The cardinal inequality α² < 2^α

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 < 2^α (in ZF, for every infinite cardinal number α, 2^α ≰ 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).