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 198:
For every set \(S\), if the only linearly orderable subsets of \(S\) are the finite subsets of \(S\), then either \(S\) is finite or \(S\) has an amorphous subset.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N2\) The Second Fraenkel Model | The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \). |
Code: 3
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