FOM: re: FOM using universes?

Colin Mclarty cxm7 at
Thu Apr 8 16:50:57 EDT 1999

    Reply to message from friedman at of Wed, 07 Apr
Harvey says:
>*The use of universes in FLT - or any serious number theory - has never,
>even remotely, been any kind of issue. Nobody who understands such proofs
>does anything but think about very small structures from the start till the
>end. The number theorists are perfectly well aware of this. And they didn't
>have to do any work to eliminate large structures.*
       	So you believe that Grothendieck is not a serious number
theorist, nor is Deligne. They won Fields Medals, sure, but that is
just politics. 
    	That is bare ignorance. Check it with your expert. Please.
       	I consider John Tate and Barry Mazur serious number 
theorists. Do you? Now look at their articles in MODULAR FORMS AND
FERMAT'S LAST THEOREM--entirely aimed at explaining the FLT proof.
Large structures come up much less often here than in some other 
parts of Mazur's work but they still come up.
       	The category of commutative Hopf algebras over a ring is no 
small structure, just to take one example from Tate's article. A 
single Hopf algebra can have arbitrarily large cardinality. This 
brings us to the real point: Wiles's proof may formally need only
some very small Hopf algebras. But quite serious number theorists, 
who understand the proof, find it simpler to work from general 
theorems on the large structure.
        A propos of the expert's reported claim that no 
reference in the body of the Wiles paper uses Universes, 
Harvey writes:
>So you are saying that the most literal interpretation is wrong. But all
>you are saying is that some book at least uses some abstract nonsense
>somewhere. That doesn't mean that it is used to establish "Grothendieck
>duality over fields," or even "Grothendieck duality over countable fields"
>or even "Grothendieck duality over fields that are sets."
       	The Grothendieck duality theorem for a finite field K, deals 
with universe-sized categories and functors based on K. The proof 
that Wiles cites, by Altman and Kleiman, is taken with acknowledgment 
from Grothendieck's SGA and it uses universe-sized sets. You could 
avoid them but Altman and Kleiman use them. If these sets exist, so
do Grothendieck universes.
       	If you think of Wiles as a serious number theorist, 
take his word for the importance of the theorem: In the introduction
to his paper he says the "turning point in the whole proof came 
in spring 1991" when he saw he could use Grothendieck duality 
theory (page 451).  
>Isn't the situation as stupidly banal as the following?:
>##Suppose I prove some facts about the complex algebraic numbers. I may
>well quote that they are the unique algebraic closure of the field of
>rational numbers. I may well provide a reference to Lang's Algebra, where
>it is proved that every field has a unique algebraic closure. Now in full
>generality, this uses some serious set theory. Not only the axiom of
>choice, but also in the form of Zorn's Lemma - with its implicit
>comprehension - and also even power set and replacement if done too
>slickly! Aha!! I am "using" serious abstract set theory - essentially all
>of Zermelo set theory. Bullshit!##
    	Here is the difference in a nutshell: Yours is an absurd
example, made up to be useless. A simpler rigorous treatment 
is well known.

	Grothendieck universes were a step in first creating the 
cohomology of schemes, especially to prove the Weil conjectures. 
They succeeded. No simpler rigorous treatment is yet known, 
though there is a familiar vastly more complicated treatment that 
avoids using universes.

	Universe-sized sets are routinely used, without comment,
in the literature including introductions to etale cohomology--
and even the standard graduate introduction to algebraic geometry 
and number theory, Hartshorne ALGEBRAIC GEOMETRY (Springer 1977). 
They occur in essentially every application of derived functor
cohomology. You may regard this as ignoring rigor, or as implicit 
use of universes.  
    	They are not formally indispensible to the number 
theoretic results. That has long been clear to mathematicians 
and I said it in my first post on this thread. They are in fact 
present in virtually all cohomological number theory today 
including at least one key reference in Wiles's proof of the 
semi-stable Taniyama-Shimura.

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