Re: singleton sets

I think nested lists are more common that nested sets.
Both can be useful in complex data structures.

In the mKR data structures, my workhorse is
    SYMBOL(stype,svalue)
I must have at least 100 different stypes,
and svalues are usually lists.
The elements of the list can be any Unicon type, including list.

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;
http://mKRmKE.org/

----- Original Message ----- 
From: "Michael Schneider" <schneid@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>
Sent: Tuesday, August 12, 2008 3:39 PM
Subject: RE: singleton sets


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
FZI Forschungszentrum Informatik Karlsruhe
Abtl. Information Process Engineering (IPE)
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Received on Wednesday, 13 August 2008 07:22:15 UTC