Implications

We have the following indirect implication of form equivalence classes:

324 \(\Rightarrow\) 80
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
324 \(\Rightarrow\) 327 clear
327 \(\Rightarrow\) 250 clear
250 \(\Rightarrow\) 80 clear

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

Howard-Rubin Number Statement
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.)

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.

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