What do we conclude from diagonalization principle?

Question

I understand $R_{fa}$ etc. I understand why the diagonals are higlighted. I understand D={a,d,f}. But I don't understand what is the conclusion we derive from this?

Answer 1

It's rather unfortunate that D is presented in the table, because the point is that D cannot possibly be in the table!

You construct a function (in this case D) that is different from all the rows in the table. The way you do that is by first taking the opposite of the first function's first element, the opposite of the second function's second element, and so on.

Then you have a function that differs from all the functions in the table. Why? Suppose that this new function D was actually in the table, in row i. But D was constructed in such a way that it differs from the ith function in the ith place.

We conclude that D is not in the table.

What does this give us? Well, if the set of functions like this was countable, then we could list them one by one in a table. But since D is a function and not in the table, it cannot be possible to list all the functions one by one. Hence the set of such functions cannot be countable.