Axiom Of Choice

We have the following indirect implication of form equivalence classes:

7 \(\Rightarrow\) 373-n given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
7 \(\Rightarrow\) 9	On the existence of large sets of Dedekind cardinals, Tarski, A. 1965, Notices Amer. Math. Soc. The Axiom of Choice, Jech, 1973b, page 161 problem 11.6
9 \(\Rightarrow\) 10	Zermelo's Axiom of Choice, Moore, 1982, 322
10 \(\Rightarrow\) 288-n	clear
288-n \(\Rightarrow\) 373-n	clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
7:	There is no infinite decreasing sequence of cardinals.
9:	Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
10:	\(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function.
288-n:	If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function.
373-n:	(For \(n\in\omega\), \(n\ge 2\).) \(PC(\aleph_0,n,\infty)\): Every denumerable set of \(n\)-element sets has an infinite subset with a choice function.

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