Cantor 3 part 2

Assertion: `The second diagonal argument is a circular argument, the conclusions are not valid´

Proof, part 2: Because the construct of Cantor couldn´t be used to have a conclusion about the countability of the irrationals in the interval [0, 1], there is to look for new methods for to have a possibility to get a bijection (one to one depiction) to N. We leave out the formulation by the dual-numbers, which rising up simply may be — by finite length — assigned to natural ones: 1 -> 0; 3 -> 00; 7 -> 000; 11 -> 100; 14 -> 0000;

2 -> 1; 4 -> 01; 8 -> 001; 12 -> 101; …

5 -> 10; 9 -> 010; 12 -> 110;

6 -> 11; 10 -> 011; 13 -> 111;

A conclusion is needed which, by development, on the one hand has a well defined beginning ready and which on the other hand shows the capturing of all numbers of the useful numerical system (based on ten) which could be described. Fractions come clearly structured and by that illustrative by the numerators to each the end of completeness of the depth of the decimal places of the depicted number in the interval. 0/10

1/10

2/10

…

10/10

The bounds 0 and 1 were included. We develop (in step-sequences from 0) the decimal places written as fractions; line-index = n : 0: 1: 2: 3: . . . n:

0/1; 1/1 (0/10); 1/10; 2/10; 3/10; 4/10; 5/10; 6/10; 7/10; 8/10; 9/10; (10/10) (0/100); 1/100; …; 9/100; (10/100); 11/100; …; 99/100; (100/100) (0/1000); 1/1000; …; 99/1000; 101/1000; …; 999/1000; (1000/1000)

(0/10n); 1/10n;

…; (10n – 1)/10n; (10n/10n)

The first line (n = 0) is built exclusively by the bounds. Number (of amount) of figures being built between the bounds: 0 = 10n – 1 . For following developing-steps (/-lines) all figures which were previous been builded escapes. Each of the following lines generates up to 10n .

1 of 2

The second line (n = 1) developed up to 10. By 0/1 = 0/10 and 1/1 = 10/10 two of twelve digits of the development be dropped. Number (of amount) of the digits which were developed between the bounds: 9 = 10n – 10(n – 1) . The third line (n = 2) `counts´ to 100 . Number (of amount) of new between the bounds developed digits: 90 = 10n – 10(n – 1) . And so on. Each line is finit and by that it could be counted by the index N which is winding it´s way through the lines. By the development to infinite all values of digits would be recorded. Th bijectivity to N is deduced. Unfortunately there is nothing more shown as Cantor fundamental had depicted it by his first diagonal argument. All decimal places in the interval, built by the digits 0 … 9 at any depth (number (of amount) of the decimal places), are type `rational´, because this is the development of the fractions of the decimal system! There couldn´t be depicted more different digits in number (of amount) by the numbers 0 … 9 at the position of the decimal number — by the style of decimals. Where do the irrationals deduce from? The one, which lead the rationals to the totality of the reals as a replenishment? Numbers in itself could be put in order by numeric-technical construct at the reals, as it is for e (neutral in derivation) given by ex ´ = ex . By geometric construct (multidimensionality) there are, for example, numbers like π , !2 and the values of the angle-functions being constructible (designable). How are these numbers — different in type — to be connected with the rationals?

Copyright 2017, Peter Kepp, dbqp-Verlag — Deutschland [Fünfter Teil der Aufsatz-Serie `Logik des Formalismus´] (translated 17. Feb. 2018) Jegliche Verbreitung, in Wort, Bild, Ton oder Rede (Lehrveranstaltung) gilt gemäß Copyright als geschützt und bedarf bei Interesse der vertraglichen Zustimmung des Verlages. Elektronische Weitergabe fällt unter `jegliche Verbreitung´.

2 of 2