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Re: antifoundation, flat, wellfounded

From: Pat Hayes <phayes@ai.uwf.edu>
Date: Fri, 15 Feb 2002 02:59:02 -0500
Message-Id: <p0510143bb8926b30ff2c@[192.168.0.6]>
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
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Received on Friday, 15 February 2002 02:59:02 GMT

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