Axiom Of Choice

We have the following indirect implication of form equivalence classes:

421 \(\Rightarrow\) 38 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 \(\Rightarrow\) 5	clear
5 \(\Rightarrow\) 38	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.

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.
5:	\(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function.
38:	\({\Bbb R}\) is not the union of a countable family of countable sets.

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