FOM: Thinking and faking with universes

[Harvey Friedman]

Response to McLarty 6:02AM 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." You probably should
respond to Mathias. 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.

In any case, I would like to delve a little bit further into some specifics
raised by your 6:02AM.

>    	The FOM discussion of Grothendieck universes may go
>better with a little more indication of how universes come into
>cohomological number theory. It is not the way that a set
>theorist's intuition might easily suggest.

I think that it may be more like "every field has an algebraic closure"
than you indicate. But we'll see.

>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.

>    	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. You use a lot of set theory
to prove that every field has an algebraic closure. But not to prove any
field you care about in a concrete setting has an algebraic closure.

>Friedman and Shipman have both expressed the
>suspicion that universes may be attractive for very large
>fields in general but not for specific finite (or countable,
>or otherwise very small) fields. But actually the size and
>'concreteness' of the field has nothing to do with it.

This is the main point that generated this posting.
>
>    	For any topological space E, a "sheaf on E"
>is a local homeomorphism h:T-->E. That is continuous
>function h such that each point x of T has some open
>neighborhood which h maps one-one onto an open set in E.
>Given sheaves h:T-->E and h':T'-->E on E, a sheaf map
>f:h--h' is just a continuous function f:T-->T' such that
>the composite h'f:T-->T'-->E equals h.

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?

>The category of
>sheaves on E is a Grothendieck topos Sh(E) so we can
>interpret the usual constructions of mathematics in a
>natural way in Sh(E). In particular an "Abelian group
>sheaf" on E is just an Abelian group in the topos Sh(E).

>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?

>    	The Cech cohomology of E has been around since
>the 1930s but it has a very neat definition in these terms:
>It is "the derived functor of the global section
>functor from AbSh(E) to Abelian groups" which Grothendieck
>defines by its universal property among "delta functors".
>For a full account see Tennison SHEAF THEORY. A "delta
>functor" is actually a certain kind of family of functors
>H^n from AbSh(E) to Abelian groups, one for each natural
>number n, and definition of the derived functor quantifies
>over delta functors.

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."

>    	This definition may look brutal if you are not
>familiar with it. Perhaps like Friedman's recent
>parody definition of complex algebraic numbers. But in
>fact it is already useful.

For a concrete situation? Give me an example where I can't trivially just
cut down the generality at will.

>More importantly it generalizes
>to other cohomology theories that were only invented by
>its help. For many of these theories this framework is
>used for proving the general theorems, even though it
>is formally dispensible, and even though one or another
>particular calculation of cohomology can be made with far
>less apparatus.

You mean general theorems like: every field has an algebraic closure.

>    	A key FOM moral is at hand: No matter how small
>and concrete the space E may be, the category AbSh(E) is
>very large. It is a proper class in the most naive
>terms, and so is any functor from it. The definition
>of cohomology quantifies over these functors. By using
>Grothendieck universes we can make everything here a
>set--specifically, if E lies in any universe U
>then we can make AbSh(E) a set in any universe that
>contains U.

But that's because you have deliberately injected, e.g., arbitrary Abelian
groups, for fluffy convenience. In my parody, I inject "every field has an
algebraic closure" even when just talking about the complex algebraic
numbers.
>
>	All that follows is some further examples that
>I may want to refer to on FOM. Group cohomology predates
>this derived functor approach, and Galois cohomology
>predates the full Grothendieck conception. But both
>commonly draw on this approach for perfectly practical
>reasons. The approach was invented as a step in creating
>etale cohomology and many others used in number theory.
>
>    	Consider any group G (an ordinary group, a set
>with multiplication et c.). Define a "sheaf on G" to
>be any set S with an action of G on it. That is for
>each g in G and each x in S we define g(x) in such a
>way that given also g' in G we have g'(g(s)) equal to
>(g'g)(s), and for e the unit element of G we have e(x)=x.
>A sheaf map from S to a sheaf S' is just an equivariant
>function f:S-->S', that is a function such that for each
>g in G and x in S we have f(g(x))=g(f(x)). These sheaves
>and maps form a Grothendieck topos Sh(G), and AbSh(G)
>is the category of Abelian groups in that topos.

Again - *arbitrary* groups. Like arbitrary fields in my parody.
>
>	The group cohomology of G is the derived
>functor of the global section functor from
>AbSh(G) to Abelian groups. See Brown COHOMOLOGY OF
>GROUPS. Again, note that no matter how small G is,
>even if it is the trivial one element group {e},
>the category AbSh(G) is very large.

Yeah, becuase you deliberately injected all of this junk in there.

Let me close with a point that may surprise some casual readers of this
debate.

***I LOVE UNIVERSES!!!***

They are essentially strongly inaccessible cardinals, more or less the
smallest large cardinals that go beyond ZFC. I have spent close to 35 years
trying to show that there is some real point in considering such things for
concrete mathematics. But I want to disassociate myself from fake reasons
for considering such things.

Nevertheless I have at least learned something from this debate. Namely how
willing mathematicians are to accept these objects (universes) for
establishing the truth of a concrete theorem of mathematics. This shows
that it is possible to convince mathematicians to accept some even higher
large cardinals for this purpose.