 whose \(i+1\)st member is defined in terms of primitive recursion of the \(i\)th member form a hierarchy of functions whose values grow increasingly quickly in proportion to their inputs. While each function in this sequence is primitive recursive, we can also consider the function \(\alpha(x,y)\) defined as \(\alpha_x(y,y)\)—a version of the so-called Ackermann-Péter function defined in  Section 1.4—whose  values are not bounded by any fixed function \(\alpha_i\).   As it can be shown that the values of   \(\alpha(x,y)\)  are not bounded by any of the functions \(\alpha_i(x,y)\), this shows that \(\alpha(x,y)\) cannot be defined by any finite number of applications of the scheme \(\mathcal{PrimRec}_1\).   This provides a constructive proof that there exist functions of type \(\mathbb{N}^2 \rightarrow \mathbb{N}\) which are not primitive recursive.