Gödel's constructible universe

Gödel's constructible universe, denoted L, is an inner model of set theory built in stages from subsets definable over earlier stages. It satisfies the axioms of Zermelo–Fraenkel set theory with choice and the generalized continuum hypothesis.

Constructible hierarchy: The transfinite sequence of sets Lα whose union is Gödel's constructible universe L. Its stages specify exactly how L is assembled.
Axiom of constructibility: The set-theoretic axiom asserting that every set belongs to Gödel's constructible universe L. It identifies the ambient universe with L, rather than merely treating L as an inner model.
Kurt Gödel: An Austrian-American logician whose incompleteness theorems and work in set theory shaped modern mathematical logic. He introduced the constructible universe and proved its principal consistency results.
Forcing (mathematics): A method for constructing and analyzing extensions of models of set theory. Forcing can produce universes unlike L, including models with different continuum behavior.
Definable subset: A subset characterized by a formula, parameters, and a structure in which the formula is interpreted. At successor stages, L admits subsets definable over the preceding stage.
Axiom of choice: The axiom asserting that every family of nonempty sets has a choice function. Gödel proved that L satisfies choice, establishing its consistency relative to the base theory.
Consistency of the axiom of choice: Gödel’s result that the axiom of choice is consistent with the other axioms of Zermelo–Fraenkel set theory, if those axioms are consistent. The model L was the central construction in this relative consistency proof.
Cohen's extension: A generic extension of a model of set theory obtained by adjoining a set through forcing. Cohen used such extensions to show that L’s continuum pattern need not hold in every universe.
First-order logic: A formal system of reasoning with quantifiers over objects and predicates describing relations among them. Definability at each stage uses first-order formulas interpreted in set-sized structures.
Generalized continuum hypothesis: The assertion that for every infinite cardinal κ, the power set of κ has cardinality κ⁺. Gödel proved that L satisfies this hypothesis alongside the axioms of choice.

Linked from 17 pages