Understanding a proof for the existance of a non-computable function
The contradiction reached is that $0 = 1$ which violates one of Peano's axioms. Assume $G$ is computed by the $j$-th Turing machine. Observe that $G$ is everywhere defined. Then
$$f_j(j) = G(j) = f_j(j) + 1$$
and by canceling $f_j(j)$ on both sides we get
$$0 = 1.$$