Subject: categories: Re: Getting rid of cardinality as an issue

From: Vaughan Pratt <[email-masked]>
Date: Sun, 25 Apr 2004 06:54:06 -0700

>From: Mike Oliver <[email-masked]>
>Surely, from time to time,
>categorists must care about genuinely ultra-first-order notions, such as
>(say) the metric completeness of the real numbers?  To me the natural
>way of getting such notions right is to make sure that each of your
>universes is closed under the (true) powerset operation.

Yes, which was why I formulated the exponential gap only as a lower bound.
The idea was that if you needed more, take it.

In retrospect I should have included the Russell paradox, viewed
constructively as a set factory rather than mysteriously as a bogeyman
under the bed, as something one might or might not need for some purpose,
e.g. as a successor function.  This reclassification (as a shift only in my
personal outlook) prompts me to withdraw my suggestion (made as much for
my benefit as anyone's but as such good to bounce off people) of imposing
any lower bound at all on size of gaps between successive n-CAT categories.

Size can certainly be an issue, whether involving rates of growth of functions
on the integers, or large cardinals.  In their CM104 book, Makkai and Pare
treat the first order model theory (as opposed to first order logic) of
accessible categories, where the goal is to characterize the behavior of
categories independently of their size as far as possible, and where not
possible to characterize the dependencies on size.  Such an enterprise is
not ordinary mathematics but foundations, and as such is *about* these gaps.
Their results (presumably with the help of a consultant) should allow those
on the consuming side of foundations, i.e. those doing ordinary mathematics
(if there really is such a thing), to judge for themselves whether a given
construction is in danger of colliding with a size paradox.  One would hope
that a few simple rules of thumb would minimize dependence on consultants,
though it did not seem to me that CM104 was organized with that economy
clearly in mind; this might be corrected with a short cheat sheet as an
addendum.

Not all paradoxes concern size.  The liar paradox and the
division-by-infinitesimal paradoxes can be turned into size paradoxes via
a suitable encoding, but they are not intrinsically size-related; well,
in the case of infinitesimals, not large sizes anyway.

Perhaps it just reflects my old-fashioned upbringing, but the foundational
role intended for CM104 is way clearer to me than any of the several topos
texts currently scattered around my desk.  Not with regard to the definitions,
examples, and (to the extent I understand their motivation) the theorems
of topos theory.  The elementary definition of a topos is crystal clear
(not to mention incredibly beautiful), as are the basic examples of toposes.

Where I run into problems is in placing topos theory as a foundation beside
say accessible categories.  I can go repeatedly through the topos texts
and just not get it.  Is there some finely honed sentence or paragraph that
explains this relationship?

I get the feeling there should be a sentence or paragraph to the effect that
one brings size under control (or makes it a non-issue) by passing from the
external logic of accessible categories to the internal logic of toposes.
Is some such clear and succinct story (not necessarily that one since it
might be totally wrong) told somewhere?  If so, one could deal with idiots
like me who rant about size as an issue by pointing them at that story,
by way of indicating how to stop worrying about inaccessible cardinals
by embracing someone else's internal logic (and making it one's own?).
Or whatever the story actually is.

What about Remark 7.1.14 in Paul Taylor's Practical Mathematics, for example?
Is this tangential, on point, or core?  What about the preface to Borceux'
Volume 3?  Does Peter Johnstone's nonconstructive theorem "There exists
an elephant" in his preface have a succinctly summarized constructive
counterpart somewhere, a sort of sharply focused photo of an elephant taken
from 50 feet away?  (Actually I suppose a sharply focused photo of a real
elephant would have very close to the same number of megabytes of data as
in the two volumes, so maybe I mean an elephant icon.)

Or is this all just a misunderstanding or misinterpretation of the real
goals of topos theory, with the truth being that there is ultimately no
way mathematicians can avoid large cardinals if they expect to be able to
prove certain theorems, even those of an ostensibly combinatorial flavor?
This is certainly the sermon that Harvey Friedman has been preaching for
a number of years; is Harvey wrong about this?

There seem to be some sunglasses and rose-colored glasses lying around but
I can't tell who they belong to.  Surely they're not all mine.

Vaughan Pratt