Understanding Cantor's diagonal argument

Question

I'm trying to grasp Cantor's diagonal argument to understand the proof that the power set of the natural numbers is uncountable. On Wikipedia, there is the following illustration:

The explanation of the proof says the following:

By construction, s differs from each sn, since their nth digits differ (highlighted in the example). Hence, s cannot occur in the enumeration.

I don't understand why the sequence s at the bottom cannot occur anywhere in the enumeration of sequences above. I have read the proof about five times, but I'm still not getting it. I think I'm having an error in reasoning. Could someone please explain me why s cannot be in the enumeration with an example?

Answer 1

The key is that it's different by construction, which means that you're choosing the digits of $s$ specifically so that it will be different from every other item in the list.

Compare $s$ to $s_1$: you see right away that they are different because the first digit is different. Now compare $s$ to $s_2$: they are different at the second digit. The same holds for the remaining $s_i$. The reason this happens is precisely because we chose the digits of $s$ to have this property.

Answer 2

Basically, the diagonalizing method is to take every possible number one at a time and say "I can make a number that is different than this and different than any number so far". !!*!IF!!!* the resulting number did exist in the list then you would have come across it and you would have said "I can make a number different than this" and you would have, so the number can't be in the list.

That number can't be the first number because it has a different first term. It can't be the second number because it's got a different second term. It can't be the nth number because it's got a different nth term.

So let's suppose the number is on the list. Well, what number listing does it have? Let's say it's the coopledinkyfudgeth number on the list. Well when we do our diagonalization and we come the coopledinkyfudgeth number we change the coopledinkyfudgeth term and get a different number. So it ISN'T the coopledinkyfudgeth number after all!

It's not on the list! The list is impossible and there is no list.

Answer 3

The reason that this works is because we are assuming that we can list every possible string in a sequence, but the point of the proof is to show that this cannot be done. If we $\textit{could}$ make a list, then $s$ has to be a term in the sequence, and thus would have to be equal to some $s_N$. But by our construction of $s$, we switch the $N$-th digit of $s_N$. Thus, $s$ is a new string of $0$ and $1$s - which is a contradiction to the assumption that we can make a list.

Answer 4

First, you were probably taught the proof incorrectly; or at least, have seen the incorrect version at some point. The Wikipedia article, as it stands today, tries to not make the mistakes in the incorrect version. But it gets re-explained incorrectly.

The first mistake is superficial only. Cantor didn't use real numbers. They can work, but need additional steps. Like Wikipedia, he used infinite-length binary strings. They can be mapped to the reals in the interval [0,1], but some real numbers are mapped by two strings (in binary, the strings 100... and 0111... both map to 1/2).

The big mistake, that confuses many, is that the first step of the correct proof is not a hypothetical. It is not "we are assuming that we can list every possible string," as another answer here put it. Cantor considered any infinite list of such strings that could exist. This isn't a hypothetical list, to be used for a proof by contradiction. Nor does it have to contain every such string. It means any list that actually exists. You are thinking that every string must be in the list, which is still hypothetical and is what leads to your confusion.

What diagonalization shows, is that for any list that can be put in front of you, there must be a string s that is not in that list. It doesn't show that s cannot be in a list, or added to the list in front of you. It shows that there must be an s that is not in the particular list in front of you.

The critical, second part of the proof is seldom taught. (It gets folded into the first part.) This is the part that requires a hypothetical. If you could list the set of all such strings, a contradiction is produced. We can construct an s that is not in the set listed, but by definition s is in the set listed.