Implications

We have the following indirect implication of form equivalence classes:

260 \(\Rightarrow\) 48-K
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
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 \(\Rightarrow\) 122 clear
122 \(\Rightarrow\) 48-K clear

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

Howard-Rubin Number Statement
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.

122:

\(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function.

48-K:

If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(WO,K)\): For every \(n\in K,\) \(C(WO,n)\).

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