 It is useful to observe that while we have just defined such a function for each \(k\), it is also possible to define a binary function \(\upsilon(i,x)\) which treats its second argument as a code for a finite sequence \(x_0,\ldots,x_{k-1}\) and then computes in the same manner as the \(k\)-ary universal function so that we have \(\upsilon(i,\langle x_0,\ldots, x_{k-1} \rangle) \simeq \upsilon^k(i,x_0,\ldots,k_{k-1})\). This provides a means of replacing the prior enumerations of \(k\)-ary partial computable functions with a single enumeration of unary functions 