FOM: Re:Thinking and faking with universes

[Harvey Friedman]

Reply to McLarty 2:03PM 4/14/99. This posting of McLarty contains a
disucssion of some set theoretic constructions which are far far weaker
than universes. But even here, they appear to be fake uses. Since the
relevant Riemann surfaces are separable, when approximating a global
function one needs to look at only unions of countably many approximates
defined on open sets. Presumably it is trivial to get rid of the hot air
and do countable constructions - i.e., pasting at most countably many
functions together. Perhaps the best way to get rid of the hot air also by
considering only a countable basis for the open sets - i.e., a countable
collection of open subsets of S such that every open subset of S is a union
of these open subsets of S - rather than all of the open sets.

McLarty again falls short of explaining anything substantial about the
relevance of doing this construction in anything like the generality
indicated with his usual mode of operation - which is to cite references
over and over again.

By the way, I was in the office today of a third expert in algebraic number
theory. He said (paraphrase)

"it is true that some of the books mention very briefly some overblown set
theoretic stuff, in order to attain a certain kind of honesty. But then
they never mention or refer to it again. Besides, things are done in great
generality only because i) it is generally quite convenient; and 2) one
might as well since one can never know what applications in the future
might really depend on it. Of course, one has never really needed any such
things, but it is comforting to know that it exists if one would. It is a
way of completely dispensing with an issue that might someday come up."

>	Yes, they [Deligne, Grothendieck, etc.] needed these structures
>[universes]. But not in the sense you use when you
>talk about needing an assumption.

No expert I talked to thinks that they "needed" universes in any reasonable
sense of the word. Sure, I agree that they needed universes to study
universes. You are coming very close to contradicting your posting of
5:35PM 4/14/99.

>	Grothendieck "needed" whatever methods he could find to solve an
>outstanding problem: the Weil conjectures.
>
>	Did he use "excess generality"? Well, "excess" to what? Obviously
>he had no
>idea at the time what generality he would need. In hindsight his method has
>"excessive" proof-theoretic strength. Some aspects (famously toposes) are
>considered excessive in practice by many number theorists. Universes are
>apparently not excessive in practice since number theorists continue using
>them.

Set theoretically, universes are far in excess of topoi, as you well know.
On a productive note, you may be suggesting that there is a way of
formalizing universes which does not prove the existence of topoi. That
would be of interest to people in real f.o.m. This would have to be a new
way of formalizing universes.

>	Try posing these two questions to your experts: "Can you name a
>textbook on
>cohomological number theory that does not refer to large structures?" And
>"Are the papers in MODULAR FORMS AND FERMAT'S LAST THEOREM written with
>excess generality?"

I gather that they may refer to large structures in the same sense that
Lang's Algebra refers to large structures. If you say "arbitrary field"
then you are referring to large structures, in a sense. Because in the
usual set theoretic formalization of mathematics, an arbitrary field may be
large. So you are asking a silly question.

>	I already answered, in the sentence after the one you quote. The
>universes function in the (now) routine parts. The theorist giving the proof
>never proves that a given field extension has a Galois cohomology. He or she
>simply uses that fact to justify particular calculations. If the theorist
>even cites a source to show these cohomologies exist, that source is likely
>to define them as derived functors on a certain category, thus referring to
>and even quantifying over universe-sized structures. The theorist will not
>think about that.

This time, I'm not even going to consult my experts, because I think I am
on to your game. I think that the number theorists know that the fields
they are interested in have "Galois cohomology", but they take that to mean
Galois cohomology in the sense of involving only reasonable objects. For
example, in set theory, let S be a set we are interested in, which might be
reasonably small. We can prove a trivial theorem that for any set X
whatsoever, the set of all maps from X into S exists. This involves large
structures, since X may be large. However, we may only think and care about
X that are, say, of the same cardinality as S.

>Lang's ALGEBRA is not a textbook on algebraic numbers. People also study
>algebraic extensions of finite fields, for perfectly concrete reasons such
>as encryption, and want to know these fields have algebraic closures. People
>also look at function fields over these fields, and thus need to know about
>inseparable extensions. Lang covers all this and more because it is used.

What gets used is: reasonable fields have reasonable algebraic closures,
and nothing more. Again, I say that Lang does arbitrary fields because it
is convenient and he sees no reason to try to delineate what a reasonable
field is or isn't. Let people decide what they regard as reasonable, and
then use "every field has an algebraic closure" in the case of the fields
they regard as reasonable. This is an example of a fake use of set theory
in Lang's book. This is not a criticism of Lang's book, of course.

>
>Sorry, no tutorial here on finite field methods in encryption. References
>are well known and well catalogued in university libraries.

References to algebraic geometry are also well known and well catalogued in
university libraries.

>Generality has real uses. Indeed it also has fake uses and you may find some
>textbook somewhere on algebraic number theory that only works on numbers
>algebraic over the rationals and yet proves every field has an algebraic
>closure. But you may not either. Certainly most texts on this neither
>prove nor
>refer to that theorem.

The issue is whether the generality you are so attached to has any real
uses. It doesn't seem to be the case - yet.

>Apparently all texts, and probably all research papers, on cohomological
>number theory rely on proofs using universes.

This appears to be totally absurd.

>Your experts have not
>mentioned any exceptions.

Hartshorne, Algebraic Geometry, for one - says my third expert in number
theory.

>Well, your first expert said Wiles's proof was an
>exception. But in fact the "turning point in the whole proof", to use
>Wiles's own words, came when he noticed a consequence of the Grothendieck
>duality theorem, and for that theorem he directly cites a proof using
>>universes.

You are repeating your misleading of the FOM over and over again. I implore
you to be more careful in how you use the word use.