>>20
First of all, I don't think you even understood the definition of f. In
>>18 you say "[you] cant define f where for every x, x is not in f(x)" which is not necessary or relevant. f, in
>>16, was an arbitrary one-to-one mapping, and the fact that, by assuming its existance, we could create the set S using ZFC axioms shows that f does not exist. Your explanation in
>>18 is a rambling incoherent mess.
Example:
"and x in N ---> f(x) in P(N) | x not in f(x)"
So x, an element of N, goes to f(x) in P(N).. ok. But your "| x not in f(x)" is non-sensical in any notation I've ever seen.
I want to reiterate that I really do not believe you understood the construction of S. S is the set of elements of X whose image in P(X) does not contain themselves. No one said S was necessarily equal to X or any such thing - if we supposed a map from f:N -> P(N) with f(0) = N, the only thing that tells us is that 0 is not an element of S, since 0 is in f(0).
The only correct thing you've said (at least among the rare statements of yours that I can actually interpret) is that S is equivalent to Russell's paradox - It is, and the fact that we can construct that set using just ZFC and the existance of f is proof that f does not exist.