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Re: [ontolog-forum] [ontology-summit] FW: [ontolog-invitation] Invitatio

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Fri, 17 Dec 2010 11:37:21 -0500
Message-id: <4D0B91C1.80200@xxxxxxxxxxx>
Dear Matthew,    (01)

There are many different versions of set theory, but the most
common ones assume the following axiom or something equivalent:    (02)

>> For any monadic predicate P, there exists a set of all x
>> such that P(x) is true.    (03)

Some logicians use the word 'class' for set-like things that
don't satisfy this axiom.    (04)

> Yes. I'm happy to drop that in favour of a set exists if its members
> exist and can be enumerated (explicitly or implicitly for infinite sets).    (05)

That raises a question about how you enumerate the elements.  ZF set
theory has a systematic layering that requires each set to be composed
of elements from the layers beneath it.  With that definition, nothing
called a "ZF set" can have itself as a member.  Therefore, ZF set theory
can retain the above axiom.    (06)

>> And by the way, Common Logic manages to avoid contradictions
>> by a method similar to what I just described:  the examples
>> that create paradoxes in some versions of logic just produce
>> CL statements in which the paradoxical entity doesn't exist.    (07)

> And what do you call that version of set theory?    (08)

Common Logic does not include any built-in set theory.  It's possible
to state axioms in CL for all the common versions of set theory.    (09)

It's also possible to use CL to state new versions.  If you want
to do so, you could call it "MW set theory".    (010)

John    (011)

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