Re: Cantor's definition of set

John Jones schrieb:
Cantor said:

'By a "set" we mean any collection M into a whole of definite,
distinct objects m (which are called the "elements" of M) of our
perception [Anschauung] or of our thought.'
Further, except in a multiset, every element of a set must be unique-
no two members may be identical.

My observation is this:
Surely, these definitions disallow a set whose individual members are
characterized by being sequenced, or heirarchical, or derive their
properties from being in a collection.

Maybe cantor had only ur-elements in mind. i.e. elements that
are not further structured. But modern set theory does not
put any constraints on the elements of the sets. Any thing that
is in the domain of discourse can also be in a set.

So if you have sequenced or hierarchical objects in your domain
of discourse these things are perfectly allowed to be elements
of sets. And if you have in your background language devices
to talk about them you might even be able to define and distinguish
them in your background language, and thus define sets of them.

Independent whether you have the appropriate language, they
can exist in a set model. But if you have the appropriate
language they will even provably exist, these sets that
contain sequenced, or hierarchical objects. The simplest language
additions you need are predicates:

S: unary predicate of the sequenced objects
H: unary predicate of the hierarchical objects

Then you have automatically:

S(x) & S(y) & x<>y -> exists z(z={x,y})
etc..

For the set elements that are itself sets you don't have to do a lot.
You have only to note two things. We have the powerset axiom, which
assures the existence of a typical set of sets. And the formula A
in the comprehension might also use the membership relation and
the equality relation. So the set theory stipulates already a
rich model of sets of sets. And you can mix and match these with
the sets that you introduce by yourself.

Example:

I_S: The graph of the identity function on S, that sends an x to x.

I_S = {<x,x> | S(x) } <--- this is a short hand for:

= { y in Pow(SxS) | exists x(y=<x,x> & S(x)) }
<-- which is a further short hand for:

= ... use your prefered pairing function background stuff ...

Best Regards





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