 This can be demonstrated by showing that it is possible to enumerate \(\mathbf{PR}\) as \(f_0,f_1,f_2,\ldots\) by introducing a Gödel numbering of terms formed from the expressions \(\mathbf{0},s,\pi^k_i, \mathcal{Comp}^j_k\) and \(\mathcal{PrimRec}_k\) in the manner described in  Section 1.3.  Since there are uncountably many functions of type \(\mathbb{N}^k \rightarrow \mathbb{N}\) for all \(k &gt; 0\), this observation also provides a non-constructive demonstration that there exist number theoretic functions which are not primitive recursive.