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Re: [ontolog-forum] First-Order Semantics

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Fri, 15 Jun 2007 20:39:49 -0400
Message-id: <46733155.90106@xxxxxxxxxxx>
Chris and Waclaw,    (01)

That is a good way to state the point:    (02)

CM> If (per impossibile) D were to contain all possible relations
 > over D then (even ignoring issues of non-well-foundedness) we'd
 > have card(D) = 2^card(D), which of course contradicts Cantor's
 > Theorem.    (03)

But it would be possible to start with a set Z, such as the
integers, and derive another set S of all relations over Z.
Then the domain D could be the union of Z and S.    (04)

vQ> My point was that a relation over a single set is a set of 1-tuples;
 > but see it as yet another example of my terminological pickyiness.    (05)

That is not the way the phrase "the relations over a set S" is
normally used in math and logic:    (06)

  1. A set of 1-tuples from S is isomorphic to some subset of S.  The
     set of all such sets is isomorphic to the set of all subsets of S.    (07)

  2. The set of all dyadic relations over S would be isomorphic to
     the set of all subsets of the Cartesian product SxS.    (08)

  3. The set of all n-adic relations over S would be isomorphic to
     the set of all subsets of SxSx...xS (with n copies of S).    (09)

The term "the set of all relations over S" normally means the
union of the above for all positive integers n.    (010)

John    (011)

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