Axiom:Axiom of Global Choice

Axiom

In the context of class theory, the axiom of global choice is as follows:

There exists a mapping $f : V \setminus \set \O \to V$, where $V$ is the universal class, such that:

$\forall x \in V: \map f x \in x$

Symbolically:

$\exists A: \map {\text{Fnc}} A \land \forall x: x \ne \O \implies \exists y: y \in x \land \tuple {x, y} \in A$

Sources

• 1940: Kurt Gödel: The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory
• 2010: Elliott Mendelson: Introduction to Mathematical Logic (5th ed.): $4$ Axiomatic Set Theory