Axiom Of Choice

We have the following indirect implication of form equivalence classes:

151 \(\Rightarrow\) 389 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
151 \(\Rightarrow\) 122	Russell's alternative to the axiom of choice, Howard, P. 1992, Z. Math. Logik Grundlagen Math. note-27
122 \(\Rightarrow\) 10	clear
10 \(\Rightarrow\) 80	clear
80 \(\Rightarrow\) 389	clear

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

Howard-Rubin Number	Statement
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.
10:	\(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function.
80:	\(C(\aleph_{0},2)\): Every denumerable set of pairs has a choice function.
389:	\(C(\aleph_0,2,\cal P({\Bbb R}))\): Every denumerable family of two element subsets of \(\cal P({\Bbb R})\) has a choice function. \ac{Keremedis} \cite{1999b}.

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