Axiom Of Choice

We have the following indirect implication of form equivalence classes:

257 \(\Rightarrow\) 423 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
257 \(\Rightarrow\) 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 \(\Rightarrow\) 39	clear
39 \(\Rightarrow\) 8	clear
8 \(\Rightarrow\) 9	Was sind und was sollen die Zollen?, Dedekind, [1888]
9 \(\Rightarrow\) 10	Zermelo's Axiom of Choice, Moore, 1982, 322
10 \(\Rightarrow\) 423	clear

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

Howard-Rubin Number	Statement
257:	\(Z(TR,P)\): Every transitive relation \((X,R)\) in which every partially ordered subset 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.
39:	\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.
8:	\(C(\aleph_{0},\infty)\):
9:	Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
10:	\(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function.
423:	\(\forall n\in \omega-\{o,1\}\), \(C(\aleph_0, n)\) : For every \(n\in \omega - \{0,1\}\), every denumerable set of \(n\) element sets has a choice function.

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