Axiom Of Choice

We have the following indirect implication of form equivalence classes:

214 \(\Rightarrow\) 4 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
214 \(\Rightarrow\) 152	note-140
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

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

Howard-Rubin Number	Statement
214:	\(Z(\omega)\): For every family \(A\) of infinite sets, there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a non-empty subset of \(y\) and \(\|f(y)\|=\aleph_{0}\).
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 \(\cong \aleph_0\).)

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