 A number-theoretic function \(\phi\) is said to be recursive if there is a finite sequence of number-theoretic functions \(\phi_1 , \phi_2 , \ldots \phi_n\) that ends with \(\phi\) and has the property that every function \(\phi_k\) of the sequence is recursively defined in terms of two of the preceding functions, or results from any of the preceding functions by substitution, or, finally, is a constant or the successor function \(x + 1\)…. A relation \(R(x_1, \ldots , x_n)\) between natural numbers is said to be recursive if there is a recursive function \(\phi(x_1 \ldots , x_n)\) such that, for all \(x_1, x_2, \ldots, x_n\) 