nLab
axiom of multiple choice

Idea

The axiom of multiple choice (AMC) is a weak version of the axiom of choice which can hold in constructive mathematics.

Statement

A set-indexed family $\{D_c{\}}_{c\in C}$ of sets is said to be a collection family if for any $c\in C$ and any surjection $E\twoheadrightarrow D_c$, there exists a $c\prime \in C$ and a surjection $D_{c\prime }\twoheadrightarrow D_c$ which factors through $E$.

The axiom of multiple choice states one of two things:

1. for every set $X$, there exists a collection family $\{D_c{\}}_{c\in C}$ such that $X\cong D_c$ for some $c$, or
2. for every set $X$, there exists a collection family $\{D_c{\}}_{c\in C}$ and a family of surjections $\{D_c\to X{\}}_{c\in C}$.

The second formulation seems to be the one originally given by M&P, below, while the first is that given by Rathjen and attributed to Aczel and Simpson.

Mike Shulman: Are these the same? If not, why are they given the same name?

Relationships to other axioms

• Note that $P$ is a projective set if and only if the singleton family $\{P\}$ is a collection family. Therefore, since AC is equivalent to “all sets are projective,” it implies AMC.

• An extension of this argument shows that COSHEP is sufficient to imply AMC.

• The Reflection Principle? (RP) is equivalent to AMC (called stong AMC by van den Berg). RP is motivated by the regular extension principle (REA) from constructive set theory. RP states that every map belongs to a representable class of small maps.

• However, AMC does not imply countable choice or any of the other usual consequences of AC.

• Rathjen proves that SVC also implies AMC. It follows that AMC holds in “most” models of set theory.

• AMC implies WISC, and therefore also implies that the category of anafunctors between two small categories is essentially small.

• A ΠW-pretopos satisfying the axiom of multiple choice if a predicative topos.

References

• Ieke Moerdijk, Erik Palmgren, Type theories, toposes and constructive set theory: predicative aspects of AST (2000) (web)

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

In

• Benno van den Berg, Predicative toposes (arXiv:1207.0959)

WISC is called the “axiom of multiple choice”.

category: foundational axiom