Implications

We have the following indirect implication of form equivalence classes:

213 \(\Rightarrow\) 271-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
213 \(\Rightarrow\) 85 clear
85 \(\Rightarrow\) 270 Restricted versions of the compactness theorem, Kolany, A. 1991, Rep. Math. Logic
270 \(\Rightarrow\) 271-n Restricted versions of the compactness theorem, Kolany, A. 1991, Rep. Math. Logic

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

Howard-Rubin Number Statement
213:

\(C(\infty,\aleph_{1})\): If \((\forall y\in X)(|y| = \aleph_{1})\) then \(X\) has a choice function.

85:

\(C(\infty,\aleph_{0})\):  Every family of denumerable sets has  a choice function.  Jech [1973b] p 115 prob 7.13.

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.

271-n:

If \(n\in\omega-\{0,1\}\), \(CT_{n}\): The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs in at most \(n\) formulas.

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