TITLE: Understanding Cantor's diagonal argument
QUESTION [10 upvotes]: 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?

REPLY [4 votes]: 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.