Re: SUO: Re: nature of organisation
Chris M, Chris P, and John --
Thanks to all for for the clarifications. I can use
whatever type of set or class the experts decide is best for
ontologies, I just want to know what its properties
are.
It seems that there is agreement that:
(1) A SUMO Set is a mathematical set (ZF and VNBG).
(2) A SUMO Class is not necessarily defined intensionally,
and can have classes as members, thus is not a VNBG class.
(3) The CYC #$Collection has no exactly corresponding
class in SUMO.
I hope I got it right this time.
Pat Cassidy
===============================
Chris Menzel wrote:
>
> > The fine points of ZF vs. VNBG are only relevant when the sets (or
> > classes) have cardinalities that go far beyond the size of anything
> > that is currently being axiomatized in SUMO.
> >
> > If the largest collections are at most countably infinite, then
> > the applicable theorems in ZF or VNBG are equivalent.
>
> Actually it goes farther than that, John. The two don't disagree on sets
> at all, regardless of cardinality. VNBG and ZF differ only in that,
> roughly speaking, VNBG quantifies over a layer of classes -- so-called
> "proper classes" -- over and above all the sets, whereas ZF only
> quantifies over the sets. Unlike sets, proper classes cannot be members
> of other classes. (Let me note that in the last couple of posts I've sent
> out I might have used "class" carelessly to mean "proper class" here and
> there -- in VNBG, all sets are classes, but some classes, viz., the proper
> classes, are not sets.)
>
> As things currently stand, SUMO classes are NOT VNBG classes, as SUMO
> classes can be members of other classes, and even themselves.
>
> > PC> A related question I have asked and do not yet know the answer to is
> > > > why a "Set" in SUMO is a subclass of "Class" when in most theories the
> > > > opposite is true.
> >
> > CM> No, in fact, in all theories I know of, and notably in the most prominent
> > > theory VNBG (for "von Neumann - Godel - Bernays), all sets are classes,
> > > but not vice versa. (Though it wouldn't be quite right to say that SET is
> > > a subclass of CLASS, as there is in VNBG no class of all classes, unlike
> > > SUMO.) I think you have been snookered by the quirky terminology of the
> > > Frame Ontology -- see below.
> >
> > Discussions like these arise from the fact that the word 'class' has
> > been grossly overused. In mathematical theories, the distinction
> > between sets and classes is only relevant when you get beyond countably
> > infinite.
>
> No, it becomes relevant when you get past cardinality entirely. Proper
> classes can in fact be defined in VNBG as collections that are "too big"
> to have a cardinality.
>
> > But for the overwhelming majority of nonmathematicians, a class is
> > a object-oriented notion from C++, Java, etc.
>
> True, if the overwhelming majority of nonmathematicians happen to be
> programmers. False otherwise.
>
> > To avoid such confusions, I recommend that we drop the word 'class'
> > from SUMO and adopt the following conventions:
> >
> > 1. The word 'set' without any other qualifiers means the mathematical
> > notion of set.
>
> Sounds good.
>
> > If the sets happen to have cardinality greater
> > than aleph0, then the word 'set' may be prefixed as 'ZF-set'
> > or 'VNBG-set', or whatever other pet theory anyone is using.
>
> Except that ZF-set = VNBG-set, as noted. We already have all the
> terminology we need for the distinction in question here, John: countable
> vs uncountable. Though some such prefixing might be necessary were one
> to want to import VNBG into the SUMO to distinguish SUMO classes from
> VNBG classes.
>
> -chris
>
> --
>
> Christopher Menzel # web: philebus.tamu.edu/~cmenzel
> Philosophy, Texas A&M University # net: chris.menzel@tamu.edu
> College Station, TX 77843-4237 # vox: (979) 845-8764
--
=============================================
Patrick Cassidy
MICRA, Inc. || (908) 561-3416
735 Belvidere Ave. || (908) 668-5252 (if no answer)
Plainfield, NJ 07062-2054 || (908) 668-5904 (fax)
internet: cassidy@micra.com
=============================================