FOM: Nice Idea! Thanks to Kanovei

Colin Mclarty cxm7 at
Sun Apr 11 17:21:11 EDT 1999

	Kanovei points out what I should have recalled (and all
the more since Penelope Maddy had to remind me of the same thing
a few years ago): 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.

	Simply define a "universe" to be any V(i) with i
a limit ordinal.

	I suspect there are no essential uses of the axiom
of replacement in the general results of the Grothendieck 
school (as in Elements de Geometrie Algebrique, and the Seminaire
de Geometrie Algebrique de Bois Marie) nor in the applications 
of cohomology in number theory (such as our icons Deligne, 
Faltings, and Wiles).

	Insofar as that suspicion is true, this formalization
lets us accept the proofs in number theory naively as stated, 
including reference to Grothendieck-style proofs in the SGAs 
(or Wiles's reference to Altman and Kleiman's proof of 
Grothendieck duality).

	Perhaps this will make Friedman as happy as it makes
me, since it lets me have Grothendieck's program while it
honors ZFC foundations. It merely limits the appeal we
could make to replacement within a cohomological argument.

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