'' is a substructure of ''M'', one oft ...

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 In set theory and mathematical logic, the Lévy hierarchy, introduced by Azriel Lévy in 1965, is a hierarchy of formulas in the formal language of the Zermelo–Fraenkel set theory, which is typically called just the language of set theory. This is ...

. Also, Devlin showed the assumption that ''X'' is transitive automatically holds when

\alpha=\omega_1

.W. Marek, M. Srebrny, "Gaps in the Constructible Universe" (1973), p.364. The lemma was formulated and proved by

Kurt Gödel Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...

in his proof that the

axiom of constructibility The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as ''V'' = ''L''. The axiom, first investigated by Kurt Gödel, is inconsistent with the pr ...

implies GCH.

References

* (theorem II.5.2 and lemma II.5.10)

Inline citations

Constructible universe Lemmas in set theory {{settheory-stub