 The former part of this characterization differs from the definition of general recursiveness given in  Section 1.5  in allowing that in addition to the initial functions \(\mathbf{0}\) and \(s(x)\), the functions \(\psi_1, \ldots, \psi_k\) can also enter into systems of equations which define the function \(\phi\). This corresponds to the assumption that the values of \(\psi_1, \ldots, \psi_k\) are available in the course of a computation without the need for further calculation.