I have an issue with Cantor's diagonal argument

[DanKnaD]

TL;DR Summary

Can't you reverse any real number to make an natural "version" so a bijection can be made, eg 0.187362729 would become 927263781 and wouldn't this work for any real number

I'm pretty bad at maths, got an A at gcse (uk 16 years old)then never went any further, I've been looking into cantors diagonal argument and I thing I found an issue, given how long its been around I'd imagine I'm not the first but couldn't any real number made using the construct by adding 1 to the nth digit of the nth number be turned into a natural "version " by removing the 0. And writing the number out backwards. 0.19746292 would become 29 264,791, 0.0072923846 would become 6,483,292,700, etc. Couldn't this be done for any real number and then each real number has a one to one correspondence with a natural number so both would be a countable infinity?

 

[anuttarasammyak]

Your argument seems good for rational numbers but not all the real numbers.
In your idea how do you get a correspondent of ##\pi##=3.141592..., an irrational number ?

 

[phinds]

@DanKnaD , I think perhaps you don't understand what real numbers include (see post #2 for example)

 

[pbuk]

Couldn't this be done for any real number

No, it can only be done for any real number with a finite decimal expansion. What natural number has a 1:1 correspondence to 1/3 = 0.333... under this proposed mapping?

and then each real number has a one to one correspondence with a natural number so both would be a countable infinity?

But this has nothing to do with the application of Cantor's diagonal argument to the cardinality of ## \mathbb R ##: the argument is not that we can construct a number that is guaranteed not to have a 1:1 correspondence with a natural number under any mapping, the argument is that we can construct a number that is guaranteed not to be on the list.

 

[PeroK]

No, it can only be done for any real number with a finite decimal expansion. What natural number has a 1:1 correspondence to 1/3 = 0.333... under this proposed mapping?

In other words, all natural numbers have a finite decimal expansion. Whereas, the real numbers can have an infinite expansion after the decimal point.

 

[Drakkith]

<snip>Mentor note: Off-topic post and reply have been excised.

couldn't any real number made using the construct by adding 1 to the nth digit of the nth number be turned into a natural "version " by removing the 0. And writing the number out backwards.

How does this method work for two real numbers with the same digits after the decimal but different digits before? For example, 1.2345 and 2.2345. Do these turn out to be 54321 and 54322? If so, then you run into an issue where multiple numbers are mapped to the same number. Both 2.2345 and 0.22345 map to 54322 in this case. That seems like a problem, as you lose injection and bijection.

If you keep the decimal after the swap then you still run into the issue of trying to make a number with infinitely many digits after the decimal turn into a number with infinitely many digits before the decimal. Both 0.3333... and 0.4444... become infinitely large if you try to reverse their digits. This is a problem. A number like 0.333... is, well, a number. It has a unique representation and a unique value. A change to any digit makes it a different number. But neither ...333 and ...444 (where the ... means the digits continue on to the left without end) are numbers as far as I can tell. They have no unique value; both ...333 and ...444 are infinitely large. I can change ...333 to ...332 and the 'value' if you want to call it that is still infinite.

This issue occurs because a number with infinite digits after the decimal converges on a value as the number of digits increases. That is, going from 0.3 to 0.33 to 0.333 and so forth makes sense. There is a single value or quantity that this process will never exceed (more accurately, it is the smallest value that this process of adding digits will not exceed), which we define as the final value of this process and is represented by 1/3 or 0.333...

But going the other way the number diverges, never reaching a unique value. 5 to 55 to 555 and so forth results in increasingly large values and there is no value that this process will fail to exceed. In other words, pick a finite number of any size and this process of adding digits will eventually exceed it, no matter how large the number you pick. So I'm not even sure these qualify as numbers.

 

[Mark44]

But neither ...333 and ...444 (where the ... means the digits continue on to the left without end) are numbers as far as I can tell.

...333 and ...444 are meaningless, which means they can't possibly be numbers.