Implications

We have the following indirect implication of form equivalence classes:

317 \(\Rightarrow\) 249
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 \(\Rightarrow\) 153 The Baire category property and some notions of compactness, Fossy, J. 1998, J. London Math. Soc.
153 \(\Rightarrow\) 10 The Baire category property and some notions of compactness, Fossy, J. 1998, J. London Math. Soc.
10 \(\Rightarrow\) 249

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.

153:

The closed unit ball of a Hilbert space is compact in the weak topology.

10:

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

249:

If \(T\) is an infinite tree in which every element has exactly 2 immediate successors then \(T\) has an infinite branch.

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