Implications

We have the following indirect implication of form equivalence classes:

400 \(\Rightarrow\) 389
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
400 \(\Rightarrow\) 401 clear
401 \(\Rightarrow\) 327 clear
327 \(\Rightarrow\) 250 clear
250 \(\Rightarrow\) 80 clear
80 \(\Rightarrow\) 389 clear

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

Howard-Rubin Number Statement
400:

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

401:

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

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.)

250:

\((\forall n\in\omega-\{0,1\})(C(WO,n))\): For every natural number \(n\ge 2\), every well ordered family of \(n\) element sets has a choice function.

80:

\(C(\aleph_{0},2)\):  Every denumerable set of  pairs has  a  choice function.

389:

\(C(\aleph_0,2,\cal P({\Bbb R}))\): Every denumerable family of two element subsets of \(\cal P({\Bbb R})\) has a choice function.  \ac{Keremedis} \cite{1999b}.

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