Re: Raatikainen's critique of Chaitin

From: Nath Rao (RnNaDthOrMao_at_yahoo.com)
Date: 09/09/04 

Date: Thu, 09 Sep 2004 11:00:03 -0400

KRamsay wrote:

> In article <chn0t8$667$1@charm.magnus.acs.ohio-state.edu>, Nath Rao
> <RnNaDthOrMao@yahoo.com> writes:
> |BTW, Changeaux, a biologist IIRC, is (I am sure) a realist with regard
> |to the phenomenal world. I suspect he didn't know enough to ask where
> |such things as "the" category of (all) C^*-algebras and K-theory
> |functors live. I just have a problem coming with an answer to such
> |questions that can be squared with the way people actually think and
> |talk. [Yes, Grothendieck universes are a way around if you don't mind
> |excessively strengthening the consensus axiom system, but as A. Levy
> |pointed decades ago, people talk as if they are referring to "the"
> |universe, rather than >a< partial universe. The other way out is alien
> |to most practicing topologists/algebraic geometers etc, and will fail
> |anyway the moment somebody 'discovers' unavoidably impredicative
> |functors 'in nature'.]
>
> Well, it seems to me that people naturally think of there being such
> things as properties of sets, and treat proper classes as being the
> extensions of those properties.

Topologists seemed to have implicitly used Morse-Kelly (to judge by
proofs given) when talking about categories and functors. I doubt that
any of them thought of underlying classes of categories as extensions of
predicates, >always<. [They would call such categories "concrete".]

> I think they feel okay about this as
> long as the argument can in principle be rephrased so as not to
> depend on treating those properties as objects in their own right.
> Many arguments can of course be rephrased by substituting
> specific predicates where references to proper classes are made.
> I'm not sure at what point such a workaround ceases to work.

> By "impredicative" I suppose you mean a functor which, regarded as
> a proper class, can't be defined by quantifying merely over sets, i.e.
> isn't expressible in the first-order language of set theory (of the
> cumulative hierarchy).
>
> I find it a little hard to imagine how this would arise in "nature".

Here is one way. I will assume that we are working in a two-sorted
theory, with sets and classes, in which classes satisfy suitably
reformulated ZFC axioms. [If you prefer, ZFC + one universe U: replace
set/small by 'is in U', and class by set. The universe need only be
an internal model of ZF and need to be Grothendieck universe to prove
the existence of the functor. If we want to make infinite
limits/colimits involving this functor, then, U needs to be a
Grothendieck universe]:

Suppose that I need a continuous functor between topological categories
[categories with a topology on the class of morphisms, with continuous
composition, source and target maps] that satisfies an infinite number
of equations. I find a compact space (a proper class) of suitable
functors and show that each equation defines a closed subspace and any
finite number of equations have a common solution. So there exists a
functor which satisfies all the equations. But if the equations do not
have a unique common solution, I don't see how can consider the functor
to be predicatively defined. [With global choice, you can have a
predicatively defined function that satisfies the equations, but it is
not clear that you make it be a functor.]

> One can prove a lot of theories of proper classes (and metaclasses
> of proper classes and so on) to be consistent relative to these same
> large cardinal hypotheses. Just pretend that the extension of ZFC
> is talking about the sets below some large cardinal, and that the
> proper classes and so on belong to the next few ranks. This shows
> that the extension of ZFC talking about all those extravagantly big
> things (way too big to be sets, even) has no more consistency
> strength than axioms "merely" asserting the existence of some
> not-so-endless infinite cardinals. I think this is in some sense the
> way people use the workarounds using Grothendieck universes: show
> that we're not running into a trap by the manner in which we're
> talking about "large" categories. Then revert to using the same
> lingo, but thinking of it as referring to properties of all sets,
> or to proper classes.

Of course, if you state what large cardinal axiom you are using [though,
I am told that there are experts who will assert that theorems proved
this way can be proved within ZFC].

A. Levy (in his chapter on the role of classes added to Ackermann's set
theory book) pointed out a potential problem with this. Suppose that you
prove that for each universe, there exists a group G with some specified
property. You cannot talk as if you have a group that works for all
sets. In practice, this is not a problem so far because universal
properties used include or imply uniqueness upto isomorphism (in a
suitable category). But homotopy theory has things where only 'weak'
analogs (i.e, we have existence, but not unique existence) exist. If we
don't keep track of the universes (people often don't. I know at least
one paper where the authors say so explicitly), this can lead to a mess.

I never liked Grothendieck universes, and after reading Levy I am
convinced that modern mathematics needs a two-sorted language to keep us
honest.

> I still remember the day someone defined a K-group to me, starting
> by saying, "take the free abelian group with a generator for each
> module over the ring R...".

I prefer "one generator for each isomorphism class of finitely generated
modules" (or fg projective). Now this can be made harmless by noting
that we can come up with a set that has exactly one representative from
each isomorphism class. But there are those who find such things
unnatural, want "the free abelian group with a generator for each
> module over the ring R..." to 'exist' before they mod out by some
equivalence relation. Now you know why I I don't like 'realism'.

Nath Rao

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