Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
344 \(\Rightarrow\) 62 clear
62 \(\Rightarrow\) 121 clear
121 \(\Rightarrow\) 401 clear

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

Howard-Rubin Number Statement
344:

If \((E_i)_{i\in I}\) is a family of non-empty sets, then there is a family \((U_i)_{i\in I}\) such that \(\forall i\in I\), \(U_i\) is an ultrafilter on \(E_i\).

62:

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  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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