mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, specifically in class theories, the axiom of global choice is a stronger variant of the

axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...

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 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

Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...

(ZF) with the

(AC), known as ZFC, 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 superth ...

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 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, FC proves thatin 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, either voluntarily or under compulsion, provides testimonial evidence, either oral or written, of what they know or claim to know. A witness might be compelled to provide testimony in court, before a grand jur ...

). 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 collec ...

(NBG) and Morse–Kelley set theory, 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 Let ''X'' be a set of sets none of which are empty. Then a choice function (selector, selection) on ''X'' is a mathematical function ''f'' that is defined on ''X'' such that ''f'' is a mapping that assigns each element of ''X'' to one of its ele ...

. * V \ has a choice function (where V is the class of all sets). * There is a well-ordering of V. * There is a

bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

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 leas ...

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.

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