- #1
Organic
- 1,224
- 0
An edited post:
Please can you show a proof that contradicts my conjecture, saying:
"By using base^power representation method we can represent a list of rational numbers (repetitions over scales, for example: 0.123123123...) where the missing rational number is based on the diagonal rational number used as an input for Cantor's function (the function that defines the rational number, which is not in the list), where the result (the rational number, which is not in the list) depends on some arbitrary order of the list and some rule, which is used by Cantor's function.
For example:
0 . 1 7 1 1 3 1 7 1 1 3 1 7 ...
1 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 4 2 1 3 4 2 1 3 4 2 1 3 ...
0 . 1 0 1 0 1 0 1 0 1 0 1 0 ...
0 . 3 3 3 3 3 3 3 3 3 3 3 3 ...
2 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 3 5 4 9 5 5 1 3 5 4 9 5 ...
3 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 6 4 1 6 4 1 6 4 1 6 4 1 ...
0 . 3 0 2 0 3 0 2 0 3 0 2 0 ...
0 . 6 1 3 6 1 3 6 1 3 6 1 3 ...
0 . 2 7 1 0 2 7 1 0 2 7 1 0 ...
...
In this case Cantor's function result is 0.0101010101010101... which is not in the list.
The rule is:
a) Every 0 in the original diagonal number is turned to 1 in Cantor's new number.
b) Every non-zero in the original diagonal number is turned to 0 in Cantor's new number.
We can add 0.0101010101010101... to the list, and then rearrenge the list in such a way that give us another rational number as cantor's function result, which means that our list is still not complete.
my conjecture is:
When we have the Aleph0 complete list of rational numbers, represented by base^power representation method, then and only then we can find only some irrational diagonal number (no repetitions over scales, for example: 0.123005497762...) as an input for Cantor's function."
I call this conjecture "Aleph0-1 conjecture" because i clime that if even 1 rational number is not in the list, we shall find it as the result of Cantor's function, but when the list is the complete list of all rational numbers (an aleph0 list), we can find only some irrational number as Contor function result.
Please try to contradict this conjecture.
Sincerely yours,
Organic
Please can you show a proof that contradicts my conjecture, saying:
"By using base^power representation method we can represent a list of rational numbers (repetitions over scales, for example: 0.123123123...) where the missing rational number is based on the diagonal rational number used as an input for Cantor's function (the function that defines the rational number, which is not in the list), where the result (the rational number, which is not in the list) depends on some arbitrary order of the list and some rule, which is used by Cantor's function.
For example:
0 . 1 7 1 1 3 1 7 1 1 3 1 7 ...
1 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 4 2 1 3 4 2 1 3 4 2 1 3 ...
0 . 1 0 1 0 1 0 1 0 1 0 1 0 ...
0 . 3 3 3 3 3 3 3 3 3 3 3 3 ...
2 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 3 5 4 9 5 5 1 3 5 4 9 5 ...
3 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
0 . 6 4 1 6 4 1 6 4 1 6 4 1 ...
0 . 3 0 2 0 3 0 2 0 3 0 2 0 ...
0 . 6 1 3 6 1 3 6 1 3 6 1 3 ...
0 . 2 7 1 0 2 7 1 0 2 7 1 0 ...
...
In this case Cantor's function result is 0.0101010101010101... which is not in the list.
The rule is:
a) Every 0 in the original diagonal number is turned to 1 in Cantor's new number.
b) Every non-zero in the original diagonal number is turned to 0 in Cantor's new number.
We can add 0.0101010101010101... to the list, and then rearrenge the list in such a way that give us another rational number as cantor's function result, which means that our list is still not complete.
my conjecture is:
When we have the Aleph0 complete list of rational numbers, represented by base^power representation method, then and only then we can find only some irrational diagonal number (no repetitions over scales, for example: 0.123005497762...) as an input for Cantor's function."
I call this conjecture "Aleph0-1 conjecture" because i clime that if even 1 rational number is not in the list, we shall find it as the result of Cantor's function, but when the list is the complete list of all rational numbers (an aleph0 list), we can find only some irrational number as Contor function result.
Please try to contradict this conjecture.
Sincerely yours,
Organic
Last edited: