Archive for Mathematical Logic 55 (3-4):415-429 (2016)

In set theory without the Axiom of Choice, 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 everyn ∈ ω, f ≺ f, 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 ACLO ↛ NDS in ZFA set theory. 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 ACWO ↛ NDS in ZF and that “for all infinite cardinals m, m + m = m” ↛ NDS in ZFA.
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DOI 10.1007/s00153-015-0472-5
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The Independence of the Continuum Hypothesis.Paul Cohen - 1963 - Proc. Nat. Acad. Sci. USA 50 (6):1143-1148.
The Independence of the Continuum Hypothesis II.Paul Cohen - 1964 - Proc. Nat. Acad. Sci. USA 51 (1):105-110.
An Independence Result Concerning the Axiom of Choice.Gershon Sageev - 1975 - Annals of Mathematical Logic 8 (1-2):1-184.

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