Axiom Of Choice

We have the following indirect implication of form equivalence classes:

231 \(\Rightarrow\) 357 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
231 \(\Rightarrow\) 32	clear
32 \(\Rightarrow\) 357	clear

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

Howard-Rubin Number	Statement
231:	\(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets is well orderable.
32:	\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets has a choice function.
357:	\(KW(\aleph_0,\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of denumerable sets: For every denumerable set \(M\) of denumerable sets there is a function \(f\) such that for all \(A\in M\), if \(\|A\| > 1\) then \(\emptyset\neq f(A)\subsetneq A\).

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