Re: SUO: On the supreme supertype
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".
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