Implications

We have the following indirect implication of form equivalence classes:

317 \(\Rightarrow\) 370
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 \(\Rightarrow\) 370 clear

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.

370:

Weak Gelfand Extreme Point Theorem: If \(A\) is a non-trivial Gelfand algebra then the closed unit ball in the dual of \(A\) has an extreme point \(e\).  Morillon [1986].

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