Re: Cantor's definition of set

On Fri, 26 Oct 2007 13:29:08 -0700, John Jones <jonescardiff@xxxxxxx>
wrote:


I'm fine with Platonic numbers [...]

Please let's keep that in mind, since it simplifies the following
discussion very much! So -for the sake of the argument- let's assume
that there _are_ some mathematical objects called (natural) numbers.

Now

1,2,3,4,5... is often portrayed as numbers. But aren't they examples
of the signs we use to portray numbers, ...

You are mixing op names with the objects denoted by this names. (-->
use-mention distinction!)

In the following I'm talking about certain numbers, namely the numbers 0
and 1:

The natural number 1 is the successor of the natural number 0.

I do this by using names for the mentioned numbers: the names "0" and
"1".

"In the sentence the name represents the object.

Objects I can only /name/. Signs represent them.
I can only speak /of/ them. I cannot /assert them/.
[...]"

(L. Wittgenstein, Tractatus Logico-Philosophicus, 1921)


and aren't these signs simply arranged in a sequence ...

Yes, if I consider the sentence "Let's consider the numbers 0, 1, 2, 3,
.... " Then in this sentence we see a sequence of names: "0", "1", "2",
"3" (in this order).

Moreover, the natural numbers actually ARE ordered in a certain
"natural" way. 0 usually is considered the first natural number. 1 is
considered to be the successor of 0, 2 is considered to be the successor
of 1, etc. (Note that we are talking about numbers here, not about
signs; but we do that using certain signs/names, of course.)


and not numbers after all?

"In the sentence the name represents the object.

Objects I can only /name/. Signs represent them.
I can only speak /of/ them. I cannot /assert them/.
[...]"

(L. Wittgenstein, Tractatus Logico-Philosophicus, 1921)

This is true for mathematical objects too. (Here the "realistic picture"
of mathematical objects is rather helpful. --> Mathematical Platonsim.)


For I cannot use 1,2,3,4 ... mathematically.

Sure you can.

1 + 2 = 3.

That's certainly a true claim. Or with other words, the sum of the two
numbers 1 and 2 equals 3.


1,2,3,4 ... does not occur in any mathematical calculation.

Well, if you say so. Of course, the NUMBERS 1, 2, 3 do not "occur" in
the /statement/ "1 + 2 = 3". On the other hand the names "1", "2". "3"
used in the statement "1 + 2 = 3" certainly refer to the numbers 1, 2
and 3.


What I am saying is, you can't pull a number out of the application
that generates it.

If you say so. But so what?


It would seem, if this is true, that a set of numbers is an impossibility.

If you say so. But even the "antecedence" of this conditional is
doubtful. Moreover a set of natural number certainly is not doubtful (if
there actually are such objects as numbers and sets.)

So I'll just repeat myself:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So at least *I* can't see any _mathematical_ problems here.


The second major problem which simply won't go away, is this: A set is
the concept of a particular collection or group.

Maybe it would be better to say: A set is a particular collection ...


At least I have seen a set described as either a collection or a group.

Right.


Now a collection does not support sequence

- at least it seems...

But actually it does, as can be shown!


...in fact if a collection could be a sequence, it would be a
sequence and not a collection.

Non sequitur. Actually, in axiomatic set theory certain sets are called
/sequents/ (and some are even called /functions/, etc.).

The following might be helpful for you:

"/Set-theoretic reductionism/ is the view that all the abstract
objects that are talked about in mathematics can be represented as
sets. These representations are called the /set-theoretic surrogates/
for the mathematical objects in question. Perhaps the best-known
example would be taking the /finite von Neumann ordinals/ as the
set-theoretic surrogates for the natural numbers. [...] When, however,
one persists in thinking or talking about mathematical objects without
conceiving of them as their set-theoretic surrogates, one is said to
be thinking or talking about those objects /sui generis/."

(Neil Tennant, A brief account of the fundamentals of set theory,
2003)

Another voice:

"Our contemporary orthodoxy: To show that there are so-and-sos
is to prove 'So-and-sos exist' from the axioms of set theory."

(Penelope Maddy, Mathematical Existence)


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.

It can. Believe me. :-)


I must establish the presence of a sequence independently of its
membership in a set.

No. If you are interested, I'll show you how to perform this task.


This, I advance, is another reason why I cannot have a set of numbers.

Wrong conclusions from false assumptions. :-)


F.

--

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

Relevant Pages

  • Re: Question about Set Theory
    ... believe that set theory itself is such ... As I understand your latest reply, you recognize that the continuum is ... whether there were mathematical objects that have no bearing on "reality". ...
    (sci.logic)
  • Re: which is more fundamental?
    ... >> I have this debate going on with a friend... ... set theory as mathematical objects. ...
    (sci.logic)
  • Re: Cantor Confusion
    ... with their notions modern set theory will perish. ... But this point of view is also entertained in many other modern books: ...
    (sci.math)