Implications

We have the following indirect implication of form equivalence classes:

391 \(\Rightarrow\) 402
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
391 \(\Rightarrow\) 392 clear
392 \(\Rightarrow\) 394 clear
394 \(\Rightarrow\) 402 clear

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

Howard-Rubin Number Statement
391:

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

392:

\(C(LO,LO)\): Every linearly ordered set of linearly orderable sets has a choice function.

394:

\(C(WO,LO)\): Every well ordered set of non-empty linearly orderable sets has a choice function.

402:

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

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