Implications

We have the following indirect implication of form equivalence classes:

256 \(\Rightarrow\) 92
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

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.

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