Implications

We have the following indirect implication of form equivalence classes:

422-n \(\Rightarrow\) 389
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
422-n \(\Rightarrow\) 111 clear
111 \(\Rightarrow\) 80 clear
80 \(\Rightarrow\) 389 clear

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

Howard-Rubin Number Statement
422-n:

\(UT(WO,n,WO)\), \(n\in \omega-\{0,1\}\): The union of a well ordered set of \(n\) element sets can be well ordered.

111:

\(UT(WO,2,WO)\): The union of an infinite well ordered set of 2-element sets is an infinite well ordered set.

80:

\(C(\aleph_{0},2)\):  Every denumerable set of  pairs has  a  choice function.

389:

\(C(\aleph_0,2,\cal P({\Bbb R}))\): Every denumerable family of two element subsets of \(\cal P({\Bbb R})\) has a choice function.  \ac{Keremedis} \cite{1999b}.

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