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 330:
\(MC(WO,WO)\): For every well ordered set \(X\) of well orderable sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). (See Form 67.)
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N15\) Brunner/Howard Model I | \(A=\{a_{i,\alpha}: i\in\omega\wedge\alpha\in\omega_1\}\) |
| \(\cal N41\) Another variation of \(\cal N3\) | \(A=\bigcup\{B_n; n\in\omega\}\)is a disjoint union, where each \(B_n\) is denumerable and ordered like therationals by \(\le_n\) |
Code: 3
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