Axiom of global choice

In class theories, the axiom of global choice is a stronger variant of the axiom of choice which applies to proper classes as well as sets.

Statement

The axiom can be expressed in various ways which are equivalent:

"Weak" form: Every class of nonempty sets has a choice function.

"Strong" form: Every collection of nonempty classes has a choice function. (Restrict the possible choices in each class to the subclass of sets of minimal rank in the class. This subclass is a set. The collection of such sets is a class.)

V { ∅ } has a choice function (where V is the class of all sets; see Von Neumann universe).

There is a well-ordering of V.

There is a bijection between V and the class of all ordinal numbers.

Discussion

In ZFC, the axiom of global choice cannot be stated as such because it involves existential quantification on classes: so it is not a statement of the language of ZFC (nor even an infinite number of statements like axiom schemes requiring universal quantification on classes). It can, however, be stated for a given "explicit" class, e.g., one can state the fact that such-or-such an explicit class-function is a choice function for V { ∅ } or that such-or-such a class-relation is a well-ordering of V: in this form (i.e., for some explicit class function that is tedious but possible to write down), the axiom of global choice follows from the axiom of constructibility.

In Gödel-Bernays, 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.

ee also

*Axiom of choice
*Axiom of limitation of size
*Von Neumann–Bernays–Gödel set theory
*Morse–Kelley set theory

References

*Jech, Thomas, 2003. "Set Theory: The Third Millennium Edition, Revised and Expanded". Springer. ISBN 3-540-44085-2.
* John L. Kelley; General Topology; ISBN 0-387-90125-6