nLab (classical) axiom of multiple choice

This article is about a classical set theory axiom. Some literature instead uses this name for an unrelated weakening of AC. For that notion, see axiom of multiple choice.

Idea

The axiom of multiple choice (AMC) weakens the axiom of choice by allowing choice functions to choose finite sets, rather than particular elements.

Statement

Recall that one common statement of the axiom of choice is:

For every set $S$ of non-empty sets, there is a function $f$ defined on $S$ such that $\forall X\in S$, $f(X)\in X$.

Such an $f$ is called a choice function for $S$.

The axiom of multiple choice weakens the axiom of choice by allowing choice functions to pick out finite subsets, rather than finite sets. It says:

For every set $S$ of non-empty sets, there is a function $f$ defined on $S$ such that $\forall X\in S$, $f(X)\subseteq X$ and $f(X)$ is finite and non-empty.

Relationships to other axioms

The axiom of multiple choice is equivalent to the axiom of choice modulo ZF set theory. However, it is strictly weaker in ZFA and other similar set theories. AMC holds in any permutation model with finite supports where each atom has a finite orbit. For a detailed proof, see Jech’s “The Axiom of Choice”, chapter 9.

The constructive axiom by the same name is not historically related, and the two axioms are independent. Any permutation model will satisfy SVC, which Rathjen proves implies the constructive axiom, but this AMC can fail in a permutation model.

References

• Jech, The Axiom of Choice (1973), ISBN : 0444104844 (New York)

• A. Lévy. Axioms of multiple choice. Fundamenta mathematicae, vol. 50 no. 5 (1962), pp. 475–483

The constructive axiom by the same name is discussed in:

• Rathjen, “Choice principles in constructive and classical set theories”

category: foundational axiom