The diagonal argument is based upon impossible conditions.

01/02/2015 - 12:01 von WM | Report spam
The diagonal argument in Cantor's original version [G. Cantor: "Über eine elementare Frage der Mannigfaltigkeitslehre", Jahresbericht der DMV I (1890-91) 75-78.] is applied to an infinite sequence of infinite binary sequences. In it's folklore version the diagonal argument is applied to an infinite sequence of real numbers represented by infinite sequences of decimal digits. In a well-known procedure always the diagonal element is replaced by a different element.

This argument assumes the representation of infinite sequences by bits or digits. The impossibility apply infinite sequences may never have been recognized because it is too obvious. (Of course, always only a finite initial segment is used. But that means that only rational numbers can be represented and can be resulting in the diagional.)

Every sequence requires a finite definition. But there are only countably many such definitions. But even if it was possible, to use infinite sequences, such a sequence would never represent an irrational number. [W. Mueckenheim: "Sequences and Limits", Advances in Pure Mathematics 5, (2015) 59-61.]
http://www.scirp.org/Journal/PaperI...M4E39LF-gR
So even in this impossible case, the diagonal argument would not show the uncountability of transcendental numbers.

Regards, WM
 

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#1 Peter Kramer
21/02/2015 - 07:16 | Warnen spam
WM wrote in
news::




Deine Schallplatte ist kaputt. Die làuft schon seit Jahren immer in der
gleichen kaputten Rille.

The diagonal argument in Cantor's original version [G. Cantor: "Über
eine elementare Frage der Mannigfaltigkeitslehre", Jahresbericht der
DMV I (1890-91) 75-78.] is applied to an infinite sequence of infinite
binary sequences. In it's folklore version the diagonal argument is
applied to an infinite sequence of real numbers represented by
infinite sequences of decimal digits. In a well-known procedure always
the diagonal element is replaced by a different element.

This argument assumes the representation of infinite sequences by bits
or digits. The impossibility apply infinite sequences may never have
been recognized because it is too obvious. (Of course, always only a
finite initial segment is used. But that means that only rational
numbers can be represented and can be resulting in the diagional.)

Every sequence requires a finite definition. But there are only
countably many such definitions. But even if it was possible, to use
infinite sequences, such a sequence would never represent an
irrational number. [W. Mueckenheim: "Sequences and Limits", Advances
in Pure Mathematics 5, (2015) 59-61.]
http://www.scirp.org/Journal/PaperI...512#.VM4E3
9LF-gR So even in this impossible case, the diagonal argument would
not show the uncountability of transcendental numbers.

Regards, WM




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