SUO: Re: Consensus?
Jon and others,
Initially, the IFF axiomatization included the axioms for topos theory.
Quoting Goldblatt in the prospectus of his book "Topoi" ["set theory
provides a general conceptual framework for mathematics. Now since category
theory, through the notion of topos, has succeeded in axiomatizing set
theory, the outcome is an entirely new *categorical foundation of
mathematics*! The category-theorists attitude that "function" rather than
"set membership" can be seen as the fundamental mathematical concept has
been entirely vindicated."] However, since the topos axioms were thought too
heavy by various SUO WG members, these have been (at least temporarily until
absolutely needed, which means probably when the axiomatization for the IFF
lower set namespace is completed) withdrawn in favor of a more bottom-up
approach aimed at trying the model the mathematical intuitions of the
"working ontologist". BTW, the IFF follows the spirit mentioned in
Goldblatt's note, that the notion of function is fundamental, whereas the
notion of relation is secondary. The IFF metalevel only uses the notion of
binary relation -- multivalent relations are modeled at the IFF metalevel,
but actually appear and are used only in the object-level ontologies. This
contrasts with the Common Logic KIFish approach that the notion of relation
is fundamental. In fact, the IFF has a _very_ elaborate, flexible and
intuitive axiomatization for multivalent relations that is centered in the
IFF Ontology meta-Ontology in the lower IFF metalevel.
The IFF is based upon a three-tiered set-theoretic backbone of small
collections (called sets), large collections (called classes), and generic
collections. The category denoted by *Set* of all small sets and all
functions between small sets is a case in point. The objects in this
category are intuitively small sets. Now in general, a category consist of
two components -- its collection of objects and its collection of morphisms
(aka arrows). The collection of all small sets, which constitutes the object
collection of *Set*, is a class. So also is the collection of all functions
between small sets. A (normal) category is a category whose object and
morphism collections are set-theoretic classes (set-theoretically larger
categories are called "quasi-categories" in the IFF and elsewhere). The
collection of all (normal) categories is a generic collection -- neither
small nor large. These comments indicate some of the intuitive notions in
category theory. And this is what the IFF axiomatization aims to represent,
at least at first cut. The three-tiered set-theoretic backbone of generic
notions, large notions, and small notions is reflected in the three
metalevels of the IFF: top, upper and lower.
From Robert "The IFF radical"
----- Original Message -----
From: "Jon Awbrey" <jawbrey@oakland.edu>
To: "Adam Pease" <apease@ks.teknowledge.com>
Cc: "Robert E. Kent" <rekent@ontologos.org>; "John F. Sowa"
<sowa@bestweb.net>; "SUO" <standard-upper-ontology@ieee.org>
Sent: Friday, May 16, 2003 6:50 AM
Subject: Re: Consensus?
> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
>
> SUO WG,
>
> I have been looking at all three proposals again,
> and I find the incompatibilities too radical for
> words. I would like to suggest a strategy for
> moving the project forward, independently of
> whatever official status these or any other
> potentials have at a given moment in time.
>
> Let each conributor to the possible merger
> first index each of their terms with the
> name of the group, for example, giving
> "Set_IFF", "Set_OC", "Set_SUMO", etc.
> for their corresponding concepts of
> a set. This would facilitate the
> formation of "disagreement sets"
> as used in standard unification
> algorithms, though, of course,
> only in a rough way at first.
> The axioms or comments for
> the several concepts could
> then be hung on the lattice
> with glee, and if any of them
> turn out to be coincident, that
> would be just wonderful, even if
> purely coincidental, but hanging on
> to the indices would also leave room
> for future extensions by other parties.
>
> Mathematicinas and others have coped with a similar situation
> for many years now, for instance, implicitly dealing with
> Set_Naive, Set_Axiomatic, and within the latter class,
> Set_HBvNG, Set_ZF, and several others. Of course,
> this was handeled informally, in a context-wise,
> discourse-wise way, and that is not a dodge
> that we can get away with as we formalize
> concepts in a computable fashion.
>
> Jon Awbrey
>
> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
>
>