Implications

We have the following indirect implication of form equivalence classes:

23 \(\Rightarrow\) 327
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
23 \(\Rightarrow\) 151 clear
151 \(\Rightarrow\) 122 Russell's alternative to the axiom of choice, Howard, P. 1992, Z. Math. Logik Grundlagen Math.
note-27
122 \(\Rightarrow\) 327 clear

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

Howard-Rubin Number Statement
23:

\((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\).

151:

\(UT(WO,\aleph_{0},WO)\) (\(U_{\aleph_{1}}\)): The union of a well ordered set of denumerable sets is well  orderable. (If \(\kappa\) is a well ordered cardinal, see note 27 for \(UT(WO,\kappa,WO)\).)

122:

\(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function.

327:

\(KW(WO,<\aleph_0)\),  The Kinna-Wagner Selection Principle for a well ordered family of finite sets: For every well ordered set \(M\) of finite 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.)

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