FOM: Russell paradox for naive category theory

[Stephen G Simpson]

To: John Isbell

Dear Professor Isbell,

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Recently the discussion turned to ``the category of all categories''.
Obviously such a thing can't exist under the usual set-theoretic
foundations of category theory (i.e. sets versus proper classes, small
categories versus large categories, inaccessible cardinals,
Grothendieck universes, etc).  But some people think that ``the
category of all categories'' may be consistent with some other kind of
foundational setup.

Sol Feferman has published some papers along these lines.  His 1996
paper ``Three conceptual problems that bug me'' contains a survey and
references and is available on-line at
<ftp://gauss.stanford.edu/pub/papers/feferman/conceptualprobs.ps.gz>.

Recently I came up with a fairly general argument which is a
category-theoretic analog of the Russell paradox.  The argument takes
place in a ``naive category theory'' setting and could easily be
transplanted to a variety of formal settings.  The argument concludes
that, not only does ``the category of all categories'' not exist, but
there is no category of categories containing *an isomorphic copy of*
every category.

My question for you is, what was already known along these lines?

In recent e-mail to me, Sol referred to you, saying:

 > When I took part in the meeting of the Midwest Category Seminar in
 > 1969 for which the paper "Set theoretical foundations of category
 > theory" is published, in the discussion I reported a paradox on the
 > assumption of the category of all categories.  I can't remember if
 > it was a Russell style paradox or a Cantor style paradox.  At any
 > rate, Isbell said then that such paradoxes were already known--I
 > think he referred to a Burali-Forti style paradox.  I don't know if
 > there are any explicit mentions of these in the literature; it may
 > just be one of those folklore things.

Can you give me any more exact information about what is known by way
of folklore, references to the literature, etc?

Sincerely,
-- Steve Simpson

Stephen G. Simpson
Department of Mathematics, Pennsylvania State University
333 McAllister Building, University Park, State College PA 16802
office 814-863-0775 fax 814-865-3735 home 814-238-2274
simpson at math.psu.edu http://www.math.psu.edu/simpson/