Date: Tue, 15 Nov 1994 19:26:31 -0400 (AST)
From: categories <cat-dist@mta.ca>
To: categories <categories@mta.ca>
Subject: history question: Grothendieck universes

Date: Tue, 15 Nov 94 10:05:59 CST
From: Juergen Koslowski <koslowj@math.ksu.edu>

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)

Date: Thu, 17 Nov 1994 10:15:09 -0400 (AST)
From: categories <cat-dist@mta.ca>
To: categories <categories@mta.ca>
Subject: Re: history question: Grothendieck universes

Date: Wed, 16 Nov 1994 11:41:49 -0500 (EST)
From: MTHISBEL@ubvms.cc.buffalo.edu


<<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

Date: Thu, 17 Nov 1994 10:11:59 -0400 (AST)
From: categories <cat-dist@mta.ca>
To: categories <categories@mta.ca>
Subject: Re: history question: Grothendieck universes

Date: Wed, 16 Nov 1994 11:20:42 -0500
From: Jim Otto <otto@rodent.math.mcgill.ca>

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

Date: Fri, 2 Dec 1994 12:32:12 -0400 (AST)
From: categories <cat-dist@mta.ca>
To: categories <categories@mta.ca>
Subject: Re: history question: Grothendieck universes

Date: Fri, 02 Dec 1994 09:29:10 -0500 (EST)
From: MTHDUSKN@ubvms.cc.buffalo.edu

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"