Hypothesis: HR 307:
If \(m\) is the cardinality of the set of Vitali equivalence classes, then \(H(m) = H(2^{\aleph_0})\), where \(H\) is Hartogs aleph function and the {\it Vitali equivalence classes} are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in {\Bbb Q})(x-y=q)\).
Conclusion: HR 234:
There is a non-Ramsey set: There is a set \(A\) of infinite subsets of \(\omega\) such that for every infinite subset \(N\) of \(\omega\), \(N\) has a subset which is in \(A\) and a subset which is not in \(A\).
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 \(\Bbb R\) |
Code: 3
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