Implications

We have the following indirect implication of form equivalence classes:

258 \(\Rightarrow\) 377
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 \(\Rightarrow\) 39 clear
39 \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 9 Was sind und was sollen die Zollen?, Dedekind, [1888]
9 \(\Rightarrow\) 376 clear
376 \(\Rightarrow\) 377 Weak choice principles, De la Cruz, O. 1998a, Proc. Amer. Math. Soc.

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.

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.

376:

Restricted Kinna Wagner Principle:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) and a function \(f\) such that for every \(z\subseteq Y\), if \(|z| \ge 2\) then \(f(z)\) is a non-empty proper subset of \(z\).

377:

Restricted Ordering Principle:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) such that \(Y\) can be linearly ordered.

Comment:

Back