Axiom Of Choice

We have the following indirect implication of form equivalence classes:

264 \(\Rightarrow\) 47-n given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
264 \(\Rightarrow\) 202	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
202 \(\Rightarrow\) 121	clear
121 \(\Rightarrow\) 33-n	clear
33-n \(\Rightarrow\) 47-n	clear

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

Howard-Rubin Number	Statement
264:	\(H(C,P)\): Every connected relation \((X,R)\) contains a \(\subseteq\)-maximal partially ordered set.
202:	\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has a choice function.
121:	\(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function.
33-n:	If \(n\in\omega-\{0,1\}\), \(C(LO,n)\): Every linearly ordered set of \(n\) element sets has a choice function.
47-n:	If \(n\in\omega-\{0,1\}\), \(C(WO,n)\): Every well ordered collection of \(n\)-element sets has a choice function.

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