Re: Cantor's definition of set

Peter_Smith schrieb:
On 25 Oct, 20:51, John Jones <jonescard...@xxxxxxx> wrote:

OK. But if the members of a set have nothing in common, then how can
we have a set of numbers, for numbers are derived from each other -
they have something in common.

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 anyway
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.

Here is an example (note the other notation with
the vertical bar):

{ object | objects is a positive integer numeral,
i.e. a sequence of digits whereby these objects
are treated the same independent on the number
of leading zeros}
{ object | object lives in new york on the 10th October 2007
and is a cat }
etc..

Now most of the set theories are based on the following:
- membership relation (there is this special open e sign for it)
- equality relation (there is this special = sign for it)
- arbitrary further predicates and relations (you can choose
what ever you want, that is the so called background
language you choose)

Now 80% percent of the axioms of a set theory only use the
membership relation and the equality relation. The only axiom
that usually referes to further predicates and relations is
the comprehension axiom, which is a usually an axiom schema.
This axiom schema looks as follows in ZF:

forall w1,..,wn forall x exists y forall z
(z in y iff z in x & A(z,w1,..,wn))
w1,..,wn distinct from x,y

Here you can replace A by an arbitrary formula with free
variables z,w1,..,wn from your background language. The
comprehension axiom states a certain richness of a set
model of the set theory for a given background language.
The richness it claims is that everything we can express
in our background language does also exist as a set.

But this existence is proclaimed in two ways. First by the
existence quantifier "exists y", which states that the
set is normally part of our domain of discourse. And further
by the membership relationship "z in y", which is then
related and constrained to your arbitrary statement A.
But when these sets exist further set theoretic formations
can be based solely by the membership relation.

Best Regards



.

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