A set-indexed family of sets is said to be a collection family if for any and any surjection , there exists a and a surjection which factors through .
Depending on the author, the axiom of multiple choice is one of the following statements:
The nLab uses the initialization AMC to cover either the first two formulations.
Mike Shulman: Are the first two the same? If not, why are they given the same name?
The third is a weaker condition, and while some may refer to as a “weak axiom of multiple choice”, van den Berg obviously does not; he calls his the AMC and the Moerdijk-Palmgren formulation rather the “strong axiom of multiple choice”.
Note that is a projective set if and only if the singleton family 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 (the one called strong 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.
Rathjen, “Choice principles in constructive and classical set theories”
WISC is called the “axiom of multiple choice”.