FOM: Nice Idea! Thanks to Kanovei

Kanovei kanovei at
Mon Apr 12 02:51:24 EDT 1999

Date: Sun, 11 Apr 1999 17:21:11 -0400 (EDT)
From: cxm7 at (Colin Mclarty)

In ZFC for any limit ordinal i, the set V(i) 
is a model of Zermelo set theory with choice. This suggests a 
very easy formalization in ZFC of the full Grothendieck approach 
to cohomology.
I needn't be thanked in this concern. 

I think the "experts" call those universes and 
corresponding categories "small" (univ. and cat.), 
that is, I have definitely seen this term in a book, 
as opposed to "large" ones, by full Grothendieck. 

However I doubt that this is enough to maintain 
the "general nonsense" in full scale and natural 
fascion, because then it would be fully unclear 
what the all the fuss is about. 

Anyway, we can require of "small universes" 
(= sets of the form V_a, a being a limit ordinal) 
something more than just ZC, namely, 
1) Replacement for all formulas of restricted 
   complexity (say all Sigma_n formulas, n fixed)
2) Replacement for all formulas, but with the 
   "domain" set of restricted cardinality 
   (say countable or \leq continuum). 


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