SUO: 18 May 2002 -- Set Theory
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SUO WG Members, and Other Interested Parties,
In one line of inquiry pursuant to our joint ontological quest,
I have persistently sought some component of ontology that can
be addressed in qualified independence from our diverse tastes
in syntax, that could be tackled a little more constructively,
and whose state of adequation to practical realities could be
advanced in a less adversarial fashion.
After a due amount of preliminary triage, it looks to me
like set theory might be the component that is presently
the most critical.
Quite a bit of my own work over the years has
been dedicated to integrating and reconciling
the set-theoretic, the category-theoretic, and
the pragmatic spins on this basic formal matter,
so I believe that taking another look at these
rudiments will be acutely relevant to any hope
that we might have of ubifying -- a typo, but
it strikes me as apt -- the IFF and the SUMO
styles of ontology, among what others arise.
Most people of my acquaintance operate with a mixture
of "formative" and "formalized" set theories, and are
often surprised when the mélange of different flavors
breaks into a mêlée of clashing ontological palettes.
Here is a start on one tradition in set theory, the
Skolem-Morse-Hilbert-Bernays-Von_Neumann-Gödel line.
It would take another such study to tease out each
of the other main lines of evolution, but perhaps
a contrast with the Zermelo-Fraenkel development
would be enough to bring out the salient points
of the differences that make any difference:
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Set Theory
01. http://suo.ieee.org/ontology/msg04082.html
Appendix. Elementary Set Theory
02. http://suo.ieee.org/ontology/msg04083.html
A.1. The Classification Axiom Scheme
03. http://suo.ieee.org/ontology/msg04084.html
04. http://suo.ieee.org/ontology/msg04086.html
05. http://suo.ieee.org/ontology/msg04088.html
A.2. Elementary Algebra of Classes
06. http://suo.ieee.org/ontology/msg04089.html
07. http://suo.ieee.org/ontology/msg04091.html
08. http://suo.ieee.org/ontology/msg04092.html
09. http://suo.ieee.org/ontology/msg04093.html
10. http://suo.ieee.org/ontology/msg04094.html
A.3. Existence of Sets
11. ...
Links 2 through 10 of the above material are
selected and transcribed into plaintext from:
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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