Implications

We have the following indirect implication of form equivalence classes:

85 \(\Rightarrow\) 285
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
85 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 285 On functions without fixed points, Wi'sniewski, K. 1973, Comment. Math. Prace Mat.

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

Howard-Rubin Number Statement
85:

\(C(\infty,\aleph_{0})\):  Every family of denumerable sets has  a choice function.  Jech [1973b] p 115 prob 7.13.

62:

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

285:

Let \(E\) be a set and \(f: E\to E\), then \(f\) has a fixed point if and only if \(E\) is not the union of three mutually disjoint sets \(E_1\), \(E_2\) and \(E_3\) such that \(E_i \cap f(E_i) = \emptyset\) for \(i=1, 2, 3\).

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