From: Dik T. Winter on
In article <1158312563.101733.242910(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes:
>
> Dik T. Winter schrieb:
>
> > > > Yes, it is not in the list. So what is the problem? All possible
> > > > indexes are in the list, so:
> > > > (1) A{p = digit position} E{q = list item} {such that q
> > > > indexes p}
> > > > which you deny.
> > >
> > > That is not an argument.
> >
> > *Why* is it not an argument? You state: "all possible indexes are in the
> > list". What is *wrong* about the argument? You always state "it is
> > wrong", but never state what part of the argument is wrong.
>
> It is wrong because not for all {p = digit position} there exists {q =
> list item} {such that q indexes p} , but only for those p, which can
> be indexed, i.e., which are present in the list.

But see above: "All possible indexes are in the list".

> As 0.111... is not in
> the list, there must be a digit position, which is not in the list. (As
> such a position cannot be identified, 0.111... cannot exist.)

Again: define 0.111... is the string with an 1 in the p-th position for
each natural number p, and with no other positions. I would think this
is a clear definition. It clearly can be indexed. And there is not a
digit position which is not in the list. *And it exists*. Your
statement: "As 0.111... is not i n the list, there must be a digit
position, which is not in the list" is unfounded and without proof.

> > No. There are "numbers" (exactly one) that can be indexed by that list
> > but that is not in that list.
>
> That is a false belief. And why should there be only one of those
> numbers? Why couldn't it be two at least or even ten? If you answer
> this question, you have a good chance to recognize that not even one
> exists that is not in that list but can be indexed by the numbers of
> that list.

Because there is only one set that contains *all* natural numbers.

> > But there is a single "number" that can be indexed by your list but not
> > covered by your list. 0.111... is one (as I defined it above). But it
> > is *not* a natural number.
>
> Therefore it cannot be indexed by natural numbers. Try to find out, why
> you insist on only one of those strange numbers. There could be
> infinitely many. Could they all be indexed by natural numbers?.

There is only *one* set of natural numbers, so there is only *one*
sequence 0.111... that can be indexed using *all* natural numbers.

> > Do you agree that the non-terminating list can index non-terminating
> > numbers? If the answer is no, why?
>
> Because it should index at least two different of those infinite
> numbers if it could index one of them.

Why?
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Han de Bruijn on
Han de Bruijn wrote:

> stephen(a)nomail.com wrote:
>
>> You are in disagreement with WM, who has clearly stated
>> that sqrt(2), pi, e and all the irrationals are not numbers.
>
> I think that you are babbling, but before saying so, I shall first read
> what WM actually says about it.

I can't find any document on VM's site where he says that sqrt(2), pi, e
and all the irrationals are not numbers. Would you mind to point me to a
reference? (I've tried "On the abundance of the irrational numbers")

Han de Bruijn

From: Dik T. Winter on
In article <1158313360.862711.19070(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes:
>
> Dik T. Winter schrieb:
>
> > > That does not exist. At least it is not described by n. III is a
> > > representation of 3.
> >
> > Now you shift back to representation, pray remain consistent.
>
> Representation is number. There is no difference. Numerals have no
> "soul".

Makes no sense to me. You may represent a natural number as you wish,
that does not change anything.

> > > If you know the positions by heart, then you need no addition actually.
> > > You had already used it or the person who devised that technique had
> > > used it. But earlier or later your knowing by heart will end and you
> > > will have to count +1.
> >
> > Nope. I will only to need to know successors. Anyhow, you read much more
> > in the successor of the Peano axioms than is present. The successor is
> > defined without even any knowledge of addition at all. So
> > succ(George V) = Edward VIII, succ(Edward VIII) = George VI and
> > succ(George VI) = Elizabeth II within the set of British kings and queens.
> > I do not think what way of addition you would propose for that. Of
> > course, this successor function does not satisfy all of Peano's axioms,
> > but I hope you get the idea.
>
> I was talking of *counting* (remember: Zahl and zaehlen) which requires
> natural numbers which require the ability to add 1 after you have run
> out of the numbers known by heart.

You stated that you needed counting to determine the successor. That is
false. The successor is defined without any reference to counting.

> > > The old Greek and othe cultures have just used their letters as numbers
> > > too.
> >
> > Yes, every culture had their representation of numbers. Sometimes base 10,
> > sometimes other bases. Base 20 is quite common in Europe. Mixed bases
> > are also used. Sometimes they were used to count from 1. Sometimes from
> > 0.
>
> Who did so before Cantor?

Some people from India. There are quite a few where you can read about
the zeroth year of the reign of somebody. Have a look at the vast number
of different calendars and year countings that have been in use in India.

> > I may note that the decimal system you appear to like most is, in
> > Europe, mostly due to Simon Stevin, and at that time it included 0 as
> > a digit.
>
> Did he count from 0?

As he never wrote about it, I have no idea.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on
In article <1158324386.661753.251440(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes:
>
> Dik T. Winter schrieb:
>
>
> > > Index = natural number. There is no infinite index, because indexing is
> > > identifying.
> >
> > That does *not* explain why the list contains all numbers that can be
> > indexed.
>
> Let m be a unary number with m 1's, and n a list number with n 1's.
> If E n >= m, then m is in the list and can be indexed completely by
> list numbers.
> If not E n >= m, then m is not in the list and cannot be indexed
> completely by list numbers.

Yes. With finite m. But a unary "number" with infinitely many
1's does not fall under the statement. So it does not follow that
it can not be indexed.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Han de Bruijn on
Han de Bruijn wrote:

> I can't find any document on VM's site where he says that sqrt(2), pi, e
> and all the irrationals are not numbers. Would you mind to point me to a
> reference? (I've tried "On the abundance of the irrational numbers")

Oops! He _says_ it, in the document "Physical Constraints of Numbers":

> Irrational numbers simply are not numbers.

Weird ...

Han de Bruijn