Help understanding Cantor's Theorem I am having trouble understanding the proof of Cantor's Theorem:
https://proofwiki.org/wiki/Cantor%27s_Theorem 
http://www.whitman.edu/mathematics/higher_math_online/section04.10.html
The part that confuses me is this set
$$A = \{ x \in S \mid x \not \in f(x) \}$$
Can someone explain what this is saying in plain English. I am reading it as the set of elements $x$ that are in $S$ but not in $P(S)$, or that are in $S$ but cannot be mapped to $P(S)$. 
My thinking is that every element in $S$ is a subset of $P(S)$, by the definition of power set.  Every element in $S$ CAN be mapped to $P(S)$, so that set $A$ must be the empty set? Why not?
Can someone help clarify this?
 A: The point is that $f$ takes an element of $S$ and returns a subset of $S$. And now we define $A$ to be the set of those which are not elements of the sets they are mapped to.
What is $A$ exactly? Well that depends greatly on the set $S$ and the function $f$. Let's consider two examples.
The first, take $f(x)=\{x\}$, then for every $x\in S$ it is true that $x\in f(x)$. Therefore $A=\varnothing$. But it is true that there is no $x$ such that $f(x)=\varnothing$!
In the second example, take $f(x)=S\setminus\{x\}$, then for every $x\in S$ it is true that $x\notin f(x)$. Therefore $A=S$. But in this case it is true that there is no $x$ such that $f(x)=S$!
In both examples we see that $A$, which depends very much on the function $f$, is not in its range. And that's the point of the theorem. If $f\colon S\to\mathcal P(S)$, then $f$ is not surjective.
A: It is a version of the Russell paradox except with a function. 
A: I think the secret to the proof is to understand that $A = \{x \in S: x \not \in f(x) \}$ can be an intelligent way to define S. In the best case scenario you have an A that includes all x in S. Under that best scenario S itself cannot be mapped to, because we have the condition $x \not \in f(x)$, which doesn't allow us to pick an f(x) where x is include (like would happen if we mapped s to the entire set S).
We assumed we had a surjection function f that mapped S to P(S). We found a subset of S called A but we cannot find a way to point from S to A and we should be able because A is a subset of S. We need to be able to point to every subset of S this is a contradiction and our assumption that we had a surjection f that mapped from S to P(S) is wrong.
