Implications

We have the following indirect implication of form equivalence classes:

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

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

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

Howard-Rubin Number Statement
399:

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

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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