Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
323 \(\Rightarrow\) 62 note-70
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
323:

\(KW(\infty,WO)\), The Kinna-Wagner Selection Principle for a family of well orderable sets: For every 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.)

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

Comment:

Back