Timeline for Where in ordinary math do we need unbounded separation and replacement?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 11, 2013 at 10:51 | comment | added | Benedict Eastaugh | @FredRohrer according to Mathias, replacement did not appear in the older editions of Bourbaki, but was added later. dpmms.cam.ac.uk/~ardm/bourbaki.pdf | |
| Feb 11, 2013 at 6:27 | comment | added | Fred Rohrer | It is not true that Bourbaki's set theory "does not include the axiom [scheme] of replacement". Replacement (or some version thereof) is included in axiom scheme S8 (E.II.1.6). (At least in the 1970 version; I do not know about older versions.) | |
| Feb 10, 2013 at 22:30 | comment | added | user9072 | For more on FLT and "large sets" (including BCnrd's opinion) mathoverflow.net/questions/35746/… | |
| Feb 10, 2013 at 21:00 | comment | added | user30035 | In short -- Andreas, your comment is absolutely correct, but I mean "use the trimmed-down version". It's just a historical accident that the person who wrote some of the things Wiles needed decided to write them in such generality that he got entangled in set-theoretic issues. There is no replacement or separation-applied-to-gigantic-sets or large cardinals or Grothendieck universes in Wiles' actual underlying argument. | |
| Feb 10, 2013 at 20:50 | comment | added | user30035 | Wiles does use many facts from EGA/SGA, but Brian Conrad has told me that he has explicitly checked that he needs none of the Grothendieck-universe business. He might (indeed he does) assert that some ring represents some functor defined on e.g. the category of all complete local Noetherian rings with residue field some fixed finite field, but he does not need that e.g. this category, or some subcategory of "small" objects, is a set, and any representing-functor arguments can simply be replaced with assertions which are not category-theoretic in nature but which do the job. | |
| Feb 10, 2013 at 20:47 | comment | added | user30180 | Grothendieck's proof of his criterion for an abelian category to have enough injectives (brilliantly generalizing the case of modules over a ring) uses transfinite induction on Hom-sets. Unrelated to that, Kisin is the real expert on trimming down the prerequisites for Wiles' proof. :) | |
| Feb 10, 2013 at 20:20 | comment | added | Andreas Blass | There is an ambiguity in "Wiles's proof uses". Do we mean literally the proof Wiles gave, together with the cited prerequisites, which include Grothendieck-style category-theoretic work? Or do we mean a trimmed-down version that replaces that general machinery with the specific instances actually needed? The former would involve Grothendieck universes, which are way beyond $V_{\omega+\omega}$. The latter needs less than $V_{\omega+\omega}$. I believe Colin McLarty is the expert on trimming down the prerequisites for Wiles's proof. | |
| Feb 10, 2013 at 20:06 | history | answered | user30035 | CC BY-SA 3.0 |