The shrinking principle and the axiom of choice

摘要

The axiom of choice is equivalent to the shrinking principle: every indexed cover of a set has a refinement with the same index set which is a partition. A simple and direct proof of this equivalence is given within an elementary fragment of constructive Zermelo–Fraenkel set theory. Variants of the shrinking principle are discussed, and it is related to a similar but different principle formulated by Vaught.