 where \(\phi^k_i(\vec{x}) \simeq \mu s T_k(i,\vec{x},s)\). Such an enumeration provides a uniform means of listing off all partial computable functions in the order of their indices. It should be noted, however, that each partial computable function has infinitely many indices. For instance, given a function \(f:\mathbb{N}^k \rightarrow \mathbb{N}\) computed by \(\phi_e(\vec{x})\), it is possible to define infinitely many extensionally coincident functions with distinct indices \(\phi_{e'}(\vec{x}), \phi_{e''}(\vec{x}), \ldots\)—e.g., by “padding” the definition encoded by \(e\) with terms that successively add and then subtract \(m\) for each \(m \in \mathbb{N}\). As this yields a definition of an extensionally equivalent function, it thus follows that infinitely many of the \(\phi^k_i(\vec{x})\) will correspond to the same function in extension.