FOM: Russell paradox for naive category theory

Carsten BUTZ butz at
Tue May 11 14:15:50 EDT 1999

Dear Steve,

> [...] Section II.5 [of MacLanes book]
> 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.

People (like maybe MacLane) being unhappy with the current foundations
(for category theory) does not mean that they abandon anything
the current foundation has achieved. Why should they? As I said before,
and I repeat it here, the distinction small/large is (at the moment)
essential (unless you want to work in a set-theory with a universal set,
with all its side effects). 

The Russell paradox for naive category theory you try to sell here as new
is at least 30 years old. People interested in foundations for category
theory tried at all times to use "a category of all category". The hope
for the existence of such a creature does not last very long, depending on
your experience as a researcher between 15 minutes and one day I guess. 

As long as category theory is based on a notion like collection it has to
deal with Russell's paradox, and at the moment it does so by
distinguishing small versus large. 

To sum up: nothing new here.

  Best regards,


Carsten Butz
Dept. of Mathematics and Statistics
McGill University, Montreal, Canada

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