Implications

We have the following indirect implication of form equivalence classes:

68 \(\Rightarrow\) 122
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 \(\Rightarrow\) 121 clear
121 \(\Rightarrow\) 122 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.

121:

\(C(LO,<\aleph_{0})\): Every linearly ordered set of non-empty finite sets has a choice function.

122:

\(C(WO,<\aleph_{0})\): Every well ordered set of non-empty finite sets has a choice function.

Comment:

Back