Implications

We have the following indirect implication of form equivalence classes:

50 \(\Rightarrow\) 313
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 \(\Rightarrow\) 311 The Banach-Tarski Paradox, Wagon, [1985]
311 \(\Rightarrow\) 313 clear

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.

311:

Abelian groups are amenable. (\(G\) is amenable if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G)=1\) and \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).)

313:

\(\Bbb Z\) (the set of integers under addition) is amenable.  (\(G\) is {\it amenable} if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G) = 1\) and \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).)

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