Axiom Of Choice

We have the following indirect implication of form equivalence classes:

68 \(\Rightarrow\) 285 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 \(\Rightarrow\) 285	On functions without fixed points, Wi'sniewski, K. 1973, Comment. Math. Prace Mat.

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.
285:	Let \(E\) be a set and \(f: E\to E\), then \(f\) has a fixed point if and only if \(E\) is not the union of three mutually disjoint sets \(E_1\), \(E_2\) and \(E_3\) such that \(E_i \cap f(E_i) = \emptyset\) for \(i=1, 2, 3\).

Comment:

Back