Implications

We have the following indirect implication of form equivalence classes:

151 \(\Rightarrow\) 48-K
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
151 \(\Rightarrow\) 122 Russell's alternative to the axiom of choice, Howard, P. 1992, Z. Math. Logik Grundlagen Math.
note-27
122 \(\Rightarrow\) 48-K clear

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

Howard-Rubin Number Statement
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.

48-K:

If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(WO,K)\): For every \(n\in K,\) \(C(WO,n)\).

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