Timeline for What happens with large singular cardinals on the far side of the HOD dichotomy?
Current License: CC BY-SA 4.0
6 events
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| Jul 11, 2019 at 14:08 | comment | added | Gabe Goldberg | Monroe's forcing is the one I had in mind. Asaf is right, of course, that I should have been more careful. The preservation of extendibility seems to require that we started with a HOD-extendible, i.e., that for all $\alpha$, there is a $j : V_{\alpha+1}\to V_{j(\alpha)+1}$ such that $j(\text{HOD}\cap V_\alpha) = \text{HOD}\cap V_{j(\alpha)}$. (This follows from somewhat stronger large cardinals, for example Vopenka's Principle.) Given this I think the lifting argument is just like the $2^\kappa = \kappa^{++}$ case. | |
| Jul 11, 2019 at 7:35 | comment | added | Monroe Eskew | @AsafKaragila What might happen is that adding some Cohen subsets resurrects extendibility, simply because it’s the tail end of an appropriate iteration. I’m certain that this works if we want to, for example, have an extendible along with $2^\kappa=\kappa^{++}$ for all regular $\kappa$. | |
| Jul 11, 2019 at 7:01 | comment | added | Asaf Karagila♦ | @Monroe: If my memory serves me right, adding Cohen subsets above an extendible will definitely violate its extendibility. Again, I am not an expert on the topic, I was just under the impression that this is an open and difficult problem, one that requires a lot more than a "this is so trivial I'm not even gonna give you the details" approach. | |
| Jul 11, 2019 at 6:50 | comment | added | Monroe Eskew | @AsafKaragila I think it works if you iterate following the rule, “At stage $\alpha$, if the iteration so far has size $\leq \alpha$, find the least $\lambda_\alpha>\alpha$ such that $\lambda_\alpha$ is singular but regular in $HOD^V$, and add $\lambda_\alpha$ Cohen subsets of $\alpha$.” Then we would want to take a $j : V_\alpha \to V_\beta$ in $V$ with critical point $\delta$ such that $\alpha > $ the least blah as above, but also where $V_\alpha$ and $V_\beta$ reflect locally the true disparity between singulars and HOD-regulars. Now I’m stuck. | |
| Jul 10, 2019 at 23:11 | comment | added | Asaf Karagila♦ | Can you do that thing with the forcing? I always had the impression that extendible cardinals are very destructible under "large forcing" (that was my first question to my advisor when I was thinking about the Bristol model in light of the AC/HOD Conjectures). You make this off-hand remark, but this appears as an open question in Usuba's preprint on extendible cardinals and the mantle (arxiv.org/abs/1803.03944). What is this forcing that you want to use, and why does it preserve extendibility? | |
| Jul 10, 2019 at 14:21 | history | answered | Gabe Goldberg | CC BY-SA 4.0 |