Re: A consideration concerning the diagonal argument of G. Cantor
WM
> In article <376a4188-8e72-4e77-be46-aa1b46862...@x41g2000hsb.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
> ...
> > Cantor's conclusion was: Every entry differs in at least one digit
> > from the AD. Therefore they cannot be equal. This conclusion is wrong.
> > In his 1892 paper (and later)Cantor did not consider cases like
> > 1.000... =3D 0.111....
>
> In his original paper on the diagonal proof Cantor did for that not
> consider numbers at all. And indeed, the sequence
> {m, w, w, w, w, ...}
> is different from
> {w, m, m, m, m, ...}
as 1.000... is different from 0.111...? The discussion about what
Cantor thought
http://groups.google.com/group/sci.math.research/browse_frm/thread/5d9926c33ee6e40f?scoring=d&hl=de
unfortunately could not settle the question.
Regards, WM
Virgil
<546f3718-5e3b-45f3...@59g2000hsb.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On 3 Okt., 14:31, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > In article
> > <376a4188-8e72-4e77-be46-aa1b46862...@x41g2000hsb.googlegroups.com> WM
> > <mueck...@rz.fh-augsburg.de> writes:
> > ...
> > > Cantor's conclusion was: Every entry differs in at least one digit
> > > from the AD. Therefore they cannot be equal. This conclusion is wrong.
> > > In his 1892 paper (and later)Cantor did not consider cases like
> > > 1.000... =3D 0.111....
> >
> > In his original paper on the diagonal proof Cantor did for that not
> > consider numbers at all. And indeed, the sequence
> > {m, w, w, w, w, ...}
> > is different from
> > {w, m, m, m, m, ...}
>
> as 1.000... is different from 0.111...?
"1.000..." differs from "0.111..." , just as {m, w, w, w, w, ...}
differs {w, m, m, m, m, ...}, which is more to the point.
Herbert Newman
>>>
>>> In his original paper on the diagonal proof Cantor did for that not
>>> consider numbers at all. And indeed, the sequence
>>>
>>> (m, w, w, w, w, ...)
>>>
>>> is different from
>>>
>>> (w, m, m, m, m, ...).
>>>
>> As 1.000... is different from 0.111...? [WM]
>>
> "1.000..." differs from "0.111...", just as (m, w, w, w, w, ...)
> differs from (w, m, m, m, m, ...) [...]
>
Or using the same sequence notation: as
(1, ., 0, 0, 0, ...)
differs from
(0, ., 1, 1, 1, ...).
WM is still mixing up names (i.e. decimal representations) with the objects
denoted by these names (i.e. numbers).
Helb
Dik T. Winter
> On 3 Okt., 14:31, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
...
> > In his original paper on the diagonal proof Cantor did for that not
> > consider numbers at all. And indeed, the sequence
> > {m, w, w, w, w, ...}
> > is different from
> > {w, m, m, m, m, ...}
>
> as 1.000... is different from 0.111...?
They *are* different if you only look at the sequence of digits. They
become the same when you consider them to be binary representations of
nubers. The argument of Cantor is about sequences, not about
representations. And the paper in question does not talk about
representation at all.
But when you want, you may consider m and w as two of the three
possible digits in a ternary representation (the Cantor set...).
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
WM
> In article <546f3718-5e3b-45f3-b2f4-2ec0a38d4...@59g2000hsb.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 3 Okt., 14:31, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> ...
> > > In his original paper on the diagonal proof Cantor did for that not
> > > consider numbers at all. And indeed, the sequence
> > > {m, w, w, w, w, ...}
> > > is different from
> > > {w, m, m, m, m, ...}
> >
> > as 1.000... is different from 0.111...?
>
> They *are* different if you only look at the sequence of digits.
That is unimportant if real numbers are in question.
> They
> become the same when you consider them to be binary representations of
> numbers. The argument of Cantor is about sequences, not about
> representations. And the paper in question does not talk about
> representation at all.
But it talks about real numbers and intervals of real numbers.
>
> But when you want, you may consider m and w as two of the three
> possible digits in a ternary representation (the Cantor set...).
No. The diagonal elements can only get m or w.
Regards, WM
Herbert Newman
>>>
>>> In his original paper on the diagonal proof Cantor did for that not
>>> consider numbers at all. And indeed, the sequence
>>>
>>> (m, w, w, w, w, ...)
>>>
>>> is different from
>>>
>>> (w, m, m, m, m, ...)
>>>
>> as 1.000... is different from 0.111...? [WM]
>>
> They *are* different if you only look at the sequence of digits. They
> become the same when you consider them to be binary representations of
> nubers.
>
With other words (for clarity):
(1, ., 0, 0, 0, ...) =/= (0, ., 1, 1, 1, ...)
but
1.000..._2 = 0.111..._2
Herb
Dik T. Winter
> On 6 Okt., 14:38, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> com> WM <mueck...@rz.fh-augsburg.de> writes:
> > > On 3 Okt., 14:31, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > ...
> t
> > > > consider numbers at all. And indeed, the sequence
> > > > {m, w, w, w, w, ...}
> > > > is different from
> > > > {w, m, m, m, m, ...}
> > >
> > > as 1.000... is different from 0.111...?
> >
> > They *are* different if you only look at the sequence of digits.
>
> That is unimportant if real numbers are in question.
They are not.
> > They
> > become the same when you consider them to be binary representations of
> > numbers. The argument of Cantor is about sequences, not about
> > representations. And the paper in question does not talk about
> > representation at all.
>
> But it talks about real numbers and intervals of real numbers.
Not in the section where the diagonal proof is given. It is later *applied*.
> > But when you want, you may consider m and w as two of the three
> > possible digits in a ternary representation (the Cantor set...).
>
> No. The diagonal elements can only get m or w.
Irrelevant. You may consider m and w as two of the three possible
digits in a ternary representation. And in that case you have a valid
proof that the Cantor set is uncountable. Something that is given more
general in the theorem on pages 237 and 238 of the collected papers. The
theorem which you cited incomplete in your paper, and so wrong.