Timeline for What strengthenings of measurability do the Mostowski collapses of ultrapowers possess?
Current License: CC BY-SA 3.0
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| Nov 7, 2017 at 23:51 | vote | accept | Keith Millar | ||
| Nov 6, 2017 at 23:52 | comment | added | Keith Millar | Aha, @JoelDavidHamkins, this new comment that you have provided is precisely my question. Thank you for taking the time to understand my poorly written question. | |
| Nov 6, 2017 at 17:23 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Nov 6, 2017 at 16:35 | comment | added | Joel David Hamkins | I was interpreting the question differently, and more simply: which large cardinal properties can be realized by ultrapower embeddings, in place of the arbitrary embeddings one might state. If you define $\kappa$ is supercompact to be: for every $\theta$ there is $j:V\to M$ with critical point $\kappa$ and $M^\theta\subset M$, then indeed it is equivalent to also insist that $j$ is an ultrapower embedding. In the case of strong, etc., this is not an equivalence, but meanwhile it is relatively consistent that strongness is witnessed by ultrapowers, e.g., by supercompactness ultrapowers. | |
| Nov 6, 2017 at 16:32 | comment | added | Miha Habič | @AndreasBlass Yes, being supercompact is properly a $\Pi_3$ statement. It cannot be $\Sigma_2$, because then the existence of a supercompact cardinal would reflect below the least supercompact cardinal, leading to big problems. | |
| Nov 6, 2017 at 16:30 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Nov 6, 2017 at 16:27 | comment | added | Andreas Blass | Although $\theta$-supercompactness can be witnessed by an ultrapower as you said, full supercompactness seems to need a lot of utrapowers --- or am I missing something here. | |
| Nov 6, 2017 at 16:23 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |