 Note that each of the sets \(I\) defined in (\ref{undecexs}) has the following property: if \(j \in I\) and \(\phi_j(x) \simeq \phi_k(x)\), then \(k \in I\) as well. Sets with this property are known as index sets as they collect together the indices of all partial computable functions which share a common “semantic” property—i.e., one which is completely determined by their graphs such as being coincident with the constant 0 function in the case of \(Z\) or being defined on all inputs in the case of \(\TOT\). An index set \(I\) is called non-trivial if \(I \neq \emptyset\) or \(I \neq \mathbb{N}\)—i.e., it fails to either include or exclude all indices. It is easy to see that all of the sets defined in (\ref{undecexs}) are non-trivial index sets. The undecidability of these sets thus follows from the following more general result: