HOD dichotomy: Difference between revisions
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* For every singular \(\gamma\), \(\gamma\) is singular in \(L\) and \((\gamma^+)^L = \gamma^+\). |
* For every singular \(\gamma\), \(\gamma\) is singular in \(L\) and \((\gamma^+)^L = \gamma^+\). |
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The first option is equivalent to [[Zero sharp|\(0^\sharp\)]]'s existence, and the latter to its nonexistence, which therefore provides two equivalents. Similarly, the HOD dichotomy theorem says that, if \(\delta\) is an [[extendible]] cardinal, either: |
The first option is equivalent to [[Zero sharp|\(0^\sharp\)]]'s existence, and the latter to its nonexistence, which therefore provides two useful equivalents. Similarly, the (weak) HOD dichotomy theorem says that, if \(\delta\) is an [[extendible]] cardinal, either: |
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* Every [[Cofinality|regular]] cardinal greater than \(\delta\) is [[measurable]] in \(\mathrm{HOD}\). |
* Every [[Cofinality|regular]] cardinal greater than \(\delta\) is [[measurable]] in \(\mathrm{HOD}\). |
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