FOM: Russell paradox for naive category theory

[Stephen G Simpson]

Thomas Forster 6 May 1999 17:42:41

 > I'm not sure what Steve means by saying that NF does not accept the
 > Russell paradox.

OK, let me rephrase it.  The Russell paradox says you can't have both
full separation and a universal set.  You must reject one or the
other.  The most widely accepted set theory, ZFC, accepts full
separation and rejects the universal set.  NF on the other hand
rejects full separation and accepts the universal set.

 > I can't see offhand why one shouldn't have a category of all
 > categories by means of some NF-like ruse - indeed Sol Feferman
 > tho'rt this years ago and circulated a manuscript (now repudiated)
 > in which he developed such a treatment in NFU (NF with Urelemente,
 > which is known to be consistent).  

Sol tells me he circulated that manuscript and published an abstract
of it in 1974.  He didn't publish the manuscript, but he didn't
repudiate it either.  So far as I know, he stands by the results.  His
more recent thoughts about ``the category of all categories'' are in
his 1996 manuscript ``Three Conceptual Problems That Bug Me'', which
is available at his web site.

 > I've assumed that Steve's manifestation of Russell's paradox in
 > Cat. theory is just what he says - Russell's pdox in Cat. theory -
 > in which case it should yied to stratification in the same way it
 > does in NF)

If my version of the Russell paradox for naive category theory works
in Sol's NFU setup, then that would imply inconsistency of NFU.  So I
guess it doesn't work.  I wonder why not?  Maybe stratified formulas
can't talk abut pseudoautistic categories.  Tell me, does
stratification prevent you from talking about isomorphic structures at
different levels?  For instance, if x and y are sets at different
levels, is there no way for a stratified formula to talk about a
bijection between x and y?  (Sorry, I'm pretty ignorant about
stratification and NFU.)

-- Steve