FOM: Russell paradox for naive category theory

Stephen G Simpson simpson at
Wed May 17 17:12:02 EDT 2000

This is a followup to my posting of May 16 on the same subject.

1. Solovay has been pressing me off-line for details of the proof of
this claim: If c is a category of categories which is isomorphic qua
category to a category of categories containing M and N, then c
contains isomorphic copies of M and N.  A relevant reference (pointed
out by Solovay) is Exercise D in Chapter 1 of Peter Freyd's book
``Abelian Categories''.  It seems to me that the results stated in
this exercise (if correct) establish the claim, but Solovay thinks

In any case, I am at the moment unable to supply the details of proof,
so I must withdraw my claimed Russell paradox for NCT, at least
temporarily.  I will keep FOM informed as this situation develops.

2. Solovay has also pointed out that I need to modify my list of
primitive predicates for NCT, as follows.  Replace

   Gxyz: x is a functor, and y corresponds to z under x


   Gxyz: x is a functor, y and z are objects, x carries y to z
   Hxyz: x is a functor, y and z are morphisms, x carries y to z

3. Again I thank Bob Solovay for his helpful remarks.

-- Steve

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