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Hypothesis: HR 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 \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).)

Conclusion: HR 183-alpha:

There are no \(\aleph_{\alpha}\) minimal  sets.  That is, there are no sets \(X\) such that

1. \(|X|\) is incomparable with \(\aleph_{\alpha}\)
2. \(\aleph_{\beta}<|X|\) for every \(\beta < \alpha \) and
3. \(\forall Y\subseteq X, |Y|<\aleph_{\alpha}\) or \(|X-Y| <\aleph_{\alpha}\).

List of models where hypothesis is true and the conclusion is false:

Name	Statement
\(\cal N27\) Hickman's Model II	Let \(A\) be a set with cardinality\(\aleph_1\) such that \(A=\{(a_{\alpha},b_{\beta}) : \alpha < \omega, \beta< \omega_1\}\)

Code: 3

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