Implications

We have the following indirect implication of form equivalence classes:

355 \(\Rightarrow\) 373-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
355 \(\Rightarrow\) 358 clear
358 \(\Rightarrow\) 288-n clear
288-n \(\Rightarrow\) 373-n clear

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

Howard-Rubin Number Statement
355:

\(KW(\aleph_0,\infty)\), The Kinna-Wagner Selection Principle for a denumerable family of sets: For every denumerable set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\).

358:

\(KW(\aleph_0,<\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of finite sets: For every denumerable set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\).

288-n:

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

373-n:

(For \(n\in\omega\), \(n\ge 2\).) \(PC(\aleph_0,n,\infty)\): Every denumerable set of \(n\)-element sets has an infinite subset with a choice function.

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