Philosophy Now Forum

Cantor found a way to build a number that cannot be a part of enumerated numbers.

E.g. If we have an infinite list of rows each from them is a decimal number starting from zero, enumerated (g. we can write them in binary, it's possible to build a number different from each, which is obviously not a part of the enumeration using the diagonal and changing each of the numbers of the diagonal.

But I don't understand why not to include diagonals in the enumeration as this:
1. Row 1
2. Diagonal 1:
x
yx
yyx
yyyx
3. Row 2
4.:
x
yyx
yyyyx
yyyyx
yyyyyyyyx
5. Row 3.
6. :
x
yyyx
yyyyyyyyx
yyyyyyyyyx
and so on.
So we can say that after all the numbers reached by diagonalization are a part of enumerated numbers.
We can rewrite all the diagonals in rows and find new diagonals an so on infinitly.
And we will get again and again enumerate set of numbers.

Im only familiar with LGBTQ at the moment - are these new types of people X Y and stuff that I need to be extremely careful about when I approach them?

(Welcome to PHN forum btw)

kouty wrote: ↑Tue May 14, 2024 7:47 am ... why not to include diagonals

I think there is a lot you haven't understood. Why, for instance, are you referring to "diagonals" plural? There is only one diagonal in Cantor's proof.

Because we can make a lot of such diagonals as I showed.

There are an infinite number of ways to assemble the diagonal that establishes Cantor's thesis (though not if the proof is laid out using binary expressions). But there is only one diagonal. What you showed doesn't resemble Cantor's proof at all.

So what?
The same question may be asked with a diagonal which will be placed in the row number 2 for instance.

So what?
So you have no idea what you are talking about.

The same question may be asked with a diagonal which will be placed in the row number 2 for instance.
Only if you have no idea what you are talking about.

The diagonal is not placed in any row. The clue is in the name "diagonal'. All the rows are horizontal (as is their nature and habit) and the diagonal cuts across them.

you are not respectful.

Not always, no.

The question is asked in the book of Tony Roy Symbolic Logic p. 48, and the answer is that no matter how many number we can add, there is always a possibility to build a number out of the list.

kouty wrote: ↑Tue May 14, 2024 7:47 am
But I don't understand why not to include diagonals in the enumeration as this:
1. Row 1
2. Diagonal 1:
x
yx
yyx
yyyx
3. Row 2
4.:
x
yyx
yyyyx
yyyyx
yyyyyyyyx
5. Row 3.
6. :
x
yyyx
yyyyyyyyx
yyyyyyyyyx
and so on.

I think your formatting went awry. I'm not finding it easy to see what you mean here

That being said, cantor's proof doesn't explicitly have to exclude any diagonals. You can include any amount of diagonally constructed numbers in your set of "all reals" if you like, it will ALWAYS be possible to construct another new diagonal, which means that your "countable set of all reals" can't be a set of all reals, because it didn't include that new diagonal.

So for any given mapping of real numbers to integers, you can always prove there are more real numbers than whatever is in your mapping, meaning you can't map all real numbers to integers, proving finally that real numbers are infinite in a deeper way than integers are infinite. They're both infinite but not in the same way.