Hypothesis: HR 86-alpha:
\(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(|X| = \aleph_{\alpha }\), then \(X\) has a choice function.
Conclusion: HR 321:
There does not exist an ordinal \(\alpha\) such that \(\aleph_{\alpha}\) is weakly compact and \(\aleph_{\alpha+1}\) is measurable.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M42\) Bull's Model | Let \(\cal M\) be a countable transitive model of \(ZFC +\) "There are uncountable regular cardinals \(\aleph_\alpha <\aleph_\beta < \aleph_\gamma\) such that \(\aleph_\alpha\) is \(\aleph_\gamma\)-supercompact; \(\aleph_\beta\) is the first measurable cardinal greater than \(\aleph_\alpha\); and \(\aleph_\gamma =|2^{\aleph_\beta}|\)." Using backward Easton forcing (which is due to Silver), Bull constructs a generic extension of \(\cal M\) |
Code: 3
Comments: