Axiom Of Choice

We have the following indirect implication of form equivalence classes:

5 \(\Rightarrow\) 5 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
5 \(\Rightarrow\) 5	An independence result concerning the axiom of choice, Sageev, G. 1975, Ann. Math. Logic

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

Howard-Rubin Number	Statement
5:	\(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function.
5:	\(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function.

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