We have the following indirect implication of form equivalence classes:
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 \(\Rightarrow\) 88 | clear |
88 \(\Rightarrow\) 140 | 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. |
88: | \(C(\infty ,2)\): Every family of pairs has a choice function. |
140: | Let \(\Omega\) be the set of all (undirected) infinite cycles of reals (Graphs whose vertices are real numbers, connected, no loops and each vertex adjacent to exactly two others). Then there is a function \(f\) on \(\Omega \) such that for all \(s\in\Omega\), \(f(s)\) is a direction along \(s\). |
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