Implications

We have the following indirect implication of form equivalence classes:

317 \(\Rightarrow\) 141
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 \(\Rightarrow\) 141 Generalizing Konigs infinity lemma, Cowen, R.H. 1977b, Notre Dame J. Formal 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.

141:

[14 P(\(n\))] with \(n = 2\):  Let \(\{A(i): i\in I\}\) be a collection of sets such that \(\forall i\in I,\ |A(i)|\le 2\) and suppose \(R\) is a symmetric binary relation on \(\bigcup^{}_{i\in I} A(i)\) such that for all finite \(W\subseteq I\) there is an \(R\) consistent choice function for \(\{A(i): i \in W\}\). Then there is an \(R\) consistent choice function for \(\{A(i): i\in I\}\).

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