RE: SUO: On the supreme supertype
Dear Robert,
> Dear Matthew,
>
> ----- Original Message -----
> From: "West, Matthew MR SSI-GREA-UK" <Matthew.R.West@is.shell.com>
> To: <standard-upper-ontology@ieee.org>
> Sent: Friday, February 23, 2001 6:42 AM
> Subject: RE: SUO: On the supreme supertype
>
>
> <snip>
>
> > MW: The way I have it in my head is that thing is a member
> of class, and
> > class is a subclass of thing, so thing is a member of thing. What I
> > understand from this (particularly following Chris M's
> recent note) is
> that
> > this means that Thing is not a set (neither is class) but a
> proper class.
>
> The IFF Foundation Ontology that I referenced in the message
> [http://suo.ieee.org/ontology/msg01159.html] has added
> ur-elements (atomic
> things) to the approach to foundations outlined in chapter 2
> of the book
> Abstract and Concrete Categories (1990) by Jirí Adámek, Horst
> Herrlich and
> George E. Strecker. This structures the set-theoretic world up into a
> hierarchy of collections of (only) 3 levels -- sets, classes and
> conglomerates, from smallest to largest -- but only needs to
> axiomatize the
> first two. Intuitively, a set is a small class and a class is a large
> collection of sets. Letting *Ur* denote the collection of all
> ur-elements,
> *Set* denote the collection of all sets and *Class* the
> collection of all
> classes, some of the set-theoretic facts are as follows.
>
> Every set is a class.
>
> The members of each class are either sets or ur-elements.
>
> There is a largest class, call it *Thing*, that is
> partitioned as *Thing* =
> *Ur* + *Set*.
>
> There are classes that are not sets (proper classes), such as
> *Set* and
> *Thing*.
>
> Every class is a conglomerate.
>
> There are conglomerates that are not classes, such as *Class*.
>
> So, this agrees with your comment that "thing is a member of
> class", but not
> the statement that "class is a subclass of thing" since
> *Thing* only has as
> members either ur-elements or sets. It would change this to either "a
> particular class is a subclass of thing" or to "set is a
> subclass of thing".
MW: But unless Class is a subclass of thing then the things that are members
of class are not members of thing, and everything is a member of thing.
MW: I agree that having a hierarchy of class-like things all of which have
members is probably a good way to partition the problem. However, I don;t
think you can have levels where the basics of membership do not apply.
> In particular, it would not be true that "thing is a member
> of thing", since
> *Thing* is not a set and only ur-elements or sets can be
> members of classes.
> It agrees with the statement that "Thing is not a set
> (neither is class) but
> a proper class", and would add the statements that *Class* is
> not even a
> class, but the next thing up, a proper or large conglomerate
> (the singleton
> collection {*Thing*} is an example of a small conglomerate).
> Other examples:
> the collection of all Information Flow classifications is a
> class, and the
> collection of all categories is a conglomerate.
>
> <snip>
>
> Robert E. Kent
> rekent@ontologos.org
>
>
Regards
Matthew
============================================
Matthew West
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