Axiom of global choice

The axiom asserts that a class function selects an element from every nonempty set. In class theories, it extends ordinary choice to a single selector defined across the entire universe of sets.

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Axiom of choice: The axiom that every set of nonempty sets has a choice function. Restricting the global selector to any set-sized family yields the ordinary axiom of choice.
Set (mathematics): A collection treated as a single object in mathematical set theory. The axiom selects from every nonempty set, not merely from one specified family.
Axiom of dependent choice: A choice principle guaranteeing sequences of successive choices under specified extension conditions. It supports many constructions without asserting one selector for every nonempty set.
Global well-order: A well-ordering of the universe of sets represented by a class relation. Choosing the least element of each nonempty set turns such an ordering into a global selector.
Choice function: A function that assigns to each set in a family one of its elements. The global axiom requires one choice function whose domain ranges over all nonempty sets.
Proper class: A collection too large to be a set in a given set theory. The totality of all sets is a proper class, explaining why a global selector is class-sized.
Axiom of countable choice: The principle that every countable family of nonempty sets has a choice function. Its countable scope is far narrower than a class-wide choice function.
Skolem function: A function selecting witnesses for existential statements in a formal language or structure. Global choice can provide class-level witness selection when constructing Skolem functions.
Class function: A function whose domain or graph is a proper class rather than a set. The selector must range over the universe, so its graph cannot generally be a set.
Zermelo-Fraenkel set theory: An axiomatic foundation for mathematics whose objects are sets. Its set-only language distinguishes ordinary choice from class-level global choice.