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From: cxm7@po.cwru.edu (Colin Mclarty)
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Subject: small universes
Date: Mon, 12 Apr 1999 09:11:10 -0400 (EDT)
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	Has anyone considered this idea for smaller Universes?
Maybe it is well known but I had not seen it.

	We can drop the "replacement" condition on a 
Grothendieck universe U: for every x in U and every onto
function h:x-->y with y a subset of U, y is also a member of
U. It is enough if a universe naturally models Zermelo set
theory. (Or even a bit less, as in the elementary theory of 
the cagtegory of sets.)

	So we could define a "universe" to be any set of the 
form V(i) for i a limit ordinal (greater than omega). Here
V(i) is the set of all sets with rank less than i.

	Then the claim "each set is member of some universe" 
is simply a theorem of ZF. This is a great deal weaker than 
Grothendieck's axiom as stated in SGA. Grothendieck's is
equivalent to extending ZF by a proper class of inaccessible
cardinals.

	Each of these "universes" models Zermelo set theory
(i.e. ZF without the replacement axiom but with separation)
and a bit more. Insofar as general category theory is 
provable in Zermelo set theory, it applies to the U-small 
categories for each universe U in this sense.

	I believe all the apparatus of Grothendieck's
TOHOKU paper, and the topos theory and cohomology of the
SGAs, is provable in Zermelo set theory with axiom of choice. 
So this definition of universes formalizes current 
cohomological number theory just as Grothendieck suggested, 
within the axioms ZFC.
	
	The only thing I want to check (as soon as I get 
to my office) is Grothendieck's proof that every AB5 category 
has enough injectives--it is an induction with a proper 
class of steps but you prove it is fixed after some set of 
steps. Possibly you need the axiom of replacement to show 
it is indeed a set of steps. I cannot think of any other 
result in this part of category theory that could require 
replacement.

	So, is this all old news? Or does anyone see a
problem here that I am missing?