Re: Cantor Confusion

On 30 Mrz., 21:07, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1175261756.864645.326...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,

mueck...@xxxxxxxxxxxxxxxxx wrote:
On 30 Mrz., 05:14, "*** T. Winter" <***.Win...@xxxxxx> wrote:
Yes, because you can revert finite sequences without consequence. You
do not even need convergence for that.

And if the series is absolutely converging, then you can exchange all
terms you like. The result is independent of the order.

But only if the result is an infinite sequence with terms indexable by,
and therefore ordered by, the naturals.

Every countable sequence can be indexed the naturals. In my
application here is no problem.



But in mathematics sequences
are defined as having a first element. On the other hand, I wonder how
you prove that the series of interchanges on the initial sequence lead
to your final "sequence".

It is the same as Cantor's "proof" that he gets ready.

The transposition of the first and second terms of a sequence followed
by the transposition of the first and third produces a different result
than the transposition of the first and third followed by the
transposition of the first and second.

abc -> bac -> cab versus abc -> cba -> bca

Thus the order of application of transpositions makes a difference.

That is not a problem at all. We can work like the cleaning service of
Hilbert hotel: For the first sequence of transposition use half an
hour, for the second sequence use quarter an hour and so on.

The replacement of members of a sequence by a rule depending only on the
value and not position of the member being replaced is independent of
the order of operations. let the rule be to replace any lower case
letter by its upper case equivalent.

abc-> Abc -> ABc -> ABC is the same as abc -> abC -> AbC -> ABC
even though the operations were differently ordered.

So the Cantor rule for building an antidiagonal for a list of binary
sequences can be applied independently to different digits

Nevertheless it cannot be applied to the n-th digit unless the
positions 1 to n-1 are known. Because otherwise position n is unknown.
This may be masked, for small n, by your knowledge acquired at school
but it will become manifest at larger n.

Regards, WM

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