Implications

We have the following indirect implication of form equivalence classes:

311 \(\Rightarrow\) 223
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
311 \(\Rightarrow\) 313 clear
313 \(\Rightarrow\) 222 The Banach-Tarski Paradox, Wagon, [1985]
222 \(\Rightarrow\) 223 clear

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

Howard-Rubin Number Statement
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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).)

222:

There is a non-principal measure on \(\cal P(\omega)\).

223:

There is an infinite set \(X\) and a non-principal measure on \(\cal P(X)\).

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