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antifoundation, flat, wellfounded

From: Jeremy Carroll <jjc@hplb.hpl.hp.com>
Date: Fri, 18 Jan 2002 12:23:30 -0000
To: <www-webont-wg@w3.org>

In the discussion on Peter's paradox I heard that a flat set theory was a
possible approach.

My understanding of the set theoretic issue is:
  Does the rdf:type relation follow anti-foundation  as in Pat's RDF Model
Theory, or a flat set theory, or a well-founded set theory.
In less technical terms:
   Does rdf:type permit cycles and infinite descent (anti-foundation)
   Does rdf:type permit *no* chains at all (flat).
   Does rdf:type permit finite descent (well-foundedness).

It seems that so far only the first two are on the table; whereas the third
is the well-established resolution of Russell's paradox. (The middle one
certainly does resolve Russell's paradox but at a high price.

If I picture it correctly well-foundedness would require us to take a
somewhat more constructive view of class creation, and there would be no
class of all classes. But using oneOf is still legal, and so we can have any
finite set of classes, which would satisfy the implementators, and allow the
TOM isa CAT isa SPECIES, rdf:type chain.

Unfortunately my set theory is not good enough to make more than a sketch of
a proposal, I defer to Peter and Pat (and anyone else who feels qualified)
to assess the validity of this proposal.

I suspect that with a lot of work an anti-foundation axiomisation of set
theory could be used to make something rigorous fairly like the current
set-up but avoiding Russell paradox. (And provably as sound as ZF). However
given the obscurity of the anti-foundation work I am not sure that it can be
seriously proposed.


PS: The anti-foundation work that I looked at (fairly casually) in the
eighties was Peter Aczel's Anti-Foundation Axiom with Atoms. I remember the
first time I met him, when I was a PhD student, was in an elevator - I was
quite over-awed.
Received on Friday, 18 January 2002 07:24:02 UTC

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