Implications

We have the following indirect implication of form equivalence classes:

270 \(\Rightarrow\) 216
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 \(\Rightarrow\) 10 clear
10 \(\Rightarrow\) 216

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.

10:

\(C(\aleph_{0},< \aleph_{0})\):  Every denumerable family of non-empty finite sets has a choice function.

216:

Every infinite tree has either an infinite chain or an infinite antichain.

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