FOM: Thinking and faking with universes
Colin Mclarty
cxm7 at po.cwru.edu
Mon Apr 12 00:32:10 EDT 1999
Reply to message from cxm7 at po.CWRU.Edu of Sun, 11 Apr
This post replies to some taunts from Friedman, and
to his surprising suspicion that sheaf theory per se is a
kind of fluff invented so mathematicians can talk about bigger
sets than they need to. Many of the issues in Friedman's
post were rerun from the "expert" thread and I had already
replied to them in my friday post to fom which crossed with
Friedman's post in the fom queue.
friedman at math.ohio-state.edu of Sat, 10 Apr wrote
>response to McLarty 6:02AM Thursday 4/9/99.
>
>First of all note that Mathias' expert - see Mathias, 6:24AM 4/9/99 -
>agrees with me and my expert on the essential points, characterizes a
>number of McLarty's wilder statements "over the top."
Both experts agree with me that universes do in fact
appear in the proofs where I said they do. That seems to me the
essential point.
Shipman started the thread by asking whether
Grothendieck universes had even been used to prove a theorem of
ordinary mathematics. I said they had, if you take published
proofs at face value, but the uses were well known to be
eliminable in principle. The experts both agree that is true.
Unfortunately Mathias's note to his expert says we are
arguing about whether or not universes are essentially involved
in the proofs. I have said from the start that they are not
essential. Mathias's interpretation makes my statements "wild"
indeed--and clearly wrong.
>McLarty probably should also respond to me when I keep
>insisting that the serious foundational advances are not these fake "uses"
>of universes that McLarty talks about, but rather what is happening in
>mainline f.o.m. Mainline f.o.m. is the real thing - no faking.
The uses are "fake" in the sense that they invoke
stronger set theoretic axioms than the ultimate result needs.
They are "real" in the sense that they actually occur in the
published record.
Obviously, since we all agree the uses are eliminable
in principle, they are not a foundational issue in Friedman's
sense. That is fine with me.
McLarty:
>>As always, recall
>>that we all agree universes are formally avoidable, but let me
>>sketch how they are actually used.
>
>Or just maybe how they are merely mentioned for expositional convenience -
>like algebraic closures of arbitrary fields.
Used for convenience. The question that interests me, and not
Friedman, is how great a convenience it is. I agree with both
experts that it is easy to see how they could be eliminated in
principle. But that does not mean it would be easy to do in
practice. No one that I know of actually uses these theories
without quantifying over universe-sized sets.
In contrast, nearly everyone working with algebraic numbers
easily forbears to state that every field has an algebraic
closure.
I will re-repeat, though, that the real action in a proof like
Wiles's does not use universes. Universes appear in the routine
parts--routine today, but not when Grothendieck et al. were
creating them.
Mclarty:
>> For example, Wiles cites Altman and Kleiman's proof
>>of the "Grothendieck duality theorem for fields". That proof
>>uses universes.
>
>Only if you have an overblown statement of it.
Altman and Kleiman's statement and proof use universes. Complain
to them, not me.
McLarty:
>> For any topological space E, a "sheaf on E"
>>is a local homeomorphism h:T-->E... [further details cut]
>
>Aha! You must mean that T is a second object that is essentially arbitrary.
>If E is tiny, what is gained from considering an arbitrary object to map
>into it? - except nice pretty fluff?
You got me! That is exactly what I mean. Your question splits in
two parts: What is gained by looking at some of this "fluff"? and
what more is gained by looking at all possible ways of adding such
"fluff"?
If E is really tiny, say a one point space, then nothing is gained,
because there is nothing to gain. A one point space by itself is
of no interest to anyone, even number theorists. But suppose E is
just a bit larger, say a compact manifold. Then a great deal is
gained by looking at some of these bits of "fluff". I am not
going to give the results in detail but give some references
and you can ask experts whether this is real math.
Look at Bott and Tu DIFFERENTIAL FORMS IN ALGEBRAIC
TOPOLOGY for many applications (as to homotopy of spheres).
More advanced applications, which I do not understand well,
are in Iverson COHOMOLOGY OF SHEAVES. And much more advanced
uses, for the topology of singular manifolds, found for example
in Goresky and MacPherson "Intersection Homology II" INVENT.
MATH. vol.72, 77-130.
As to looking at all possible ways of adding fluff, Tennison
SHEAF THEORY shows the uses, and Bott and Tu bring you up to
where it is just about to become useful. The advantage is
precisely that it lets you define the cohomology of spaces
as a derived functor. This, and further evolutions of it
as "derived categories" are used in the Goresky and MacPherson
already cited, or see Beilinson, Bernstein, Deligne "Perverse
Sheaves" Asterisque 100 (1982). Many more references are
discussed in the Forward and the Reference Guide to Gelfand
and Manin METHODS OF HOMOLOGICAL ALGEBRA.
>>Let AbSh(E) name the category of Abelian group sheaves on
>>E.
>
>I guess this is gigantic because the number of Abelian groups is gigantic.
>I.e., you are considering arbitrary Abelian groups. Again, what is the
>point of this except nice pretty fluff?
Yes, that is why AbSh(E) is big. As to the point in this case
see Serre GALOIS COHOMOLOGY and references to it (for class
field theory) in books like Cassel's and Frohlich ALGEBRAIC
NUMBER THEORY. Please ask around to see whether Serre is
considered a serious number theorist.
Friedman:
>But in any concrete situation, why don't you just as well restrict the
>Abelian groups you look at? Would trivial cardinality conditions suffice to
>get rid of the hot air in any concrete situation? E.g., every countable
>field has a countable algebraic closure. This cuts out virtually all of the
>hot air set theory involved in "every field has an algebraic closure."
For a concrete situation you could restrict them, and specific
calculations are often made by sharply restricting them, but
also general results are often humanly indispensible for specific
calculations. Often a general result can itself be proved
under one or another cardinality restriction. This is what
your expert meant by "controlling the cardinality" of structures.
It is necessary for many calculations, no matter how many large
sets you are will to use in principle.
On my earlier post today I point out that rank restrictions
will work quite generally--assuming the axiom of replacement
is not essentially used to prove existence of any of the groups
you need. This seems a VERY safe assumption.
No one seems to have discussed the rank restriction, certainly
because it is already far too generous for parts of the proof you
actually need to control.
This could lead to a more subtle discussion of what is actually
going on, and how it relates to what is known as possible in
principle. So far the fom discussion has featured an unsubtle
indifference to what actually goes on as opposed to what is
possible.
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