FOM: Russell paradox for naive category theory
Stephen G Simpson
simpson at math.psu.edu
Tue May 4 14:07:26 EDT 1999
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/
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