Re: An uncountable countable set

In article <1152384019.621477.88880@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
*** T. Winter schrieb:

> > > And what is the consequence of this?
> >
> > That the calculation of the digits can be done in parallel?
>
> It is done in zero time. It is determined from the beginning. Why
> should something "be done"?

It was you who remarked on the calculations...

The result of a calculation does not depend on the time you need to
obtain it.

No, but it may depend on the ordering of the calculations. So there is
an inherent difference between calculations that are independent of each
other and calculations where some calculation can not be done before
all previous calculations are done. So when calculating the digits of
the diagonal number it does not matter what digit you calculate first.
In your re-ordering sequence it makes a strong difference what re-ordering
you do first.

> But he needs to consider every digit with equal weight. That is not a
> limit process.

What did he *mean* when he wrote "with equal weight"?

He did not write that. I used this description to express that all
digits must be distinguishable from the exchanged digits and that the
resulting number must be distinguishable from the number with one digit
left unexchanged, be it the first or the last one (which can be
recognized).

Yes, that is what I wrote in some of my articles (that you have read).
The limit process is *not* needed to distinguish the diagonal number from
all other numbers of the list. I is needed to show that the diagonal
number is a real number.

By the way, Virgil's arguing leads to the "paradox": There is no last
number and not that one next to the last and so on. Which one is the
first one that does not exist?

Eh? Care to explain? What is the paradox? There is no last numbers.
Hence, there is also no predecessor of the last number. That seems
pretty clear to me. There is no predecessor of something that does not
exist. What makes you think that there should be a first one that does
not exist? This is (to me) close to gibberish.

No. You do *not* and he does *not*. In the Cantor diagonal the number
obtained is a real number using limits (in the Cauchy sense) with the
definition of real number by many others (that can be proven to all be
equivalent with each other). The limit used by Cantor is precisely the
epsilon argument, together with majoration and minoration on the ordered
set of rational numbers. By Cauchy any sequence of decimal digits has a
limit, but it is not certain whether that limit is in the defined set of
numbers. By Cantor, Dedekind, Weierstrass and a host of others (using
various formulations), such a limit (when starting with rational numbers)
is defined as a real number. That is the place where Cantor uses the
limit (although he may not have expressed it as such), i.e. showing that
the resulting number is a real.

Cantor does not at all use real numbers but merely infinite sequences
with only two different symbols w and m (which might be interpreted as
binary representations but were not). But that there is no satisfactory
limit consideration becomes clear from the following: We know that
0.999... = 1.000... This leads to the result that a change of 1 in the
limit where the digit number goes to oo does not have the effect which
would be required in order to distinguish the diagonal number from the
list numbers.

You still keep with Cantor's original papers. Perhaps you are right. But
since that time the arguments have been improved (and more understood).
Arguing against Cantor's original papers is futile. But let me take one
point. You write:
with only two different symbols w and m (which might be interpreted as
binary representations but were not).
And you continue with taking them to be binary representions. This is
dishonest. You state, explicitly, that they were not binary representations,
and argue against them as if they were binary representations.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

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