diagonalization, enumeration.
Posted: Wed May 15, 2024 3:23 pm
If I understand Cantor demonstrated by using a strategy using the diagonal of a matrix, that there are decimal numbers which cannot be a part of an enumeration.
We can build a Table, the rows corresponding to natural numbers from zero to greater numbers top down:
1 2 3 . . .
Let's see numbers with 0 as unity and decimals. We assume that every number has his immediate successor in the row below. For convenience we can write the numbers in binary (with only zeroes an ones).
We observe the number corresponding to the diagonal and replace each zero by one and each one by zero. The result is a number out of the enumeration.
But on the first glance I feel that it's possible to include this diagonal into an enumeration.
We can enumerate as this.
First row
First diagonal
Second row
Second diagonal.
We call first diagonal the diagonal built as this
x
yx
yyx
and so on. The second
x
yyx
yyyyx
The third
x
y
yx
yy
yyx
And so on.
We have an enumeration of numbers containing all the numbers build by changing diagonal numbers.
So the demonstration of Cantor finds a number out of the count but we can build a new count.
We can transform these diagonals in new horizontal lines numerated as I described in the question and upon the new table we can build a new diagonal and so the problematic diagonal will recur anyway.
Therefore what is the end step to find a non enumerable number in this condition?
We can transform these diagonals in new horizontal lines numerated as I described in the question and upon the new table we can build a new diagonal and so the problematic diagonal will recur anyway.
(But we can everytime expand the countable set and include every new number.).
So I don't really understand why.
Note: I already asked this question in an other site and the question were closed because not enough centered on philosophy. I think this as a philosophical issue.
We can build a Table, the rows corresponding to natural numbers from zero to greater numbers top down:
1 2 3 . . .
Let's see numbers with 0 as unity and decimals. We assume that every number has his immediate successor in the row below. For convenience we can write the numbers in binary (with only zeroes an ones).
We observe the number corresponding to the diagonal and replace each zero by one and each one by zero. The result is a number out of the enumeration.
But on the first glance I feel that it's possible to include this diagonal into an enumeration.
We can enumerate as this.
First row
First diagonal
Second row
Second diagonal.
We call first diagonal the diagonal built as this
x
yx
yyx
and so on. The second
x
yyx
yyyyx
The third
x
y
yx
yy
yyx
And so on.
We have an enumeration of numbers containing all the numbers build by changing diagonal numbers.
So the demonstration of Cantor finds a number out of the count but we can build a new count.
We can transform these diagonals in new horizontal lines numerated as I described in the question and upon the new table we can build a new diagonal and so the problematic diagonal will recur anyway.
Therefore what is the end step to find a non enumerable number in this condition?
We can transform these diagonals in new horizontal lines numerated as I described in the question and upon the new table we can build a new diagonal and so the problematic diagonal will recur anyway.
(But we can everytime expand the countable set and include every new number.).
So I don't really understand why.
Note: I already asked this question in an other site and the question were closed because not enough centered on philosophy. I think this as a philosophical issue.