- From: Pat Hayes <phayes@ai.uwf.edu>
- Date: Fri, 15 Feb 2002 02:59:02 -0500
- To: "Jeremy Carroll" <jjc@hplb.hpl.hp.com>
- Cc: webont <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. ? What has foundation got to do with Russell? You could have a nonwellfounded ramified type theory if you really wanted it. In any case, Aczel provides a firm relative consistency result for AFA against ZF. >(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. Separate issue. >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. Its still not fashionable, but its been a firm result now since 1988 and has been incorporated into quite a lot of other work., eg Barwise's situation semantics. It hardly deserves to be called obscure. And in any case, the IEXT mapping trick in the RDF MT doesn't presuppose AFA; it can be done in ZF. (That's why I used it.) I havn't had a chance yet to fully grok Peter's paradox, but Im sure its got nothing deeply to do with antifoundation in RDF and wouldnt be fixed by having finite rdf:type chains. Pat -- --------------------------------------------------------------------- IHMC (850)434 8903 home 40 South Alcaniz St. (850)202 4416 office Pensacola, FL 32501 (850)202 4440 fax phayes@ai.uwf.edu http://www.coginst.uwf.edu/~phayes
Received on Friday, 15 February 2002 02:59:02 UTC