No decreasing sequence of cardinals

作者：Howard Paul; Tachtsis Eleftherios^*

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

DOI：10.1007/s00153-015-0472-5

• Axiom of Choice
• Weak axioms of choice
• Cardinals
• Decreasing sequence of cardinals
• Dedekind-finite sets
• Dedekind sets
• Permutation models of ZFA
• Symmetric models of ZF
• Jech-Sochor Embedding Theorems

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 omega of natural numbers such that for every n is an element of omega, 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) negated right arrow 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) negated right arrow NDS in ZF (hence, "every Dedekind-finite set is finite" negated right arrow NDS in ZF, either) and that "for all infinite cardinals m, m + m = m" negated right arrow NDS in ZFA.

• 出版日期2016-5