Axiom Of Choice

We have the following indirect implication of form equivalence classes:

68 \(\Rightarrow\) 18 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
68 \(\Rightarrow\) 62	Subgroups of a free group and the axiom of choice, Howard, P. 1985, J. Symbolic Logic
62 \(\Rightarrow\) 10	clear
10 \(\Rightarrow\) 80	clear
80 \(\Rightarrow\) 18	clear

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

Howard-Rubin Number	Statement
68:	Nielsen-Schreier Theorem: Every subgroup of a free group is free. Jech [1973b], p 12.
62:	\(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function.
10:	\(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function.
80:	\(C(\aleph_{0},2)\): Every denumerable set of pairs has a choice function.
18:	\(PUT(\aleph_{0},2,\aleph_{0})\): The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.

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