FOM: Russell paradox for naive category theory
Stephen G Simpson
simpson at math.psu.edu
Mon May 17 19:29:57 EDT 1999
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
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