FOM: Grothendieck universes

[Colin McLarty]

reply to Simpson

>At any rate, so far as I can tell, Grothendieck never suggests that
>what the number theorists came up with is mathematically meaningless.

        Here the phrases "le sens de ces taches elle-memes comme parties
d'un vaste Tout, sont oublies par tous" and "arrache d'une vision qui lui
donne tout son sens" are crucial.


>What happens if we replace Grothendieck toposes by elementary toposes?
>Does this lead to a recasting of Grothendieck's general theorems about
>derived functor cohomology, thereby eliminating Grothendieck universes
>while keeping the applications to number theory intact?

        I've mentioned this several times as one goal of elementary topos
theory. It is already routine if you will use V(w+w). No one has yet unified
the general theorems with the number theoretic results in some really weak
foundation. Of course very little is yet known about just how weak a system
suffices for the number theoretic theorems alone. 

       On another topic: it is a misunderstanding to say Grothendieck

>even attacks Bourbaki for failing to incorporate
>Grothendieck toposes into the Bourbakian foundational scheme, thereby
>giving up the Bourbakian ambition to provide *the* foundation for all
>of contemporary mathematics.

        The issue with Bourbaki was not toposes. It was primarily
homological algebra which Bourbaki members routinely used but could into get
into their system. See Leo Corry MODERN ALGEBRA AND THE RISE OF MATHEMATICAL
STRUCTURES (esp. pp.376-383) on this debate in Bourbaki.