Re: SUO: Re: nature of organisation
Pat and Chris,
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.
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.
But for the overwhelming majority of nonmathematicians, a class is
a object-oriented notion from C++, Java, etc.
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. 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.
2. Another term should be adopted for the collections uses in
mereology, which do not support the distinction between subset-of
and element-of -- i.e., subcollection is the only notion of
parthood for collections. I recommend the word 'collection'
for this notion, but I am willing to go along with any other
term that receives a consensus.
3. The Cyc distinction between 'set' and 'collection' is incoherent,
since they are mainly interested in countable sets (or collections)
for which every predicate defines a set, and every set defines
a predicate. If anyone wants to talk about two predicates with the
same extension in the actual world, but different extensions in
some possible world, then I suggest they use the word 'predicate'.
4. For the kinds of organizations that have been considered in SUMO,
there is always some predicate that specifies the goal, purpose,
or intention (with a "t"). That predicate is the chief concern
of any axiomatization, and the collection of people, dogs, cats,
or other entities in the organization is determined by the
defining predicate.
John Sowa