Implications

We have the following indirect implication of form equivalence classes:

50 \(\Rightarrow\) 227
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
50 \(\Rightarrow\) 14 A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.
14 \(\Rightarrow\) 227 Variants of Rado's selection lemma and their applications, Rav, Y. 1977, Math. Nachr.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
50:

Sikorski's  Extension Theorem: Every homomorphism of a subalgebra \(B\) of a Boolean algebra \(A\) into a complete Boolean algebra \(B'\) can be extended to a homomorphism of \(A\) into \(B'\). Sikorski [1964], p. 141.

14:

BPI: Every Boolean algebra has a prime ideal.

227:

For all groups \(G\), if every finite subgroup of \(G\) can be fully ordered then \(G\) can be fully ordered.

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