Implications

We have the following indirect implication of form equivalence classes:

327 \(\Rightarrow\) 48-K
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
327 \(\Rightarrow\) 250 clear
250 \(\Rightarrow\) 48-K clear

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

Howard-Rubin Number Statement
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.

48-K:

If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(WO,K)\): For every \(n\in K,\) \(C(WO,n)\).

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