FOM: Using Universes?

[Harvey Friedman]

Reply to McLarty 5:19PM 4/7/99:

I haven't gone back to bother the expert about this. But let me assure you
that this expert is such that he could quickly and authoritatively
straighten you out on this matter.

The main point is:

*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.*

>Friedman's anonymous expert was simply wrong.
>
>        Wiles's article "Modular elliptic curves and Fermat's last theorem"
>uses Grothendieck duality over fields, and cites Altman and Kleiman
>INTRODUCTION TO GROTHENDIECK DUALITY THEORY on page 486, just about in the
>middle of the body of the paper.

Without consulting the expert again, I would surmise that all that is used
is finite fields. See ## below.

>Altman and Kleiman use sets whose existence
>is equivalent to existence of a Grothendieck Universe. So Friedman was
>misinformed when he says:
>
>>>I have been told that there is absolutely no trace back from the references
>>>used in the body of the Wiles paper to Universes (of Grothendieck).

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."

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!##

>        This proves two things about the expert: He or she is not devoted to
>memorizing bibliographic references. And he or she is not obsessed with
>finding who uses Grothendieck universes. Hardly surprising.

Another grossly misleading statement. The expert assured me that he/she
would find any serious use of Universes in serious number theory as of
truly great interest. And I was also assured that such a use would be
considered sufficient for establishing the truth of the number theoretic
consequence, but there would be great interest in obtaining a proof that
avoids universes.

In other words, this expert is as interested in the real foundational
issues as I am. But above all: He/she does not make up foundational issues
where they do not exist.

Now let me return to a point that I made in my last reply to McLarty. I wrote:

>>NOTE: The first serious application of Universes to the integers is going
>>to be in my numbered series of postings on the FOM. And these will be
>>demonstrably unremovable.

No serious application of Universes to the integers has come out of the
categorical foundations community. But it is coming out of the mainstream
FOM community. Are you trying to rewrite history?