Axiom Of Choice

We have the following indirect implication of form equivalence classes:

152 \(\Rightarrow\) 374-n given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
152 \(\Rightarrow\) 4	Russell's alternative to the axiom of choice, Howard, P. 1992, Z. Math. Logik Grundlagen Math. note-27 note-27 note-27
4 \(\Rightarrow\) 9	clear
9 \(\Rightarrow\) 10	Zermelo's Axiom of Choice, Moore, 1982, 322
10 \(\Rightarrow\) 423	clear
423 \(\Rightarrow\) 374-n	clear

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

Howard-Rubin Number	Statement
152:	\(D_{\aleph_{0}}\): Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. (See note 27 for \(D_{\kappa}\), \(\kappa\) a well ordered cardinal.)
4:	Every infinite set is the union of some disjoint family of denumerable subsets. (Denumerable means \(\cong \aleph_0\).)
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.
423:	\(\forall n\in \omega-\{o,1\}\), \(C(\aleph_0, n)\) : For every \(n\in \omega - \{0,1\}\), every denumerable set of \(n\) element sets has a choice function.
374-n:	\(UT(\aleph_0,n,\aleph_0)\) for \(n\in\omega -\{0,1\}\): The union of a denumerable set of \(n\)-element sets is denumerable.

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