Axiom Of Choice

We have the following indirect implication of form equivalence classes:

259 \(\Rightarrow\) 196-alpha given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
259 \(\Rightarrow\) 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 \(\Rightarrow\) 86-alpha	clear
86-alpha \(\Rightarrow\) 196-alpha	Successive large cardinals, Bull Jr., E. L. 1978, Ann. Math. Logic

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

Howard-Rubin Number	Statement
259:	\(Z(TR\&C,W)\): If \((X,R)\) is a transitive and connected relation in which every well ordered subset has an upper bound, then \((X,R)\) 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.
86-alpha:	\(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(\|X\| = \aleph_{\alpha }\), then \(X\) has a choice function.
196-alpha:	\(\aleph_{\alpha}\) and \(\aleph_{\alpha+1}\) are not both measurable.

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