It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis".
[1] 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.
Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.
can be thought of as being built in "stages" resembling the construction of the von Neumann universe,
In von Neumann's universe, at a successor stage, one takes
, one uses only those subsets of the previous stage that are: 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.
under a collection of nine explicit functions, similar to Gödel operations.
, such as the set of (natural numbers coding) true arithmetical statements (this can be defined from
is a transitive class and the interpretation uses the real element relationship, so it is well-founded.
is a model of ZFC, which means that it satisfies the following axioms: Notice that the proof that
be a model of ZF, i.e. we do not assume that the axiom of choice holds in
is any standard model of ZF sharing the same ordinals as
is the smallest class containing all the ordinals that is a standard model of ZF.
This set is called the minimal model of ZFC.
Using the downward Löwenheim–Skolem theorem, one can show that the minimal model (if it exists) is a countable set.
, properties of ordinals that depend on the absence of a function or other structure (i.e.
, then there is a closed unbounded class of ordinals that are indiscernible in
, which was first described by Ronald Bjorn Jensen in his 1972 paper entitled "The fine structure of the constructible hierarchy".
is the formula with the smallest Gödel number that can be used to define
itself by a formula of set theory with no parameters, only the free-variables
(some other standard model of ZF with the same ordinals) and we will suppose that the formula is false if either
It is well known that the axiom of choice is equivalent to the ability to well-order every set.
) is equivalent to the axiom of global choice, which is more powerful than the ordinary axiom of choice because it also covers proper classes of non-empty sets.
By the downward Löwenheim–Skolem theorem and Mostowski collapse, there must be some transitive set
There is a formula of set theory that expresses the idea that
Actually, even this complex formula has been simplified from what the instructions given in the first paragraph would yield.
Sometimes it is desirable to find a model of set theory that is narrow like
This gives rise to the concept of relative constructibility, of which there are two flavors, denoted by
is the intersection of all classes that are standard models of set theory and contain
, the smallest model that contains all the real numbers, which is used extensively in modern descriptive set theory.