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singleton sets

From: Richard H. McCullough <rhm@pioneerca.com>
Date: Tue, 12 Aug 2008 14:05:09 -0700
Message-ID: <A948E0C9187447B1AE90DA40211871F1@rhm8200>
To: "Pat Hayes" <phayes@ihmc.us>
Cc: "KR-language" <KR-language@YahooGroups.com>, "Semantic Web at W3C" <semantic-web@w3.org>, "Adam Pease" <adampease@earthlink.net>

Here's someone else who doesn't like singleton sets,
and hence doesn't like classes which are individuals.

John Barwise & John Etchemendy (1992), "The Language of First-Order Logic",
Third Edition, Revised & Expanded, Center for the Study of Language and 
Information, Stanford, Page 212

    Suppose there is one and only one object x satisfying P(x).  According 
to the
Axiom of Comprehension, there is a set, call it a, whose only member is x. 
That is,
a = {x}.  Some students are tempted to think that a = x..  But in that 
direction lies,
if not madness, at least dreadful confusion.  After all, a is a set (an 
abstract object)
and x might have been any object at all, say Stanford's Hoover Tower. 
Hoover is
a physical object, not a set.  So we must not confuse an object x with the 
set {x},
called the singleton set containing x.  Even if x is a set, we must not 
confuse it with
its own singleton.  For example, x might have any number of elements in it, 
but {x}
has exactly one element: x.

Dick McCullough
Ayn Rand do speak od mKR done;
mKE do enhance od Real Intelligence done;
knowledge := man do identify od existent done;
knowledge haspart proposition list;
Received on Tuesday, 12 August 2008 21:09:16 UTC

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