FOM: Russell paradox for naive category theory
Stephen G Simpson
simpson at math.psu.edu
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
otherwise.
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
by
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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