 Proof: Let \(I\) be a non-trivial index set and suppose for a contradiction that \(\chi_I(x)\) is computable. Consider the everywhere undefined unary function \(u(x)\)—i.e., \(u(n) \uarrow\) for all \(n \in \mathbb{N}\). Since \(u(x)\) is partial computable, there is an index \(b\) such that \(\phi_b(x) \simeq u(x)\). We may suppose without loss of generality that \(b \not\in I\). (If it is the case that \(b \in I \neq \mathbb{N}\), then we can switch the role of \(I\) with its complement \(\overline{I}\) in the following argument and obtain the same result). Since \(I \neq \emptyset\), we can also choose an index \(a \in I\) and define a function as follows: 