Timeline for Non smallness of the set of anafunctors without AC?
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Apr 2, 2017 at 11:45 | comment | added | David Roberts♦ | @SimonHenry actually, there's still a slight mismatch: even with autology, the best my result gets is equivalent to ZFA+!WISC, not ZF+!WISC, since toposes of sets with group actions give rise to models of material set theory with atoms. Joel Hamkins and I thought about finding a Jech-Sochor transfer theorem for class-forced models of ZFA, but we didn't get there. | |
| Apr 1, 2017 at 23:12 | comment | added | David Roberts♦ | @AsafKaragila I agree this should be written up, and from both points of view. In my mind, the property that you and aws have proved independent of ZF is the weakest possible choice principle (though I'd like to be proved wrong!). | |
| Apr 1, 2017 at 23:09 | comment | added | David Roberts♦ | @SimonHenry ah, excellent. I'd like to know how you go about this. I did talk to Mike Shulman a fair bit at CT2013 about proving autology in the framework I presented there and at IHES in 2015, but I was aiming for a general result and didn't quite get there. Please drop me an email. | |
| Apr 1, 2017 at 20:00 | comment | added | Asaf Karagila♦ | @Simon: I admit that I find it weird when people use my full name. But also, I'm quite happy with how much effort has been put into this page. On account of all the users involved, I think. I consider this to be a moderate success of MathOverflow (with the only thing better would be a paper resulting of the whole thing). | |
| Apr 1, 2017 at 14:51 | comment | added | Simon Henry | @DavidRoberts thanks for that remarks. You are right : I was struggling a bit to understand aws and Asaf Karagila constructions, but your comment made it totaly obivous to me that it works ! (also, maybe I'm wrong but I believe that the topos of action of a large group is "autological" in Mike Shulman sense as soon as the topos you started with is, and so in your paper you did prove that ECTS + unbounded selection/replacement is consistent witht the failure of WISC, which I guess is hence equivalent to Asaf Karagila result and not weaker as you said in the introduction.) | |
| Apr 1, 2017 at 3:42 | comment | added | Asaf Karagila♦ | @David: Yes! You should definitely do that! | |
| Apr 1, 2017 at 2:04 | vote | accept | Simon Henry | ||
| Mar 31, 2017 at 23:21 | comment | added | David Roberts♦ | @aws this might be able to be phrased more topos-theoretically, which is probably what Simon is more comfortable with, using techniques as in my paper dx.doi.org/10.1007/s11225-015-9603-6 (identical free version: arxiv.org/abs/1311.3074 - note that it was written in parallel with Asaf's paper on the same topic), using the large topological group (Z/2)^ORD in place of your (Z/2)^\kappa for inaccessible \kappa | |
| Mar 31, 2017 at 17:32 | comment | added | Asaf Karagila♦ | Okay, it took longer than expected. | |
| Mar 30, 2017 at 19:34 | comment | added | Asaf Karagila♦ | I will try to do it tonight. If not, then tomorrow morning. | |
| Mar 30, 2017 at 18:52 | comment | added | aws | I'm pretty sure that that part works now. I've just noticed the comments at the end of the question apply because a set admits a $\mathbb{Z}/2$-action satisfying the "saturated" condition given if and only if it has exactly 2 elements. So, I would be interested to see a sketch of the class forcing proof. | |
| Mar 30, 2017 at 18:12 | comment | added | Asaf Karagila♦ | If there you are correct, I'd be happy to supplement your answer with a sketch of a construction. | |
| Mar 30, 2017 at 18:11 | comment | added | aws | Yes; if I'm correct that would be sufficient to show local smallness is independent of $\mathbf{ZF}$. | |
| Mar 30, 2017 at 18:03 | comment | added | Asaf Karagila♦ | So you want a proper class of Russell sets, such that there is no set of "all Russell cardinals". This is doable. You don't need inaccessible cardinals or whatnot. I can sketch a class forcing argument if you want. | |
| Mar 30, 2017 at 9:48 | history | answered | aws | CC BY-SA 3.0 |