FOM: Grothendieck universes; Friedman/Grothendieck analogy

[Stephen G Simpson]

Colin Mclarty 17 Apr 1999 18:21:37 writes:
 > >For instance, what about the category of countable Abelian groups?  Is
 > >that an Abelian category?  What is the ``size'' of that category?  Is
 > >that category ``universe-sized''?
 > 
 > 	We need to be precise here. ...

Yes, you need to be precise here.  When are you going to precisely
answer my questions of 16 Apr 1999 20:04:02?  

One of my questions was, ``is the category of countable Abelian groups
an Abelian category''.  Another of my questions was, ``is that
category universe-sized'' (your terminology).  I'm amazed that you
apparently can't or won't answer these questions without further
information.

It doesn't make a difference, but just to play along with your game,
let me specify the category of *hereditarily* countable Abelian
groups, i.e. Abelian groups belonging to HC, the set of hereditarily
countable sets.  (That category is of course equivalent to the
category of *all* countable Abelian groups, for most if not all
mathematical purposes.)  Once again, is that category Abelian?  Is
that category ``universe-sized'' (your terminology)?  I think you will
ultimately have to admit that that category is Abelian and is not
``universe-sized'', though it may take a while to make you admit that.

(A set X is said to be hereditarily countable if the set consisting of
X plus all elements of X plus all elements of elements of X plus all
elements of elements of elements of X plus ... is countable.)

Now, another of my questions was, ``Are you saying that ZFC does not
prove the existence of any derived functors on that and similar-sized
categories?''  Do you understand the question?  The question is, does
ZFC prove the existence of at least one derived functor on the
category of hereditarily countable Abelian groups.  I would appreciate
a straightforward answer.  Please don't try to evade this question.

To remind you of the context, this is a follow-up to your assertion
``Derived functors between Abelian categories are themselves universe
sized sets'' in 16 Apr 1999 16:41:20.  In order to get to the bottom
of your assertion, I asked you to rehearse the relevant part of the
definition of ``derived functors between Abelian categories'', but you
didn't do so.  Let me remind you that we need to get to the bottom
this, because your assertion is hard to believe and is apparently the
only mathematical basis for your implicit assumption that all uses of
derived functors in the literature rely on Grothendieck universes.

 > Whatever restriction you place you must make sure it is compatible
 > with all the constructions you want to do.

Perhaps you have lost the thread of what we are talking about.  If you
think about the context of the discussion, I think you will
immediately realize that I don't want to do any particular
category-theoretic constructions.  What I want to do is get to the
bottom of what seem to be your exaggerated claims about the need for
Grothendieck universes.

 > But when he says which categories he will use, they are all
 > universe sized. The simplest is the category of (all) Abelian
 > groups.

Is the category of *countable* Ablian groups universe sized?

By the way, you never produced the quotes from Grothendieck that you
promised to back up your 16 Apr 1999 11:44:32 statement ``Grothendieck
regards those proofs as essential uses of universes since the proofs
which eliminate universes have no genuine mathematical meaning.''  Did
Grothendieck ever say anything like this, or is this merely your
interpretation of Grothendieck?

 > To give all the proofs, from the beginning, with no reference to
 > universes would produce a jumble with no mathematical meaning.

Is there any evidence of this, other than your say-so?  In particular,
does Grothendieck say this in any of his writings, published or
unpublished?

Re: your Friedman/Grothendieck analogy of 16 Apr 1999 11:44:32.

I have already mentioned (in 16 Apr 1999 15:57:48) one remarkably
inappropriate aspect of your analogy -- the fact that you are using
two diametrically opposed senses of ``genuine mathematical meaning''
in the two halves of the analogy.

Now let me mention another even more inappropriate aspect of your
analogy.  That is the fact that Friedman's transition between (1)
large cardinals and consistency statements involving them, (2)
Friedman's finite combinatorial statements, encompasses and relies on
major conceptual and technical advances.  By contrast, the technology
of eliminating Grothendieck universes is mathematically routine, well
known to number theorists, and involves no conceptual advance.

-- Steve