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 285:
Let \(E\) be a set and \(f: E\to E\), then \(f\) has a fixed point if and only if \(E\) is not the union of three mutually disjoint sets \(E_1\), \(E_2\) and \(E_3\) such that \(E_i \cap f(E_i) = \emptyset\) for \(i=1, 2, 3\).
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N22(p)\) Makowski/Wi\'sniewski/Mostowski Model | (Where \(p\) is aprime) Let \(A=\bigcup\{A_i: i\in\omega\}\) where The \(A_i\)'s are pairwisedisjoint and each has cardinality \(p\) |
Code: 3
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