Re: SUO: Re: nature of organisation
> 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