Re: An uncountable countable set


*** T. Winter schrieb:

> 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.

And that is all you have to object against my proof? We have a fixed
scheme of transpositions.
But we could apply the transpositions even without any law, applying
only the rule that a pair which is already ordered shall not be treated
for a second time. Then we can apply countably many arbitrary
transpositions until all elements are ordered by magnitude. The limit
to which every path will lead is always the same: The set of rationals
ordered by magnitude and by natural indices.

But there is no limit process. The *only* criterion about manipulations
on countable infinite sets is whether one can determine *precisely* at
which natural number something happens: After how many steps in a
well-order of |Q the fraction 4711/235537 will appear, for instance, or
in which line of Cantor's list a certain diagonal element will be
placed and so on. And I can determine *precisely* after how many steps
the number 4711/235537 will be inserted in the order by magnitude with
all of its predecessors of the initial well-order. This is fixed and
can be calculated for any rational number. Therefore all rational
numbers are covered and will successively appear in the well-order. The
argument that there remain always infinitely many other rationals is
wrong, because by definiton the fate of each and every rational is
determined and can be calculated. It is not necessary to really carry
out any transposition.

> > > 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. It is needed to show that the diagonal
number is a real number.

But that is not interesting. It is easy to see that the diagonal is a
sequence of the same sort as are the list entries. Whether they are
real numbers is uninteresting. Interesting is that such sequences are
uncountable.

But in order to prove that the diagonal differs from every entry, there
a limit is required but not available.

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).

The missing limit has not been remedied but has only been put aside. My
objection remains: 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.

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.

Sind nämlich m und w irgend zwei einander ausschließende Charaktere,
so betrachten wir den Inbegriff M von Elementen E = (x1, x2, ..., x,
....), welche von unendlich vielen Koordinaten x1, x2, ... x, ...
abhängen, wo jede dieser Koordinaten entweder m oder w ist.

I think one can safely interpret this sentence as describing binary
numbers, though they were not called so.

Regards, WM

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