Axiom Of Choice

We have the following indirect implication of form equivalence classes:

270 \(\Rightarrow\) 308-p 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 \(\Rightarrow\) 308-p	Maximal p-subgroups and the axiom of choice, Howard-Yorke-1987 [1987, 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.
308-p:	If \(p\) is a prime and if \(\{G_y: y\in Y\}\) is a set of finite groups, then the weak direct product \(\prod_{y\in Y}G_y\) has a maximal \(p\)-subgroup.

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