In class theories

Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...

, 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
  Von Neumann universe
  In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets...
  
  ).

• There is a well-ordering of V.

• There is a bijection between V and the class of all ordinal number
  Ordinal number
  In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
  
  s.

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

Axiom of constructibility

The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and the constructible universe, respectively.- Implications :The axiom of...

.

In Gödel-Bernays

Von Neumann–Bernays–Gödel set theory

In the foundations of mathematics, von Neumann–Bernays–Gödel set theory is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC. A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes...

, 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

Axiom of limitation of size

In class theories, the axiom of limitation of size says that for any class C, C is a proper class, that is a class which is not a set , if and only if it can be mapped onto the class V of all sets....

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See also

• Axiom of choice
• Axiom of limitation of size
  Axiom of limitation of size
  In class theories, the axiom of limitation of size says that for any class C, C is a proper class, that is a class which is not a set , if and only if it can be mapped onto the class V of all sets....
  
  
• Von Neumann–Bernays–Gödel set theory
  Von Neumann–Bernays–Gödel set theory
  In the foundations of mathematics, von Neumann–Bernays–Gödel set theory is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC. A statement in the language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes...
  
  
• Morse–Kelley set theory
  Morse–Kelley set theory
  In the foundation of mathematics, Morse–Kelley set theory or Kelley–Morse set theory is a first order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory...