mgb wrote:
It doesn't matter. All the numbers have an infinity of digits. The zeros are only there to facilitate the drawing of the diagonal.

You are still ignoring the fact that with every digit you add you need to add 10^n row if you want to claim that your list is complete.

Since when has mathematics become a game of just making unsubstantiated claims as you are doing?

Again, you don't seem to understand the problem.

Try it again with the simplest example possible, binary. And we can place a leading zero and a decimal point after the leading zero if you like. We will also assume that the leading zero is just a place holder, in other words, will will never assign it a value of 1. In this way we can ignore the leading zero.

In binary, if we go out just one digit we have the following complete list.

0.0

0.1

Notice that the list is already rectangular from the decimal point to the right. If we cross out the the first zero and replace it with a 1 we have a number that begins with one after the decimal place. We can claim that this number cannot be on our NEW list. However it would be wrong to claim that it isn't on the original list because it is.

0.[strike]0[/strike]

0.1

Our new number is 0.1 but that number is already on the list.

Now let's add another digit so we can continue with our diagonal method of creating a new number. When we do this we must also add two more rows to our list.

0.00

0.10

0.01

0.11

Now we cross out the first two zeroes and claim to have a new number that is 0.11.

0.[strike]0[/strike]0

0.1[strike]0[/strike]

0.01

0.11

But as you can see that number already exists. It's the fourth row down. A row that we could not have possibly gotten do by drawing a diagonal line that crosses out every new digit.

Let's continue on and add yet another digit. In order to do that our list now grows by four more rows.

0.000

0.100

0.010

0.110

0.011

0.100

0.101

0.111

But we are creating new numbers by drawing a diagonal line remember? We we have the following:

0.[strike]0[/strike]00

0.1[strike]0[/strike]0

0.01[strike]0[/strike]

0.110

0.011

0.100

0.101

0.111

If we replace all these zeroes with 1's then we end up with 0.111 which we claim is not on "our" list. That's fine, the only problem is that it's obviously already on the list at row number 8.

In short, because it's the innate property of numerical representations of numbers to always require far more rows than columns, the list grows exponentially faster in rows with every column we add. Therefore to claim that any new numbers we have created by using a diagonal cross-off method aren't on the list, is a bogus claim to make. It's simply false. We aren't in a position to be making such an irrational absurd claim.

The situation only continues to get worse with every digit we cross off, because our diagonal line method demands that to cross off another digit we must always add a new column. So we can never truly say anything at all about what numbers might be on this list. The numbers we have created are clearly on the lists above. In fact, it would be impossible for us to ever create a new number that isn't on the list.

~~~~~~~

What you want to do is ignore this truthful fact about the nature of numerals and proclaim, without proof or reason, the if you pretend to have an infinite list it will suddenly and miraculously become square.

But why should that happen? At what point will it happen? If you stop the process at any point the real list of numbers will always contain many more rows than what you were able to cross off with a diagonal line. So when does this list magically become square?

Has mathematics become nothing more than a game of magic?

~~~~~~~~

**1. I have shown why any compete list of numerals must necessarily contain more rows than columns, and can therefore not be treated as a square list.**
Do you disagree with this? Is so please address that point and give an example of a complete list of numerals that is square. Proclaiming that the list will somehow magically become square at infinity is nonsense.

**2. I have also demonstrated that it is the nature of these numerical lists to grow exponentially in the number of rows required to record every numeral with every column appended.**
Do you disagree with this? If so place address that point.

**3. I have shown that crossing off digits using a diagonal line that crosses off the next digit in the next row necessarily creates a square list of crossed off numerals. Therefore that list cannot be a complete list of numerals, nor can it be used to prove what may or may not be on a complete list of numerals.**
Do you disagree with this? Is so please explain why without resorting to the absurd claim that when taken to infinity a complete list of numerals will magically become square. Where is there any rationale for such an absurd claim? I have already shown that completed lists must necessarily grow exponentially in rows with every digit column added. Therefore there is no justification for claiming that these innately rectangular lists will magically become square at infinity. The situation only gets worse with every digit crossed off the list. What's going to magically make the situation get better as you approach infinity?

You need to explain how that works. You can't just claim that it will magically become square at infinity. Mathematics doesn't stand on unsubstantiated claims, or does it?

If it does, then it doesn't have much credibility.

Therefore I have successfully demonstrated why Cantor's diagonal proof does not prove what he claims.