Could someone please tell me where the term "Grothendieck universes" comes from? Some books connect Grothendieck's name with this particular approach to avoid set-theoretical difficulties, others don't. What is the current status on these foundational matters anyway? Has anything else emerged besides MacLanes one universe approach, or Feferman's use of reflection principles? best regards, J"urgen -- J"urgen Koslowski | If I don't see you no more in this world | I meet you in the next world koslowj@math.ksu.edu | and don't be late! nhabkosl@rrzn-user.uni-hannover.de | Jimi Hendrix (Voodoo Child)
J"urgen: I don't remember for sure since it's been more than 20 years, but I'm of the opinion that Grothendieck invented his universes to deal with functor categories without antinomies. I would start with Grothendieck and Verdier, Springer Lecture Notes 269-70, 1972 in articles by them mentioning the word "topos". egm
hi Juergen it's good to see you're around. for some idea of what i'm up to you might see ftp://triples.math.mcgill.ca/pub/otto/home.html here's some thoughts on 1. universes in practice 2. what is needed 3. the hierarchy paradox 1. one has functor P: set^o --> set X |-> 2^X and adjunction X --> P Y in set ------------------- X x Y --> 2 in set ------------------- Y --> P X in set ------------------- P X --> Y in set^o with unit n: id --> P^2 thus one has meta-function V: ordinals --> sets by 1. V(0) = {} 2. n(V(i)): V(i) --> V(i+1) = P^2 V(i) 3. colimit at limit ordinals then, for limit ordinals i > omega, the sets and functions in V(i) form a topos with NNO. and, for (strongly) inaccessible cardinals i, the V(i) form Grothendieck universes. e.g. see Makkai-Pare as well as Jech or Kunen. 2. one needs big enough colimits to construct e.g. (weak) reflectors to orthogonality and injectivity classes in categories of presheaves. e.g. see Adamek-Rosicky as well as work by Makkai on sketches. 3. in Rose, subrecursion, one finds that n+1 nested `for' loops are needed to reason deterministically about n > 1 nested `for' loops. yet humans reason about themselves, and living creatures reproduce. i conjecture, from work on interactive proof, that it takes probability and residual error to resolve the hierarchy paradox. bon jour, Jim Otto
<<Could someone please tell me where the term "Grothendieck universes" comes from? Some books connect Grothendieck's name with this particular approach to avoid set-theoretical difficulties, others don't. What is the current status on these foundational matters anyway? Has anything else emerged besides MacLanes one universe approach, or Feferman's use of reflection principles?>> I don't have the Grothendieck reference, but it is not attributed to him by accident. Hopefully someone will supply particulars. I enter this to observe (1) One other thing that has emerged is Blass' position, that it doesn't matter (assuming we wish to do mathematics, not theory of languages). That is clearly and effectively stated by Barr and Wells in the preface or whatever it is to <Toposes Triples and Theories>. Note further that this treatment of the <foundational difficulties> is in the spirit of Grothendieck. MacLane, sorry, Mac Lane is in the spirit of Plato: the one real universe is out there and we should be discovering its properties. It has more impressive adherents than Plato, e.g. G"odel, but it is not viable. John Isbell
The term was popularised by students and colleagues of Grothendieck in the 60's.A discussion of it can be found in SGA 4 SLN#269 in an appendix attributed to "N.Bourbaki"
participants (5)
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Ernie Manes -
Jim Otto -
Juergen Koslowski -
MTHDUSKN@ubvms.cc.buffalo.edu -
MTHISBEL@ubvms.cc.buffalo.edu