Timeline for Is "all categorical reasoning formally contradictory"?
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Aug 19, 2011 at 2:53 | comment | added | David Roberts♦ | A quick comment regarding derived functors: they form a finitely axiomatised first-order theory. Assuming one has derived functors is surely not too much worse than assuming one has a model of set theory. | |
| May 6, 2010 at 10:27 | vote | accept | José Figueroa-O'Farrill | ||
| May 6, 2010 at 1:05 | comment | added | Peter LeFanu Lumsdaine | @AndrewL: Grothendieck's universes axiom is a farily mild addition to ZFC; it's an easy consequence of most of the large cardinal axioms that set theorists happily throw around, and there's no particular reason to suspect them inconsistent. So he wasn't being irresponsible! The danger is that later mathematicians want to do without them, as they're both rather cumbersome and a bit hacky. Cartier's point is about this: we can't just ignore them; we either need to accept them, or think hard about what alternatives we're using. | |
| May 5, 2010 at 22:15 | comment | added | Kevin Buzzard | I should clarify. Firstly I was just joking when I said he said "oh sod it"---he of course never said that. Secondly the point I was trying to make is that I got the impression that rather than trying to formalise what he wanted to do in ZFC, he decided that in fact the fact that he was having trouble formalising what he wanted to do was a deficiency of ZFC, so he added a new axiom which fixed things because he didn't want to get bogged down here, he just wanted to get on with the alg geom. That was what I meant when I made the flippant remarks. | |
| May 5, 2010 at 21:20 | comment | added | The Mathemagician | If Grothendeik DID think something to that effect,it's really disappointing one of the greatest minds of the last century was that metaphysically irresponsible. Imagine what might have happened to mathematics if Cantor or Dedekind or any of the rigorizors of calculus had said to themselves,"Oh,sod it all,we can do Newtonian mechanics and that's what's important,not all this other rubbish about infinities." We need to face the fact that category theory and set theory don't work as well together to create a foundation as we'd like.Even if it results in several contrary solutions. | |
| May 5, 2010 at 12:33 | comment | added | Kevin Buzzard | Hmm, the "sod it all" bit might be editorialising. I can give you a reference---Grothendieck's Algiers notes from 65---where he doesn't say "sod it all" but does explain why he is going to introduce the notion of a universe in his first lecture---but unfortunately the link to the notes has been taken down by Leila Schneps, presumably because of the Grothendieck Jan 10 letter. | |
| May 5, 2010 at 11:28 | comment | added | Pete L. Clark | (P.S.: I find your guess very plausible. +1.) | |
| May 5, 2010 at 11:27 | comment | added | Pete L. Clark | "'Sod it all', thought Grothendieck"? Blimey, I never realized he spoke to himself like that. | |
| May 5, 2010 at 11:26 | comment | added | Wadim Zudilin | Your "guess" is just wonderful! And Cartier is The Last of the Bourbaki... | |
| May 5, 2010 at 10:56 | comment | added | Kevin Buzzard | I quoted this post here once before: cs.nyu.edu/pipermail/fom/2007-December/012359.html It's Bill Messing pointing out that one has to be careful. | |
| May 5, 2010 at 10:52 | history | answered | Kevin Buzzard | CC BY-SA 2.5 |