Subject: categories: Consistency of the category of all categories

From: Vaughan Pratt <[email-masked]>
Date: Tue, 20 Apr 2004 21:11:09 -0700

Having thought about this consistency issue for the "category of
all categories" from time to time over the years, I had not seen any
satisfactory resolution save that of resolving to stop worrying about it.
But then Sol's talk was advertised for this Thursday, renewing my interest.

One thing was clear: the "category" of all categories is in fact the
2-category 1-CAT (or just CAT) of all categories.  There is no reason
to expect every 2-category to show up in the category of all 1-categories
(though all the large 2-discrete ones will, where "large" is the cardinality
bound for categories).

Furthermore the category n-CAT of all n-categories had to be bigger than any
n-category, since n-categories had no evident property letting one circumvent
Russell's paradox differently from the case n=0 where one concludes that Set
is bigger than any set.  But (n-1)-CAT is an n-category.  Hence we *must*
have a strict hierarchy.

But how much bigger?  Well, certainly an exponential gap between n-CAT and
(n+1)-CAT (meaning the latter being at least the power set of the former)
suffices to dispose of Russell's paradox.

Can one reliably do better and just take *any* strictly increasing
sequence of cardinals?  I'm not sure how one would argue that.
But in any event the cardinals scattered in between the beth numbers
beth_0,beth_1,beth_2,... constituting Cantor's beanstalk are merely nematodes
on the beanstalk produced by the Cohen nematode factory, and it's not clear
to me what additional benefit derives from basing the spacing of the n-CAT
hierarchy on artificially manufactured nematodes.

This is all the more clear when one considers that exponential gaps are
to inaccessible cardinals as nuclear radii are to intergalactic space.
A billion of them, or even epsilon_0 of them, are no more than the layers
creating the iridescence in a butterfly wing.  These are tiny gaps, and it
is pointless trying to shrink them down based on fictional entities if they
can't be shrunk all the way to zero.

It may be distressing that one can't take all n-CAT's to be uniformly the
same size.  But a rate of growth just sufficient to ward off Russell's
paradox is so tiny in the grand scheme of things axiomatized via ZFC,
Grothendieck, or whoever, that one can practically think of them that way.

I can understand that not everyone necessarily cares about cardinality
issues, considering it unhealthy to take them too seriously or examine
them too closely.  Among those on this list that do care, is the above
attitude reasonable?  If so, can it be further elaborated and improved?
If not, what is wrong with it?

Vaughan Pratt