Understanding a proof for the existance of a non-computable function For school, we have a proof that some functions are not Turing computable.
The example is:
$$ G(k) = \begin{cases} f_k(k) + 1 &amp; \text{ if $f_k(k)$ is defined}, \\ 1 &amp; \text{ otherwise}.\end{cases} $$

Claim: $G$ is non computable.

Proof: In view of obtaining a contradiction, let's say $G$ is computable, say by the $k$th Turing machine.
Give the encoding of this $k$th Turing machine as an argument for $G$.
This leads to a contradiction: if $f_k(k)$ is defined, then $f_k(k)$ is not equal to $g(k) = f_k(k) + 1$. Else $f_k(k)$ is undefined and not equal to $g(k) = 1$.

I don't understand the contradiction, help please...
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
$$0 = 1.$$