Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
256 \(\Rightarrow\) 255	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
255 \(\Rightarrow\) 260	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
260 \(\Rightarrow\) 40	Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
40 \(\Rightarrow\) 337	clear
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
256:	\(Z(P,F)\): Every partially ordered set \((X,R)\) in which every forest \(A\) has an upper bound, has a maximal element.
255:	\(Z(D,R)\): Every directed relation \((P,R)\) in which every ramified subset \(A\) has an upper bound, has a maximal element.
260:	\(Z(TR\&C,P)\): If \((X,R)\) is a transitive and connected relation in which every partially ordered subset has an upper bound, then \((X,R)\) has a maximal element.
40:	\(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.
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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