Re: Cantor's definition of set

On Fri, 26 Oct 2007 14:45:48 -0700, John Jones <jonescardiff@xxxxxxx>
wrote:


'2' is a numeral. 2 is a number.

Please try to get that straight before engaging in even more idiotic
babble, ok?

"2" is a name we use to denote the number 2.

Of course, there are other names too, "1 + 1" for example. Indeed,

1 + 1 = 2.

(See!)

Now you claim:


The number 2 is always generated by an application, so I cannot
propose simply 2. I cannot say 2 is a number and leave it at that. I
must specify, and not simply assume, an application that generates it.

Let's -for the sake of the argument- accept that point of view.

YES, We DO specify the number 2 in math. (And especially set theory).

Ever heard of the so called Peano axioms? There we take (exactly) ONE
natural number, namely 0, for grated.

Axiom: 0 is a natural number.

Then we can define the number 1 as successor of 0:

1 =df s(0),

and the number 2 as successor of 1:

2 =df s(1).

Hence:

2 = s(s(0)).

(With other words, the number 2 is the successor of the successor of 0.
This is the specification you asked for.)

Now in axiomatic set theory we do not even take the number 0 "for
granted". But we _define_ it the following way:

0 =df {},

where {} denotes the so called empty set. And we can PROVE in axiomatic
set theory that there is an (actually exactly one) empty set.


Even if numbers existed in a third Platonic realm as stand-alone real
entities, I would not be able to recognize them as numbers except by
employing an application, like counting.

Let's -for the sake of the argument- accept that point of view. See,
comment above.


But in that case, I make numbers myself.

How would you know? (!)


So there is no Platonic realm of numbers.

No, sorry. This certainly won't work as a "refutation" of Platonism. :-)

You might like to read some of Gödel's comments concerning "Realism".

"[...] despite their remoteness from sense experience, we do have
something like a perception also of the objects of set theory, as is
seen from the fact that the axioms force themselves upon us as being
true. I don't see any reason why we should have less confidence in
this kind of perception, and more generally, in mathematical intuition
than in sense perception."

(Kurt Gödel--quoted in: Wang, Hao (1996). /A logical journey: From
Gödel to philosophy/. Cambridge, MA: The MIT Press.)

"Mathematical realism holds that mathematical entities exist
independently of the human mind. Thus humans do not invent
mathematics, but rather discover it [...]."

Source: http://en.wikipedia.org/wiki/Mathematical_realism


A sequence is a certain KIND of set. There's no conflict in a set
being of a certain KIND.

A sequence and a collection are mutually exclusive I would have
thought - a collection is indifferent to order.

Yes, one might think so, agree. But this thought a _provable_ wrong.

For a starter you might google for the term "ordered pair". (Yes, we can
define a so called /ordered pair/ in set theory.)


F.

--

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

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