 Here the first \(k\) arguments \(x_0,\ldots,x_{k-1}\) to \(g\) are the parameters, the \(k+1\)st argument \(y\) is the recursion variable, and the \(k+2\)nd argument \(h(x_0,\ldots,x_{k-1},y)\) gives the prior value of \(h\). An elementary set theoretic argument shows that for each \(k \in \mathbb{N}\), if \(f\) is \(k\)-ary and \(g\) is \(k+2\)-ary, then a there is a unique \(k+1\)-ary function \(h\) satisfying (\ref{prscheme})—see, e.g., (Moschovakis 1994, ch. 5).