Timeline for When must it be sets rather than proper classes, or vice-versa, outside of foundational mathematics?
Current License: CC BY-SA 3.0
13 events
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| Feb 4, 2013 at 9:16 | comment | added | Adam Epstein | There is also Dehornoy's work on braids and distributive algebra, which while not actually requiring large cardinals was somehow revealed by them. | |
| Feb 4, 2013 at 9:15 | comment | added | Adam Epstein | Asaf - That's just the sort of thing I'm trolling for :) The 'small object argument' mentioned by Martin involves a long-running transfinite recursion, but I gather that the whole point is that the small object hypothesis guarantees that the recursion (which might not teminate on its) can be shut off, at least for the purpose intended (construction of factorization systems). The cardinalities involved might well be large in comparison with those ordinarily encountered in topology, but not large in the sense of set theory. So actual large cardinals...that's intriguing! | |
| Feb 3, 2013 at 23:10 | comment | added | Asaf Karagila♦ | Adam, I was talking to a renowned set theorists a year and a half ago, and he said that he is going to a conference on homotopy theory. I was surprised, what a large cardinals expert has to do there? Apparently, he said, there are transfinite recursions which require very large cardinals to provably terminate. Inaccessible cardinals are so small they are "nearly finite" compared to the ones he mentioned, which is an interesting and very unexpected connection. | |
| Feb 3, 2013 at 18:07 | history | made wiki | Post Made Community Wiki by François G. Dorais | ||
| Feb 3, 2013 at 8:46 | comment | added | Adam Epstein | I also have the impression that now and again, various homotopy theoretic transfinite recursions can go on arbitrarily long. If this should somehow turn out to require inaccessible cardinals, the scenario of B) kicks in. A few years back I did see remarks of Feferman (who has himself proposed a conservative extension of ZFC with 'fake' universes, all secured by the Reflection principle, as a means of defusing size issues in category theory) concerning some potentially long-running construction of Rao.I had a look once, but I'm not expert enough to extract an opinion. | |
| Feb 3, 2013 at 8:40 | comment | added | Adam Epstein | I don't think I know enough homotopy theory to tell if this qualifies as an answer. What I do know is that when people give introductory lectures about derived categories, a few will own up to the set theoretic difficulties involved in the localization construction, but they too prefer not to dwell on the workaound. | |
| Feb 2, 2013 at 23:20 | comment | added | Tom Goodwillie | No, if I understood right, because the OP made it clear that he is not looking for that kind of answer. | |
| Feb 2, 2013 at 21:58 | comment | added | Jason Rute | Tom, would your example be something that should go in an answer? | |
| Feb 2, 2013 at 20:03 | comment | added | Tom Goodwillie | I imagine that in analysis one rarely if ever encounters the issues that Adam is asking about, because one rarely has much occasion to mention any proper classes. Well, you might mention the category of Banach spaces and so, implicitly, the "set" of all Banach spaces, but probably you won't be tempted to do anything illegal with it -- unlike the unnamed algebraists in the OP's Example A with their "set" of all stable curves. (And unlike me, when I invert the natural weak equivalences in the category of all functors from Top to Top. I have to bargain with the reader, like those algebraists.) | |
| Feb 2, 2013 at 18:20 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| Feb 2, 2013 at 17:52 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| Feb 2, 2013 at 17:39 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| Feb 2, 2013 at 17:26 | history | answered | Terry Tao | CC BY-SA 3.0 |