Re: Cantor's definition of set

On Oct 26, 2:59?am, G. Frege <nomail@invalid> wrote:
On Thu, 25 Oct 2007 13:37:58 -0700, John Jones <jonescard...@xxxxxxx>
wrote:



Anyway, if the members of a set are unique, then I would not like to
say that they have anything in common. So if numbers are unique, then
I would not like to say that they have anything in common.

Look, man: if a, b are elements in the set M, then at least they have in
common that they are elements in M; won't you think so? And that seems
to be inevitable.

Actually, if we consider the set w (as defined in set theory) we usually
refer to the elements in w as "natural numbers". Hence any element in w
is a natural number (by definition). With other words, all elements in w
have in common to be natural numbers.

So where's the problem? Don't you like natural numbers? Or do you think
there's a problem with the concept of set? :-o

-----------------

Now let's consider Cantors "definition" again:

"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."

So if we assume that there actually are natural numbers

1, 2, 3, 4, 5, ...

(say in some Platonic sense) we might consider the collection into a
whole of the numbers

1, 2 and 3

(which are definite distinct objects).

Hence let M be the collection containing exactly the numbers

1, 2, and 3.

In symbols:

M = {1, 2, 3}.

Where's the problem?

F.

--

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

I was going to give a general response to all, will still do that.

The same two problems keeps coming up, which you have brought up. I'm
fine with Platonic numbers - inasmuch as it does not address these
problems:

1,2,3,4,5... is often portrayed as numbers. But aren't they examples
of the signs we use to portray numbers, and aren't these signs simply
arranged in a sequence and not numbers after all? For I cannot use
1,2,3,4 ... mathematically. 1,2,3,4 ... does not occur in any
mathematical calculation. I can find 1, and 2 and 3 and 4 in a
calculation, but in finding them don't I merely find the signs and not
the numbers themselves? What I am saying is, you can't pull a number
out of the application that generates it. It would seem, if this is
true, that a set of numbers is an impossibility.

The second major problem which simply won't go away, is this: A set is
the concept of a particular collection or group. At least I have seen
a set described as either a collection or a group. Now a collection
does not support sequence: in fact if a collection could be a
sequence, it would be a sequence and not a collection. Don't get me
wrong here. I can have a set of sequences, but the the set itself, on
its own merits, cannot support a sequence. I must establish the
presence of a sequence independently of its membership in a set. This,
I advance, is another reason why I cannot have a set of numbers.
Numbers in a collection are like numerals in a sack, like lottery
'numbers'.

.

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