Hypothesis: HR 369:
If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\).
Conclusion: HR 169:
There is an uncountable subset of \({\Bbb R}\) without a perfect subset.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M5(\aleph)\) Solovay's Model | An inaccessible cardinal \(\aleph\) is collapsed to \(\aleph_1\) in the outer model and then \(\cal M5(\aleph)\) is the smallest model containing the ordinals and \(\Bbb R\) |
| \(\cal M12(\aleph)\) Truss' Model I | This is a variation of Solovay's model, <a href="/models/Solovay-1">\(\cal M5(\aleph)\)</a> in which \(\aleph\) is singular |
Code: 3
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