Implications

We have the following indirect implication of form equivalence classes:

202 \(\Rightarrow\) 401
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
202 \(\Rightarrow\) 121 clear
121 \(\Rightarrow\) 401 clear

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

Howard-Rubin Number Statement
202:

\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function.

121:

\(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function.

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 \(\emptyset\neq f(A)\subsetneq A\).

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