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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 10:39:53 -0500
Message-id: <4D0B8449.4000102@xxxxxxxxxxx>
On 12/17/2010 10:03 AM, Matthew West wrote:
> ISO 15926 does say that a class can be a member of itself for reasons
> Chris P has alluded to, but it is not committed to a particular form of
> non-well foundedness.    (01)

As I have said before, I have no idea what a *class* is supposed to be.
If a class is supposed to be a set, then it should be called a set.
Therefore, I assume a class might be set-like, but it's not a set.    (02)

If a class is not a set, then there is no reason why any particular
axioms for sets should apply to it.  The following statement (or its
translation to any version of logic) does not, by itself, create any
problems or contradictions:    (03)

    "A class may be a member of itself."    (04)

The contradiction that arises in common versions of set theory is
caused by the following axiom:    (05)

   For any monadic predicate P, there exists a set of all x
   such that P(x) is true.    (06)

The contradiction arises with the following choice of P:    (07)

   P = (lambda x)(x is not a member of x).    (08)

If you don't assume that some set (or class) must exist
that is determined by this P, then there no contradiction.    (09)

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.    (010)

John    (011)

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