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.

See 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