Axiom Of Choice

We have the following indirect implication of form equivalence classes:

1 \(\Rightarrow\) 140 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
1 \(\Rightarrow\) 140

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
1:	\(C(\infty,\infty)\): The Axiom of Choice: Every set of non-empty sets 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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