Regarding Cantor's diagonal proof

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  • #76
I think you are extremely confused. Digits and rows of WHAT? Cantor's proof is a proof by contradiction: You ASSUME that there are as many real numbers as there are digits in a single real number, and then you show that that leads to a contradiction. You want a proof of something that Cantor proves was false.
You know very well what digits and rows. The diagonal uses it for goodness' sake. Please stop this nonsense.

When you ASSUME that there are as many real numbers as there are digits in a single real number, this isn't true for N either. It's a given that it isn't true. If it's not true for N, what does it matter that it's not true for R?
 
  • #77
What? That is just obviously wrong. I will leave this discussion to others.
Let me ask you this. How many rows in my list match a row in the infinite identity matrix when comparing digit by digit? Yet, this is not my entire list.
 
  • #78
stevendaryl
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You know very well what digits and rows. The diagonal uses it for goodness' sake. Please stop this nonsense.
It's a proof by contradiction. You assume something, then you show that that leads to a contradiction. That proves that the assumption is false.

The assumption that Cantor started with was that there are as many real numbers as there are digits in a single real number. He proved that that is false.

Now you seem to be agreeing with Cantor's conclusion.

When you ASSUME that there are as many real numbers as there are digits in a single real number, this isn't true for N either.
You are very confused. A real number has as many digits as there are natural numbers. Every real number ##r \geq 0## can be represented by the form:

##r = K + \sum_{n=0}^{\infty} r_n##

where ##r_n## is the ##n^{th}## digit of ##r## and ##K## is the integer part of ##r##. That notation assumes that there is exactly one digit of ##r## for every natural number ##n##.
 
  • #79
@stevendaryl You're ignoring what I'm saying and throwing insults. I'll ask my same question again.

How many rows in my list match a row in the infinite identity matrix when comparing digit by digit?

That's all of the rows from the identity matrix. There are infinitely many of them. They are even N of them. But it is not the entirety of MY list. This does not mean that |N| > |N|. But for some reason, it's enough to prove that |R| > |N|. That's nonsense.

I'm not going to quote your comment because you're still using two different sets that don't have a bijection.
 
  • #80
stevendaryl
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I'm not going to quote your comment because you're still using two different sets that don't have a bijection.
What do you think a real number is? What do you think a "decimal expansion" means? You are confused about the most basic concepts of mathematics.

Have you never learned how to represent a real number as an infinite sum?
 
  • #81
@stevendaryl You're again using the same variable n to index into two sets that don't have a bijection. You're just avoiding the issue now.
 
  • #82
How many rows in my list match a row in the infinite identity matrix when comparing digit by digit?
Are there not infinitely many?
Yet it is not my entire list N which consists of only strings in base 2.

How can this be if there is a bijection between the rows and digits of my list?

Just answer these questions.
 
  • #83
stevendaryl
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Do you really not understand how to compute the ##n^{th}## decimal of a real number, when ##n## is any nonnegative integer?
 
  • #84
stevendaryl
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How many rows in my list match a row in the infinite identity matrix when comparing digit by digit?
Are there not infinitely many?
Yet it is not my entire list N which consists of only strings in base 2.

How can this be if there is a bijection between the rows and digits of my list?

Just answer these questions.
You don't understand the very basics. Your questions are not relevant until you understand the basics.
 
  • #85
stevendaryl
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If ##r## is a real number greater than or equal to 0, then let's define ##floor(r)## to be the largest integer that is less than or equal to ##r##. Now, we define the ones place of ##r## to be:

##ones(r) = floor(r) - floor(r/10) \cdot 10##

Then we can define the ##n^{th}## place of ##r## via:

##r_n = ones(r \cdot 10^n)##

That's a map from the naturals ##n## to the digits of ##r##.
 
  • #86
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This thread needs to be closed. It is a waist of time, energy, and expertise.
 
  • #87
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This thread needs to be closed. It is a waist of time, energy, and expertise.
This thread is obviously running in - unpleasant - circles. There is nothing wrong with Cantor's argument.
It seems as if there is no common basis for a discussion anymore and we had to delete a couple of post which became personal.

Thread closed.
 
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