Axiom Of Choice

We have the following indirect implication of form equivalence classes:

270 \(\Rightarrow\) 132 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 \(\Rightarrow\) 132	Sequential compactness and the axiom of choice, Brunner, N. 1983b, Notre Dame J. Formal Logic

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.
132:	\(PC(\infty, <\aleph_0,\infty)\): Every infinite family of finite sets has an infinite subfamily with a choice function.

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