Re: Cantor's definition of set

On Sat, 27 Oct 2007 00:15:14 +0200, G. Frege <nomail@invalid> wrote:


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 in 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),

or 1 = 0'

and the number 2 as successor of 1:

2 =df s(1).

or 2 = 1'

Hence:

2 = s(s(0)).

or 2 = 0''


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

"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."

Now we might use the symbol "0''" instead of the name "2" for denoting
the number 2.

This way we would actually _express_ (or specify) the way how this
number is "generated" (starting with the number 0).

It's just a historic fact that we usually denote the number 0'' by the
symbol "2".

(One might consider "0''" to be a description in disguise, while "2"
obviously is only a proper name - with no description -but convention-
involved.)


F.

--

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

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