Implications

We have the following indirect implication of form equivalence classes:

264 \(\Rightarrow\) 24
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
264 \(\Rightarrow\) 202 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
202 \(\Rightarrow\) 40 clear
40 \(\Rightarrow\) 39 clear
39 \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 24 clear

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

Howard-Rubin Number Statement
264:

\(H(C,P)\): Every connected relation \((X,R)\) contains a \(\subseteq\)-maximal partially ordered set.

202:

\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function.

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:

\(C(\aleph_{0},\infty)\):

24:

\(C(\aleph_0,2^{(2^{\aleph_0})})\): Every denumerable collection of non-empty sets each with power \(2^{(2^{\aleph_{0}})}\) has a choice function.

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