Implications

We have the following indirect implication of form equivalence classes:

213 \(\Rightarrow\) 270
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

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.

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