The cardinal inequality α2 < 2α

Abstract

In ZFC set theory (i.e., Zermelo-Fraenkel set theory with the Axiom of Choice (AC)) any two cardinal numbers are comparable. However, this may not be valid in ZF (i.e., Zermelo-Fraenkel set theory modulo AC). In this paper, we study the strength of the inequality α2 < 2α (in ZF, for every infinite cardinal number α, 2α ≰ α2; see [13]), where α is either the cardinality of special sets (see Definition 1 below) which are expressed as disjoint unions of finite, pairwise equipotent, sets lacking choice functions, or the cardinality of sets in specific permutation models of ZF0 set theory (i.e., ZF without the Axiom of Regularity).