FOM: Russell paradox for naive category theory

[Stephen G Simpson]

Carsten Butz 11 May 1999 11:15:50

 > 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).

I still say that the small/large distinction is *not* essential for
category theory, because you could simply work within the usual
set-theoretic foundational framework (formalized as ZFC) and restrict
your attention to set-size categories.  The small/large distinction
would then be replaced by set-theoretic cardinality considerations.
In this way you would lose little or nothing, at least insofar as
applications of category theory are concerned.

Why don't category theorists fully accept this?  My tentative answer
is that many category theorists harbor some sort of smoldering
resentment against the set-theoretic foundational framework.  They
long for some sort of alternative foundational setup which they feel
would liberate them from what they feel are unnecessary strictures
imposed by set-theoretic foundations.  The existence of this kind of
resentment would explain various observable phenomena, including (i)
Johnstone's endorsement of the ``Mathematician's Liberation
Movement''; (ii) Johnstone's deprecation of set theory and
set-theoretic foundations; (iii) the phony idea of ``categorical
foundations'' as advocated by certain topos theorists; (iv) MacLane's
talk of the category of all categories; (v) the insistence on the need
for large categories and Grothendieck universes.

 > The Russell paradox for naive category theory you try to sell here
 > as new is at least 30 years old. ...

You seem to be saying that my version of the Russell paradox for naive
category theory (details in my posting of 11 May 1999 01:11:25) has
been known for a long time.  I feel that there is some reason to doubt
what you are saying (see below).  Therefore, I would appreciate a
reference, even if the reference is only an informal allusion in some
published paper, conference proceedings, newsgroup, oral history,
report of an informal conversation, folklore, or whatever.  If you
don't have a reference handy, perhaps you could ask your friends on
some category theory newsgroup?

If my result is more than 30 years old, then it must have been known
to MacLane when he wrote ``Categories for the Working Mathematician''.
Yet MacLane did not mention any sort of Russell paradox anywhere in
the book, even in the sections entitled ``Foundations'' and ``The
Category of All Categories''.  I wonder why not?

Anyway, I think I have said what I want to say on this topic.  Perhaps
we can move on to something new.

-- Steve