Implications

We have the following indirect implication of form equivalence classes:

317 \(\Rightarrow\) 412
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 \(\Rightarrow\) 410 note-23
410 \(\Rightarrow\) 411 clear
411 \(\Rightarrow\) 412 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.

410:

RC (Reflexive Compactness): The closed unit ball of a reflexive normed space is compact for the weak topology.

411:

RCuc (Reflexive Compactness for uniformly convex Banach spaces): The closed unit ball of a uniformly convex Banach space is compact for the weak topology.

412:

RCh (Reflexive Compactness for Hilbert spaces): The closed unit ball of a Hilbert space is compact for the weak topology.

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