Implications

We have the following indirect implication of form equivalence classes:

68 \(\Rightarrow\) 378
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 \(\Rightarrow\) 378 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.

378:

Restricted Choice for Families of Well Ordered Sets:  For every infinite set \(X\) there is an infinite subset \(Y\) of \(X\) such that the family of non-empty well orderable subsets of \(Y\) has a choice function.

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