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RE: singleton sets

From: Michael Schneider <schneid@fzi.de>
Date: Wed, 13 Aug 2008 00:39:30 +0200
Message-ID: <0EF30CAA69519C4CB91D01481AEA06A0AD9744@judith.fzi.de>
To: "Richard H. McCullough" <rhm@pioneerca.com>
Cc: "KR-language" <KR-language@YahooGroups.com>, "Semantic Web at W3C" <semantic-web@w3.org>, "Adam Pease" <adampease@earthlink.net>, "Pat Hayes" <phayes@ihmc.us>
Hi Dick!

Richard H. McCullough wrote:

>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}.  

Ok!

Some students are tempted to think that a = x.. But in that
>direction lies, if not madness, at least dreadful confusion.  

Agreed!

After all, a is a set (an abstract object)

Yes!

>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.

Indeed!

>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.

Wait, now I'm confused! How can there be a singleton for the *set* x? Isn't
it crazy to talk about sets, which are themselves included in sets?

Very interested in your answer!

Michael

--
Dipl.-Inform. Michael Schneider
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Received on Tuesday, 12 August 2008 22:40:12 GMT

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