Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
322 \(\Rightarrow\) 324 clear
324 \(\Rightarrow\) 327 clear
327 \(\Rightarrow\) 358 clear

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

Howard-Rubin Number Statement
322:

\(KW(WO,\infty)\), The Kinna-Wagner Selection Principle for a well ordered family of sets: For every  well ordered set \(M\) there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\) then \(\emptyset\neq f(A)\subsetneq A\).  (See Form 15).

324:

\(KW(WO,WO)\), The Kinna-Wagner Selection Principle for a well ordered family of well orderable sets: For every well ordered set \(M\) of well orderable sets, there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)

327:

\(KW(WO,<\aleph_0)\),  The Kinna-Wagner Selection Principle for a well ordered family of finite sets: For every well ordered set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\)  then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)

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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