Axiom Of Choice

We have the following indirect implication of form equivalence classes:

152 \(\Rightarrow\) 342-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 \(\Rightarrow\) 342-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.
342-n:	(For \(n\in\omega\), \(n\ge 2\).) \(PC(\infty,n,\infty)\): Every infinite family of \(n\)-element sets has an infinite subfamily with a choice function. (See Form 166.)

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