Implications

We have the following indirect implication of form equivalence classes:

258 \(\Rightarrow\) 294
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
258 \(\Rightarrow\) 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 \(\Rightarrow\) 231 Abzählbarkeit und Wohlordenbarkeit, Felgner, U. 1974, Comment. Math. Helv.
231 \(\Rightarrow\) 294 clear

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

Howard-Rubin Number Statement
258:

\(Z(D,L)\): Every directed relation \((X,R)\) in which linearly ordered subsets have upper bounds, 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.

231:

\(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets is well orderable.

294:

Every linearly ordered \(W\)-set is well orderable.

Comment:

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