Implications

We have the following indirect implication of form equivalence classes:

4 \(\Rightarrow\) 288-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
4 \(\Rightarrow\) 9 clear
9 \(\Rightarrow\) 10 Zermelo's Axiom of Choice, Moore, 1982, 322
10 \(\Rightarrow\) 288-n clear

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

Howard-Rubin Number Statement
4:

Every infinite set is the union of some disjoint family of denumerable subsets. (Denumerable means \(\cong \aleph_0\).)

9:

Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.

10:

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

288-n:

If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function.

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