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
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| Feb 4, 2013 at 17:52 | comment | added | David White | In a related vein, there is something called Bousfield localization, which takes a model category and a set of morphisms, and spits out another model category where that set is contained in the weak equivalences (these are the maps which can be inverted in my comment above). In this case you really need a set, not a class, unless you're willing to invoke Vopenka's Principle from set theory. You can find this discussed on MO a lot, especially by Harry Gindi. Recent work of Chorny creates hypotheses on model categories which let you localize at a class ("class-combinatorial") | |
| Feb 4, 2013 at 17:50 | comment | added | David White | Hi Adam. I don't really have time for a full answer, but one place is when you're localizing a collection $S$ of morphisms in a category (I saw you mentioned this on a comment to Terry Tao below). The set theoretic issue is that constructing the localization you end up trying to place an equivalence relation on a class rather than a set (the class of zigzags $A\gets \bullet \to \dots \gets \bullet \to B$ where the backwards arrows are in $S$). The fix is to invent model categories. See also: mathoverflow.net/questions/92929 | |
| Feb 3, 2013 at 16:31 | answer | added | Martin Brandenburg | timeline score: 15 | |
| Feb 3, 2013 at 16:30 | answer | added | Colin McLarty | timeline score: 11 | |
| Feb 3, 2013 at 13:35 | answer | added | Joel David Hamkins | timeline score: 22 | |
| Feb 3, 2013 at 12:43 | comment | added | Adam Epstein | (continued) However if Holmes is correct and NF is consistent, then the pragmatic reasons for the restriction to wellfounded sets - which was always artificial - evaporates and we start having to ask seriously whether these collections can be sets according to NF or a consistent extension thereof." | |
| Feb 3, 2013 at 12:43 | comment | added | Adam Epstein | Thomas Forster writes: It is clear that most (and I suspect practically all) of these results that say that something cannot be a set are really results that say that that thing cannot be a wellfounded set. Antifoundation axioms don't change this, of course, since there are senses in which they give you the same mathematics - the new sets they give are all small. | |
| Feb 3, 2013 at 11:34 | history | edited | Adam Epstein | CC BY-SA 3.0 |
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| Feb 3, 2013 at 9:47 | history | edited | Adam Epstein | CC BY-SA 3.0 |
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| Feb 3, 2013 at 9:18 | history | edited | Adam Epstein | CC BY-SA 3.0 |
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| Feb 2, 2013 at 19:34 | history | edited | Adam Epstein | CC BY-SA 3.0 |
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| Feb 2, 2013 at 18:02 | history | edited | Adam Epstein | CC BY-SA 3.0 |
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| Feb 2, 2013 at 17:51 | comment | added | Adam Epstein | Thanks Graham. (Probably at some point I should scrub these meta comments.) | |
| Feb 2, 2013 at 17:40 | history | edited | Graham Leuschke | CC BY-SA 3.0 |
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| Feb 2, 2013 at 17:27 | history | edited | Adam Epstein | CC BY-SA 3.0 |
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| Feb 2, 2013 at 17:26 | answer | added | Terry Tao | timeline score: 24 | |
| Feb 2, 2013 at 17:21 | history | edited | Adam Epstein | CC BY-SA 3.0 |
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| Feb 2, 2013 at 17:11 | history | edited | Adam Epstein | CC BY-SA 3.0 |
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| Feb 2, 2013 at 16:55 | history | edited | Adam Epstein | CC BY-SA 3.0 |
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| Feb 2, 2013 at 16:47 | history | edited | Adam Epstein | CC BY-SA 3.0 |
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| Feb 2, 2013 at 16:36 | history | asked | Adam Epstein | CC BY-SA 3.0 |