Implications

We have the following indirect implication of form equivalence classes:

380 \(\Rightarrow\) 358
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
380 \(\Rightarrow\) 132 clear
132 \(\Rightarrow\) 10 Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung
10 \(\Rightarrow\) 358 clear

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

Howard-Rubin Number Statement
380:

\(PC(\infty,WO,\infty)\):  For every infinite family of non-empty well orderable sets, there is an infinite subfamily \(Y\) of \(X\) which has a choice function.

132:

\(PC(\infty, <\aleph_0,\infty)\):  Every infinite family of finite  sets has an infinite subfamily with a choice function.

10:

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

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 \(\emptyset\neq f(A)\subsetneq A\).

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