In mathematics, the constructible universe (or Gödel's constructible universe), denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1940 paper Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory. In this, he proved that the constructible universe is an inner model of set theory, and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Kurt Gödel Kurt Gödel [kurt gøːdl], (April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ... 1940 was a leap year starting on Monday (link will take you to calendar). ... In mathematical logic, suppose T is a theory in the language . If M is a model of describing a set theory and N is a class of M such that is a model of T then we say that N is an inner model of T (in M). ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, the axiom of choice is an axiom of set theory. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ...

Gödel's universe can be thought of as being built in "stages" resembling von Neumann's universe. The stages are indexed by ordinals; unlike von Neumann's construction, where one takes at a successor stage α+1 the full power set of the previous stage α (i.e., the set of all subsets of the previous stage) in Gödel's construction one uses only the subsets of the previous stage definable by a formula, possibly with parameters, in the language of set theory with the quantifiers interpreted to range over the sets of the previous stage. A separate article covers Saint John Neumann, the American priest. ... In axiomatic set theory and related branches of mathematics, the Von Neumann universe, or Von Neumann hierarchy of sets is the class of all sets, divided into a transfinite hierarchy of individual sets. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...