# Re: singleton sets

From: Frank Manola <fmanola@acm.org>
Date: Wed, 13 Aug 2008 10:59:39 -0400
To: Richard H. McCullough <rhm@PioneerCA.com>
Message-Id: <9F4132D4-45EC-4D9A-B058-552D114200B9@acm.org>
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On Aug 12, 2008, at 5:05 PM, 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}.  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.
>

Whoa!  What we were originally talking about wasn't singleton sets, it
was the following question:

>>>>>
>>>>> 2. X  type  Y;  X  subClassOf  Z;
>>>>> Another neat property: X is an individual and a class.
>>>>> Now I can ... What?  I don't know.
>>>>> Why do you want to do that?

Wanting to be able to treat a class X as an individual may or may not
be a good idea, but this isn't the same as wanting to treat a
singleton set as *the same* individual as its only member.  To
paraphrase your quotation above, in the direction of subtle subject