I copy in John’s comment ( he uses ‘elementOf’ where you use ‘membership’) :
The inconsistencies lie in the choice of axioms. All versions of set theory are based on two dyadic relations: subsetOf and elementOf.
The differences lie in the axioms that are asserted in each theory.
You could call subsetOf and elementOf primitives, but they don't behave the way that you have been claiming for the kinds of primitives you want. In particular, their "meaning" is determined by the axioms and each version of set theory has a different set of axioms.
That is one of the main reasons why I keep saying that this search for primitives is misguided. It's totally irrelevant what set of words (or predicates or relations or types or concepts or whatever) you start with -- because all the serious work is done by the axioms.
As soon as you add more axioms to a theory, the "meaning" of the so-called "primitives" changes.
From: Matthew West [mailto:dr.matthew.west@xxxxxxxxx]
Sent: 04 February 2010 14:10
To: mail@xxxxxxxxxxxxxxxxxx; '[ontolog-forum] '
Subject: RE: [ontolog-forum] Foundation ontology, CYC, and Mapping
Could you elaborate please.
Though, as I think John Sowa pointed out in general (apologies if it was someone else), the ‘root primitive’ membership has different senses / meanings in the two cases – so it is not exactly the same.
Tel: +44 560 302 3685
Mobile: +44 750 3385279
This email originates from Information Junction Ltd. Registered in England and Wales No. 6632177.
Registered office: 2 Brookside, Meadow Way, Letchworth Garden City, Hertfordshire, SG6 3JE.