Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
51 \(\Rightarrow\) 337	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
337 \(\Rightarrow\) 92	clear
92 \(\Rightarrow\) 170	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
170 \(\Rightarrow\) 93	Zermelo's Axiom of Choice, Moore, [1982]

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

Howard-Rubin Number	Statement
51:	Cofinality Principle: Every linear ordering has a cofinal sub well ordering. Sierpi\'nski [1918], p 117.
337:	\(C(WO\), uniformly linearly ordered): If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\).
92:	\(C(WO,{\Bbb R})\): Every well ordered family of non-empty subsets of \({\Bbb R}\) has a choice function.
170:	\(\aleph_{1}\le 2^{\aleph_{0}}\).
93:	There is a non-measurable subset of \({\Bbb R}\).

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