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

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"