FOM: Russell paradox for naive category theory

Lawrence N. Stout lstout at
Tue May 11 10:03:31 EDT 1999

Stephen G Simpson wrote:
> Simpson 6 May 1999 11:29:10
>  > > 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?
>  ...
> Butz 6 May 1999 14:38:04
>  > Who does this? I for myself have not seen this. And I would not
>  > trust a paper doing this unless the typing is easy to restore
>  > (i.e., that it is meant the (large) category of all (small)
>  > categories, i.e., Cat).
> According to section I.6 of MacLane's ``Categories for the Working
> Mathematician'', set-theoretic foundations of category theory are not
> entirely satisfactory because they require artificial maneuvers like
> the small/large distinction, Grothendieck universes, etc.  

MacLane's book has a copyright of 1971.  From it one can determine that
the need for the small/large distinction in category theory has been
well known for nearly 30 years.  I think further evidence is needed that
"people keep talking about the category of all categories" without
taking care for size distinctions.  Can you cite an example since 1990?
At the very least categorists will be talking about 2-categories of
small categories. 

> One gets
> the strong impression that MacLane is wishing for an alternative
> foundation which removes the need for these maneuvers.  Section II.5
> is entitled ``The Category of All Categories'' and grudgingly invokes
> a small/large distinction in order to state the results.
> MacLane and other category theorists who talk this way are perhaps not
> aware how much would be lost by abandoning set-theoretic foundations
> and insisting on something like the category of all categories.  My
> Russell paradox for naive category theory brings this point home in an
> apparently new way.

If I remember correctly MacLane attacked modern set theory in the 1980's
for abandoning the study of foundations and going off into arcane
considerations of large cardinal axioms.  He was/is concerned about
proper foundations.  He also encourages the development of category
theoretic ideas as mathematics, not specifically as a foundation for
mathematics.  A lot of the recent work in category theory has more to do
with foundations of computer science than it does with fom:  it is more
concerned with domain thoery and type theory (particularly polymorphic
types) than with sets.
Lawrence Neff Stout
Professor of Mathematics
Illinois Wesleyan University

"Fiddling is a viol habit." Anon?
"Dancing is necessary for a well ordered society." Thoinot Arbeau

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