In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
class theories, the axiom of global choice is a stronger variant of the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
that applies to
proper classes of
sets as well as sets of sets. Informally it states that one can simultaneously choose an element from every
non-empty
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...
set.
Statement
The axiom of global choice states that there is a
global choice function τ, meaning a function such that for every non-empty set ''z'', τ(''z'') is an element of ''z''.
The axiom of global choice cannot be stated directly in the language of
ZFC (
Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic se ...
–Fraenkel set theory with the axiom of choice), as the choice function τ is a proper class and in ZFC one cannot quantify over classes. It can be stated by adding a new function symbol τ to the language of ZFC, with the property that τ is a global choice function. This is a
conservative extension In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a superthe ...
of ZFC: every provable statement of this extended theory that can be stated in the language of ZFC is already provable in ZFC . Alternatively,
Gödel showed that given the
axiom of constructibility
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible universe, constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the von Neumann unive ...
one can write down an explicit (though somewhat complicated) choice function τ in the language of ZFC, so in some sense the axiom of constructibility implies global choice (in fact, (ZFC proves that) in the language extended by the unary function symbol τ, the axiom of constructibility implies that if τ is said explicitly definable function, then this τ is a global choice function. And then global choice morally holds, with τ as a
witness
In law, a witness is someone who has knowledge about a matter, whether they have sensed it or are testifying on another witnesses' behalf. In law a witness is someone who, either voluntarily or under compulsion, provides testimonial evidence, e ...
).
In the language of
von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a collect ...
(NBG) and
Morse–Kelley set theory
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely ...
, the axiom of global choice can be stated directly , and is equivalent to various other statements:
* Every class of nonempty sets has a
choice function
A choice function (selector, selection) is a mathematical function ''f'' that is defined on some collection ''X'' of nonempty sets and assigns some element of each set ''S'' in that collection to ''S'' by ''f''(''S''); ''f''(''S'') maps ''S'' to ...
.
* V \ has a choice function (where V is the
class of all sets).
* There is a
well-ordering
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-o ...
of V.
* There is a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
between V and the class of all
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...
s.
In von Neumann–Bernays–Gödel set theory, global choice does not add any consequence about ''sets'' (not proper classes) beyond what could have been deduced from the ordinary axiom of choice.
Global choice is a consequence of the
axiom of limitation of size
In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes.; English translation: . It formalizes the limitation of size principle, which avoids the paradoxes encountered in earli ...
.
References
*
*
Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. .
*
John L. Kelley;
General Topology;
{{Set theory
Axioms of set theory
Axiom of choice