No decreasing sequence of cardinals

Abstract

In set theory without the Axiom of Choice (AC), we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ? of natural numbers such that for everyn ? ?, f (n + 1) ? f (n), where for sets x and y, x ? y means that there is a one-to-one map g : x ? y, but no one-to-one map h : y ? x. It is a long standing open problem whether NDS implies AC. In this paper, among other results, we show that NDS is a strong axiom by establishing that AC LO (AC restricted to linearly ordered sets of non-empty sets, and also equivalent to AC in ZF, the Zermelo---Fraenkel set theory minus AC) ? NDS in ZFA set theory (ZF with the Axiom of Extensionality weakened in order to allow the existence of atoms). The latter result provides a strongly negative answer to the question of whether "every Dedekind-finite set is finite" implies NDS addressed in G. H. Moore "Zermelo's Axiom of Choice. Its Origins, Development, and Influence" and in P. Howard---J. E. Rubin "Consequences of the Axiom of Choice". We also prove that AC WO (AC restricted to well-ordered sets of non-empty sets) ? NDS in ZF (hence, "every Dedekind-finite set is finite" ? NDS in ZF, either) and that "for all infinite cardinals m, m + m = m" ? NDS in ZFA.