Timeline for On the large cardinals foundations of categories
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 14, 2013 at 17:17 | comment | added | Zhen Lin | Perhaps, if all you are doing is really just bookkeeping. But for other purposes (e.g. the one considered in my preprint) it is absolutely crucial that no new functions be added. | |
| May 14, 2013 at 15:14 | comment | added | Asaf Karagila♦ | Zhen Lin, the point of the question was that if you are doing some bookkeeping, then it seems possible to reduce the inaccessible to this chain of models. Even if you do add sets or subsets, since you know which model they came from you can return there when you need to. Andrej answer is somewhat affirmative to this idea, although I still haven't processed it through and through. | |
| May 14, 2013 at 14:26 | comment | added | Asaf Karagila♦ | Stefan, that has crossed my mind, but for some reason (I blame staying up all night) I decided it's fine. Thanks for pointing that out. | |
| May 14, 2013 at 9:44 | comment | added | Stefan Geschke | Asaf, can you say more precisly what you mean by end-extension? Even if your models are, say, countable $L_{\alpha}$'s, $\omega$ is still getting more subsets as you go up the chain. If you want a chain of $L_\alpha$'s where this does not happen, you want them to be closed under taking constructible subsets of elements. But then maybe you also want the $\alpha$'s to be regular so that you don't get new cofinal maps later on. And voila, here is your inaccessible cardinal (in $L$, at least). | |
| May 14, 2013 at 8:22 | comment | added | Zhen Lin | @Asaf A chain of end-extensions does sound better, although I am unfamiliar with the technical definition of that. | |
| May 14, 2013 at 7:44 | comment | added | Asaf Karagila♦ | Zhen Lin, one can require that the models are end-extensions (which is equiconsistent with the chain of models, I believe, if we take them all to satisfy $V=L$ as well, for example). Would that put your mind at ease? | |
| May 14, 2013 at 7:25 | comment | added | Stefan Geschke | Zhen Lin's comment is spot on, I think. You don't want new subsets of old sets show up as you go from one model to the next! | |
| May 14, 2013 at 7:21 | comment | added | Zhen Lin | As a practising categorist, let me emphasise how mind-bending it is to work in an arbitrary chain of models: when I think about expanding the universe, I don't expect cardinals to suddenly collapse together, or for sets to gain new subsets, or any of the other things that set theorists are used to! Rather, my basic minimum criterion is that hom-sets are preserved, and that implies there are no new functions and no new subsets. | |
| May 14, 2013 at 6:52 | comment | added | Asaf Karagila♦ | Well, if you work within the first model, then your categories are elements of the second model, and so on and so forth. As I remarked in the final paragraph, it's more than reasonable that large cardinals just simplify things; but can we get away with that much less? And in this breath, let me point out that the distance between a proper class of inaccessible, and Vopenka's principle is quite the huge gap in strength, and philosophically much more than the gap between the sequence of models and universes. | |
| May 14, 2013 at 6:46 | history | answered | Stefan Geschke | CC BY-SA 3.0 |