Hypothesis: HR 65:
The Krein-Milman Theorem: Let \(K\) be a compact convex set in a locally convex topological vector space \(X\). Then \(K\) has an extreme point. (An extreme point is a point which is not an interior point of any line segment which lies in \(K\).) Rubin, H./Rubin, J. [1985] p. 177.
Conclusion: HR 344:
If \((E_i)_{i\in I}\) is a family of non-empty sets, then there is a family \((U_i)_{i\in I}\) such that \(\forall i\in I\), \(U_i\) is an ultrafilter on \(E_i\).
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: 5
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