HOD dichotomy: Difference between revisions
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* There is a regular cardinal \(\gamma > \delta\) which is not \(\omega\)-strongly measurable in HOD. |
* There is a regular cardinal \(\gamma > \delta\) which is not \(\omega\)-strongly measurable in HOD. |
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Then we have a strong dichotomy: if \(\delta\) is an |
Then we have a strong dichotomy: if \(\delta\) is an extendible cardinal, either: |
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* Every [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. |
* Every [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. |
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* No [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. |
* No [[Cofinality|regular]] cardinal greater than \(\delta\) is \(\omega\)-strongly measurable in HOD. |
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None of these three statements are particularly hard to prove. The HOD hypothesis says that there is a proper class of cardinals \(\lambda\) which are not \(\omega\)-strongly measurable in HOD: therefore, if there is an extendible cardinal and the HOD hypothesis holds, then \(\mathrm{HOD}\) is close to \(V\). |
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Neither of these three statements are particularly hard to prove. |
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