Axiom Of Choice

We have the following indirect implication of form equivalence classes:

68 \(\Rightarrow\) 33-n 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 \(\Rightarrow\) 61	clear
61 \(\Rightarrow\) 45-n	clear
45-n \(\Rightarrow\) 33-n	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.
61:	\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element sets has a choice function.
45-n:	If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function.
33-n:	If \(n\in\omega-\{0,1\}\), \(C(LO,n)\): Every linearly ordered set of \(n\) element sets has a choice function.

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