In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an inner model of ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), 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, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.
What L is
can be thought of as being built in "stages" resembling the construction of von Neumann universe, . The stages are indexed by ordinals. In von Neumann's universe, at a successor stage, one takes to be the set of all subsets of the previous stage, . By contrast, in Gödel's constructible universe , one uses only those subsets of the previous stage that are:
definable by a formula in the formal language of set theory, with parameters from the previous stage and, with the quantifiers interpreted to range over the previous stage.
By limiting oneself to sets defined only in terms of what has already been constructed, one ensures that the resulting sets will be constructed in a way that is independent of the peculiarities of the surrounding model of set theory and contained in any such model.
Define the Def operator:
is defined by transfinite recursion as follows:
If is a limit ordinal, then Here means precedes . Here Ord denotes the class of all ordinals.
If is an element of , then . So is a subset of , which is a subset of the power set of . Consequently, this is a tower of nested transitive sets. But itself is a proper class.
The elements of are called "constructible" sets; and itself is the "constructible universe". The "axiom of constructibility", aka "", says that every set (of ) is constructible, i.e. in .
Additional facts about the sets
An equivalent definition for is:
For any finite ordinal , the sets and are the same (whether equals or not), and thus = : their elements are exactly the hereditarily finite sets. Equality beyond this point does not hold. Even in models of ZFC in which equals , is a proper subset of , and thereafter is a proper subset of the power set of for all . On the other hand, does imply that equals if , for example if is inaccessible. More generally, implies = for all infinite cardinals .
If is an infinite ordinal then there is a bijection between and , and the bijection is constructible. So these sets are equinumerous in any model of set theory that includes them.
As defined above, is the set of subsets of defined by formulas (with respect to the Levy hierarchy, i.e., formulas of set theory containing only bounded quantifiers) that use as parameters only and its elements.
Another definition, due to Gödel, characterizes each as the intersection of the power set of with the closure of under a collection of nine explicit functions, similar to Gödel operations. This definition makes no reference to definability.
All arithmetical subsets of and relations on belong to (because the arithmetic definition gives one in ). Conversely, any subset of belonging to is arithmetical (because elements of can be coded by natural numbers in such a way that is definable, i.e., arithmetic). On the other hand, already contains certain non-arithmetical subsets of , such as the set of (natural numbers coding) true arithmetical statements (this can be defined from so it is in ).
All hyperarithmetical subsets of and relations on belong to (where stands for the Church–Kleene ordinal), and conversely any subset of that belongs to is hyperarithmetical.