Implications

We have the following indirect implication of form equivalence classes:

262 \(\Rightarrow\) 322
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
262 \(\Rightarrow\) 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 \(\Rightarrow\) 322 clear

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

Howard-Rubin Number Statement
262:

\(Z(TR,R)\): Every transitive relation \((X,R)\) in which every ramified subset \(A\) 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.

322:

\(KW(WO,\infty)\), The Kinna-Wagner Selection Principle for a well ordered family of sets: For every  well ordered set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).  (See Form 15).

Comment:

Back