Timeline for Is {Ø,{Ø},{Ø,{Ø}}, ... } the only known universe?
Current License: CC BY-SA 2.5
22 events
| when toggle format | what | by | license | comment | |
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| Jul 31, 2010 at 21:25 | comment | added | Andrej Bauer | I stand corrected. | |
| Jul 31, 2010 at 20:33 | comment | added | Peter LeFanu Lumsdaine | @Andrej: I don't think Grothendieck is at all dubious about infinite sets there! His definition of a universe is a set with certain properties; and he's saying that HF (see Joel's answer) is a universe, so he certainly believes it's a set! He's expressing the perfectly classical fact that assuming infinite sets exist doesn't imply you can find a (set) universe containing any. | |
| Jul 31, 2010 at 17:09 | answer | added | Andreas Blass | timeline score: 19 | |
| Jul 31, 2010 at 15:08 | vote | accept | fosco | ||
| Jul 31, 2010 at 14:05 | answer | added | Joel David Hamkins | timeline score: 21 | |
| Jul 31, 2010 at 13:29 | history | edited | Pete L. Clark |
added tag ag -- perhaps this will be a helpful reorientation
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| Jul 31, 2010 at 13:09 | answer | added | Pete L. Clark | timeline score: 4 | |
| Jul 31, 2010 at 13:05 | answer | added | nickname | timeline score: -2 | |
| Jul 31, 2010 at 12:27 | comment | added | fosco | I think I have clearly exposed my question; neverthless I apologize if it seemed an "ill-posed" problem. Actually, I simply made the A. Bauer's conclusion that "[Grothendieck is] not sure infinite sets actually exist", and I wanted to clarify if I guessed the right meaning between the lines of SGA... | |
| Jul 31, 2010 at 11:45 | comment | added | Pete L. Clark | I don't understand where all these querulous comments are coming from. Like PYG, I think the question is perfectly clear. And indeed, it seems to be understood and answered correctly below (modulo some trivial quibbling about the empty set). But also the sentence "I assume you are referring to Grothendieck universes" in the answer seems to suggest that there is some doubt in the matter -- but the OP refers to a specific passage from SGAIV and this pasage is talking about [what are now called] Grothendieck universes. So what's the problem? | |
| Jul 31, 2010 at 11:21 | comment | added | Pierre-Yves Gaillard | Dear Martin: What do you mean? Don't you think it's well known that Grothendieck's work is based on Bourbaki's treatise. | |
| Jul 31, 2010 at 11:11 | comment | added | Martin Brandenburg | @Pierre-Yves: Ok, but then both Grothendieck's assertion and the OP's question have to be made precise. | |
| Jul 31, 2010 at 10:58 | comment | added | Pierre-Yves Gaillard | Dear Andrej and Martin: First of all, the quotation is from Grothendieck and Verdier. They adhered to Bourbaki's set theory, and made their claim within this theory. From this viewpoint, I don't think the situation has changed since 1963. I think Grothendieck and Verdier knew already the contents of Andrej and Martin's comments, but didn't find them meaningful from their viewpoint. (Grothendieck certainly admits the existence of infinite sets ...) | |
| Jul 31, 2010 at 10:52 | answer | added | Anon | timeline score: 8 | |
| Jul 31, 2010 at 10:32 | comment | added | Andrej Bauer | You should understand Grothendieck as saying "I am not sure infinite sets actually exist". This of course is a matter of opinion, but most mathematicians don't have a problem with the existence of the set of natural numbers. | |
| Jul 31, 2010 at 10:31 | comment | added | Martin Brandenburg | Every inner model of ZF is a universe; for example Gödel's L. Perhaps the question has to be made more precise. | |
| Jul 31, 2010 at 10:26 | comment | added | Andrej Bauer | This question is very vague. What do you mean by "discover"? I think it is likely that you're taking Grothendieck's quotation a bit too literally. Sure there are other universes, for example a model of ZFC. Or the initial topos. Or the effective topos. OR a model of ZFC + measurable cardinals. And there are permutation models. And so on. | |
| Jul 31, 2010 at 10:24 | comment | added | Pierre-Yves Gaillard | The question looks very interesting to me. I don't see Ryan Budney's point. | |
| Jul 31, 2010 at 10:13 | comment | added | Ryan Budney | So you want to use some particular meaning for the word "universe"? Do you bring up Grothendieck because you want a particular form of the word that he uses in one of his unpublished manuscripts? | |
| Jul 31, 2010 at 10:02 | comment | added | fosco | I bring Grothendieck into this because the quotation I wrote comes from SGA... I'm only asking if, from 1963 to now, someone found out a universe different from {Ø,{Ø},{Ø,{Ø}}, ... }. | |
| Jul 31, 2010 at 10:00 | comment | added | Ryan Budney | I don't understand why you bring Grothendieck into this. I mean sure he wrote something but it's a repetition of others' ideas. I'm confused as to what the point is. | |
| Jul 31, 2010 at 9:08 | history | asked | fosco | CC BY-SA 2.5 |