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