Axiom of global choice


In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice 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 is an element of z.
The axiom of global choice cannot be stated directly in the language of 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 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 choice function τ in the language of ZFC, so in some sense the axiom of constructibility implies global choice.
In the language of von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory, the axiom of global choice can be stated directly, and is equivalent to various other statements:
In von Neumann–Bernays–Gödel set theory, global choice does not add any consequence about sets beyond what could have been deduced from the ordinary axiom of choice.
Global choice is a consequence of the axiom of limitation of size.