Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
270 \(\Rightarrow\) 62 Restricted versions of the compactness theorem, Kolany, A. 1991, Rep. Math. 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
270:

\(CT_{\hbox{fin}}\): The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs only in a finite number of formulas.

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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