Implications

We have the following indirect implication of form equivalence classes:

421 \(\Rightarrow\) 357
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
421 \(\Rightarrow\) 338 clear
338 \(\Rightarrow\) 32 clear
32 \(\Rightarrow\) 357 clear

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

Howard-Rubin Number Statement
421:   \(UT(\aleph_0,WO,WO)\): The union of a denumerable set of well orderable sets can be well ordered.
338:

\(UT(\aleph_0,\aleph_0,WO)\): The union of a denumerable number of denumerable sets is well orderable.

32:

\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets  has a choice function.

357:

\(KW(\aleph_0,\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of denumerable sets: For every denumerable set \(M\) of denumerable sets 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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