FOM: Russell paradox for naive category theory

[Stephen G Simpson]

Todd Wilson 11 May 1999 15:34:41 

 > it appears as an elementary exercise (3L, p. 37), in the book
 > "Abstract and Concrete Categories" by Adamek, Herrlich, and
 > Strecker:
 > 
 >     3L.  Quasicategories as Objects.  Show that one cannot form the
 >     "quasicategory of all quasicategories".  [Hint: Russell's
 >     paradox appears again.]

Wait a minute.  I think you are mistaken.  This exercise is not the
same as my Russell paradox for naive category theory.  There are two
differences:

1. While my argument takes place in a naive (i.e. pre-formal) theory
   of categories, Adamek-Herrlich-Strecker are working in a ZFC-like
   set-theoretic framework, formalizable as VNBG with one inaccessible
   cardinal.  Therefore, the exercise of Adamek-Herrlich-Strecker to
   which you refer to follows trivially from the usual set-theoretic
   Russell paradox, viz ``there is no set of all sets'' or ``there is
   no class of all classes'', because a quasicategory is at bottom a
   set or a class in the sense of ZF or VNBG.  My argument on the
   other hand is not limited to this kind of ZFC-like context.  In
   particular, my argument can be formalized in NFU, where it shows
   that the concept ``pseudoautistic category'' cannot be defined by a
   stratified formula.

2. My argument shows that, not only is there no category of all
   categories, but there is no category of categories containing all
   categories *up to isomorphism of categories*.  Perhaps you are
   under the impression that this additional conclusion falls out of
   the usual Russell argument, but I think it requires a non-trivial
   additional argument.  The details of the additional argument are
   sketched in my posting of 11 May 1999 01:11:25.

In fact, I am vaguely thinking of submitting my argument for
publication.  Does anybody want to comment for or against that idea?

-- Steve