Re: Cantor Confusion

In article <1171011850.731985.236670@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
On 8 Feb., 13:48, "Dik T. Winter" <Dik.Win...@xxxxxx> wrote:
....
> Why not. IIIII can be abbreviated by 5 or V or your proposal or ...

Again, still no definition. So IIIIII can be abbreviated 5I, I5, VI or
IV? And IIIIIIIIII can be abbreviated 55 or VV?

That is a matter of convention. Therefore the unary represantation,
which is not a matter of convention, is preferable.

Also unary representation is a matter of convention in my opinion.

> > No. Numbers are abstract entities. Their representations are
> > concretisations of those abstract entities.
>
> Where and what are these entities?

Abstract entities.

Where are they? Further your answer does not distinguish between
numbers and impressions of colour or beauty. If you say a number
exists, then you must be able to distinguish it from *everything* else
(including other numbers. This is certainly not done by saying:
abstract entity.

But the distinctions can be performed. I know precisely how to compare
the abstract entities behind sqrt(2) and sqrt(3).

> You have to use some agrrement if you want to use abbreviations. For
> IIII you need no agreement.

Oh. I think one is needed.

I believe that without any other cultural contact two persons could
start communicating by unary numbers.

Yes, your belief.

> If you use "natürliche Zahl" or "N" does not matter. You use something
> not yet defined (and never defined by Peano).

You apparently do not know how a recursive definition works. (1), (2)
(when properly corrected) define the natural numbers. (3) defines the
set of natural numbers.

I know it. Cantor already used it (see p. 128 of my book). That does
not mean that it is good.
(1), (2) have not to be corrected (they appear in several text books).
Your assertion that " x is in N" and "x is a natural number" were
different is wrong. Both has to be defined.

But, in (1) you state "is a natural number", in (2) you use "is in N",
without stating anywhere in that article that the two things mean the
same. So the combination of those two is *not* a proper definition.

> N is only an abbrevation of set of natural numbers.

An undefined abbreviation. When doing definitions you should do it the
correct way. See:
We are going to define natural numbers:
(1) 1 is a natural number
(2) if a is a natural number, the successor of a is also a natural
number
These two together define the natural numbers.

Not yet. According to your definition also -7 and pi could be natural
numbers.

How do you come to that conclusion? At which step is either -7 or pi
generated by (2)?

But we could write your two axioms also as:
We are going to define N:
(1) 1 is in N
(2) if a is in N, the successor of a is also in N.
These two together define N.

Right. But not as you did:
(1) 1 is a natural number
(2) if a is in N, the successor of a is also in N.
These two together define the natural numbers.
By this definition only 1 is a natural number, because you can not
even start with statement (2).

> Correct. There is no infinite line existing. All we assume is a line
> longer than any line we have measured yet.

Such lines also do not have physical existence. Each and every physical
line has a width smaller than anything measured yet. And I do not think
that physical lines are really straight either.

The physical existence of a line is "a measurable distance" between
two points. The points exists as sets of coordinates.

Oh. What is a circle? What is a parabola? Anyhow, what you do is not
plane geometry, but analytic geometry.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
.

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