FOM: Grothendieck universes

[Kanovei]

> Date: Sat, 17 Apr 1999 18:21:37 -0400 (EDT)
> From: cxm7 at po.cwru.edu (Colin Mclarty)

>	I have mentioned a restriction that will work for all
> of Grothendieck's purposes, namely to V(w+w) but ... 

I do not know much about what things like 
*derived functor cohomology* are, but still some 
conclusions from the ongoing discussion can be 
drawn.

1) To carry out some complicated proofs in 
*serious number theory* in natural way, one needs 
complicated systems of algebraic objects, closed 
under certain operations. 

2) Any V_a, where a>w is a limit ordinal, can be taken 
as a *universe* within which all necessary operations 
can be carried out. 

3) Therefore, the results obtained are ZFC results, 
that4s it, which keeps the integrity of mathematics. 

4) A mathematician 
(perhaps except of those who are just interested 
in the category theory for its own sake) 
would like to obtain practically more 
elementary proofs of number theoretic results, 
at least such proofs which do not involve 
more complicated objects than Borel sets in 
Polish spaces, or ultimately pure PA proofs. 

Please comment.

V.Kanovei