Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
317 \(\Rightarrow\) 14 Limitations on the Fraenkel-Mostowski method of independence proofs, Howard, P. 1973, J. Symbolic Logic
14 \(\Rightarrow\) 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
317:

Weak Sikorski Theorem:  If \(B\) is a complete, well orderable Boolean algebra and \(f\) is a homomorphism of the Boolean algebra \(A'\) into \(B\) where \(A'\) is a subalgebra of the Boolean algebra \(A\), then \(f\) can be extended to a homomorphism of \(A\) into \(B\).
14:

BPI: Every Boolean algebra has a prime ideal.

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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