Implications

We have the following indirect implication of form equivalence classes:

399 \(\Rightarrow\) 102
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
399 \(\Rightarrow\) 323 clear
323 \(\Rightarrow\) 62 note-70
62 \(\Rightarrow\) 102 The Axiom of Choice, Jech, 1973b, page 162 problem 11.12

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

Howard-Rubin Number Statement
399:

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

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.

102:

For all Dedekind finite cardinals \(p\) and \(q\), if \(p^{2} = q^{2}\) then \(p = q\). Jech [1973b], p 162 prob 11.12.

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