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No decreasing sequence of cardinals

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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 ω of natural numbers such that for every n ∈ ω, 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.

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Authors and Affiliations

  1. Department of Mathematics, Eastern Michigan University, Ypsilanti, MI, 48197, USA

    Paul Howard

  2. Department of Mathematics, University of the Aegean, 83200, Karlovassi, Samos, Greece

    Eleftherios Tachtsis

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  1. Paul Howard
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  2. Eleftherios Tachtsis
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Correspondence to Eleftherios Tachtsis.

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Howard, P., Tachtsis, E. No decreasing sequence of cardinals. Arch. Math. Logic 55, 415–429 (2016). https://doi.org/10.1007/s00153-015-0472-5

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  • DOI: https://doi.org/10.1007/s00153-015-0472-5

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