Re: Cantor's definition of set

On Fri, 26 Oct 2007 00:27:00 +0200, Jan Burse <janburse@xxxxxxxxxxx>
wrote:


Where on earth do you get the idea that the members of a set need have
nothing in common?? Take some set of people: the members of the set
have something in common (being human beings, for a start).

The set membership is defined by the membership relation.
This relation need not be dependent on any relationship
or predication of the members itself. Its just a relationship.

Namely if you have a set:

{123, apple, fritz the cat}

Then all that we know is:

123 member of {123, apple, fritz the cat}
apple member of {123, apple, fritz the cat}
fritz the cat member of {123, apple, fritz the cat}
orange not member of {123, apple, fritz the cat}
12.76 not member of {123, apple, fritz the cat}
...

But this membership relationship does not refer in any way
to properties of 123, fritz the cat or orange. The only
properties it referes to is equality. So we could have
as well:

elmar member of {123, apple, fritz the cat}

When elmar equals fritz the cat.

On the other hand when we "define" sets, we might refer
to properties of the elements. But the resulting set itself
is just defined by the member ship relation, and nothing more
and nothing less. But the definition might involve
relationships and/or predicates of the members.

That's right. BUT we might claim that all elements in the set

{123, apple, fritz the cat}

have a certain "property" in common, namely to belong to the set

{123, apple, fritz the cat}.

I guess that's what Halmos meant with his statement:

"It is ubiquitous mathematical practice to identify a property with a
set, namely with set of all objects that possess the property [...]."

(Paul R. Halmos, Naive Set Theory)


F.

--

E-mail: info<at>simple-line<dot>de
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