Axiom Of Choice

We have the following indirect implication of form equivalence classes:

323 \(\Rightarrow\) 93 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
323 \(\Rightarrow\) 62	note-70
62 \(\Rightarrow\) 61	clear
61 \(\Rightarrow\) 88	clear
88 \(\Rightarrow\) 93	The Axiom of Choice, Jech, 1973b, page 7 problem 10

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

Howard-Rubin Number	Statement
323:	\(KW(\infty,WO)\), The Kinna-Wagner Selection Principle for a family of well orderable sets: For every set \(M\) of well orderable sets there is a function \(f\) such that for all \(A\in M\), if \(\|A\| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)
62:	\(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function.
61:	\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element sets has a choice function.
88:	\(C(\infty ,2)\): Every family of pairs has a choice function.
93:	There is a non-measurable subset of \({\Bbb R}\).

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