Axiom Of Choice

We have the following indirect implication of form equivalence classes:

23 \(\Rightarrow\) 250 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
23 \(\Rightarrow\) 151	clear
151 \(\Rightarrow\) 122	Russell's alternative to the axiom of choice, Howard, P. 1992, Z. Math. Logik Grundlagen Math. note-27
122 \(\Rightarrow\) 250	clear

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

Howard-Rubin Number	Statement
23:	\((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(\|\bigcup A\| = \aleph _{\alpha }\).
151:	\(UT(WO,\aleph_{0},WO)\) (\(U_{\aleph_{1}}\)): The union of a well ordered set of denumerable sets is well orderable. (If \(\kappa\) is a well ordered cardinal, see note 27 for \(UT(WO,\kappa,WO)\).)
122:	\(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function.
250:	\((\forall n\in\omega-\{0,1\})(C(WO,n))\): For every natural number \(n\ge 2\), every well ordered family of \(n\) element sets has a choice function.

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