Implications

We have the following indirect implication of form equivalence classes:

15 \(\Rightarrow\) 356
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
15 \(\Rightarrow\) 323 clear
323 \(\Rightarrow\) 356 clear

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

Howard-Rubin Number Statement
15:

\(KW(\infty,\infty)\) (KW), The Kinna-Wagner Selection Principle: For every  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 81(\(n\)).  

323:

\(KW(\infty,WO)\), The Kinna-Wagner Selection Principle for a family of well orderable sets: For every 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.)

356:

  \(KW(\infty,\aleph_0)\), The Kinna-Wagner Selection Principle for a family of denumerable sets: For every set \(M\) of denumerable 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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