FOM: Russell paradox for naive category theory

[Stephen G Simpson]

Simpson 3 May 1999 12:08:23

 > >Set theorists stopped talking about ``the set of all sets'' a very
 > long time ago.

Forster:
 >   No we didn't!

OK, sorry, I overstated my point.  What I intended (and this is clear
from the context) is:

Set theorists who accept the Russell paradox stopped talking about
``the set of all sets'' a very long time ago.

Forster works with NF set theory, a somewhat off-beat set theory in
which the set of all sets does indeed exist.  NF does not accept the
Russell paradox.  The key point of NF is to restrict comprehension by
looking at so-called stratified formulas.  The formula ``x is an
element of x'', which occurs in the Russell paradox, is not
stratified.  In this way NF evades the Russell paradox.

The real question that I was asking is: Why do people keep talking
about ``the category of all categories'', when there is (my version
of) the Russell paradox for naive category theory?  Are they unaware
of this paradox?  Perhaps they are aware of it but don't accept it?
If they don't accept it, why not?  Have they considered ways of
evading it?  I would like answers to these questions.

It seems to me that (my version of) the Russell paradox for naive
category theory is fairly hard to evade.  In order to evade it, you
would have to evade a very simple category-theoretic notion, what I
called pseudoautistic categories.  A category C is said to be
pseudoautistic if there exists a category of categories C1 and a
category C2 belonging to C1 such that C is isomorphic to both C1 and
C2.  This definition involves only perfectly clear and straightforward
notions of naive category theory.  The conclusion of my argument is
that there is no category of categories containing an isomorphic copy
of every category.  Because categories are considered only modulo
isomorphism, stratification does not seem to provide a way out.

-- Steve