# Why is the set of all real numbers uncountable?

I understand Cantor's diagonal argument, but it just doesn't seem to jive for some reason.

Lets say I assign the following numbers ad infinitum...

1. $1\to 0.1$
2. $2\to 0.2$
3. $3\to 0.3$ ...
4. $10\to 0.10$
5. $11\to 0.11$

and so on... How come there's supposedly at least one more real number than you can map to a member of $\mathbb{N}$?

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Because... Cantor explicitly tells us what that "one" real number is... ??? – The Chaz 2.0 Nov 14 '12 at 16:39
@agent154: By the way, where is $\frac13$ in your list? – Dejan Govc Nov 14 '12 at 16:40
@agent154: Not exactly; $\frac13=0.333333333\ldots$ has an infinite number of 3's in its decimal expansion. – Dejan Govc Nov 14 '12 at 16:44
...are you sure you understand Cantor's diagonal argument? – camccann Nov 14 '12 at 16:45
@MakotoKato That's a bit of a useless argument. Presumably, if he knew what the Lebesgue measure was, he would already be comfortable with Cantor's argument. – Thomas Andrews Nov 14 '12 at 16:45

If we apply the Cantor diagonal argument to the list you have there, and assign the $n$th digit in our constructed number to be 0 if the $n$th digit of the $n$th number in the list is 1, and 1 if it isn't, we construct the real number $0.0111111\ldots$ (which happens to be equal to $\frac1{90}$) and in fact is not anywhere in your list, since there is no integer that looks like $0111111\ldots$. So the construction worked fine: you presented what you claim is a list of all real numbers, and then the construction produced a real number that you omitted. Where's the problem?

(If you're not happy with that, perhaps it's enough to point out that your list not only omits all the repeating decimals such as $\frac 13, \frac23, \frac 16, \frac 17, \frac 27\ldots$, but also all the numbers less than 1/10. So it's not even a list of all the rational numbers.)

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the fact that you can inject $\mathbb R$ into $\mathbb N$ doesn't prove that there are more real numbers then natural ones. It just shows that there are no less real numbers then natural ones. – Golob Nov 14 '12 at 21:26
@mjd That actually helps make a bit more sense. At least my assignment doesn't work since it doesn't account for numbers less than 1/10 as you state. But I guess my issue is with the fact that I don't understand this well enough yet. I only just learned about countability, and even then very poorly, since my prof is not a good teacher. – agent154 Nov 15 '12 at 19:41

Let's look at the interval $0 < x < 1.$ Let's assume all of its numbers are countable, i.e. they are listable. Let's say the list goes like:

1. 0.12345...
2. 0.32145...
3. 0.87654...
4. 0.28374...
5. 0.67676...
6. etc...

From the list, I make a new number. Its first decimal place is the same as the first number's, its second decimal place is the same as the second number's, its third decimal place is the same as the third number's, etc. The new number I get is:

$$0.12676...$$

Using this number, I create a new number. If a digit is a $1$ then I change it to a $2$. If a digit is anything else then I change it to a $1$. Thus:

$$0.12676... \mapsto 0.21111...$$

This new number, $0.21111...$ is not on my list. It's not the first number because it's different in the first place, it's not the second number because it's different in the second place, it's not the third number because it's different in the third place, it's not the fourth number because it's different in the fourth place, etc. It follows that my new number is not in my list, and so I could not have listed all of the numbers $0 < x < 1$. Thus, by contradiction, the interval is not countable.

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As stated, I do understand the basic theory of the diagonalization method... but despite it saying that the new number couldn't possibly exist in the list, I just couldn't see a logical reason why I couldn't enumerate it as stated above. But I hadn't noticed that doing it my way at the very least omits all the numbers less than 1/10 – agent154 Nov 15 '12 at 19:44
@agent154 To be blunt: you obviously didn't understand the basic theory. If you had then you wouldn't have posted the question that you did. – Fly by Night Nov 15 '12 at 20:26
well that's not fair. Many smart people claim to have some kind of proof that Cantor was wrong, but it doesn't mean they don't understand his theory. I just can't take something at face value sometimes. I understand the logic behind the diagonal argument, but neglecting that, I just can't wrap my head around why you cannot possibly enumerate all the real numbers. – agent154 Nov 16 '12 at 6:09
Good example. Most proofs on this are too long. – Beached Whale Jan 9 at 19:31

How come there's supposedly at least one more real number than you can map to a member of $\mathbb N$?

Well, suppose there isn't - that Cantor's conclusion, his theorem, is wrong, because our enumeration covers all real numbers. Wonderful, but let us see what happens when we take our enumeration and apply Cantor's diagonal technique to obtain a real number that can't be in this sequence. But that contradicts our supposition! Hence our supposition - that we can have an enumeration of all reals - is false. That the argument can be applied to any enumeration is what it takes for Cantor's theorem to be true.

Wilfred Hodges wrote an excellent survey of wrong refutations of Catonr' argument, his An Editor Recalls Some Hopeless Papers (Postscript). Section 7, dealing with how the counterfactual assumption confuses, might be of particular interest.

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Cantor's diagonal is a trick to show that given any list of reals, a real can be found that is not in the list.

First a few properties:

• You know that two numbers differ if just one digit differs.
• If a number shares the previous property with every number in a set, it is not part of the set.

Cantor's diagonal is a clever solution to finding a number which satisfies these properties. The number which is the diagonal is transformed s.t. it doesn't share the first digit of the first number nor the second digit with the second and so on. Thus the number is unique to the list.

This is why Cantor's diagonal as a method proves the result that the reals are uncountable.

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