 Recall that the language of first-order arithmetic \(\mathcal{L}_a\) contains the primitive symbols \(\{&lt;,',+,\times,0\}\) respectively intended to denote the less than relation, successor, addition, and multiplication functions on the natural numbers as well as the first natural number 0. A first-order arithmetical formula is one built up from these expressions using variables, propositional connectives, and the first-order quantifiers \(\forall x, \exists x\) where the variables are intended to range over the natural numbers \(\mathbb{N}\). Recall also that the standard model of first-order arithmetic is the structure \(\mathfrak{N} = \langle \mathbb{N},0,&lt;,s,+,\times \rangle\) in which the symbols of \(\mathcal{L}_a\) receive their intended interpretations. Finally we say that an \(\mathcal{L}_a\)-formula \(\varphi(\vec{x})\) defines a relation \(R \subseteq \mathbb{N}^k\) just in case \(R = \{\langle n_1,\ldots,n_k \rangle : \mathfrak{N} \models  \varphi(n_1,\ldots,n_k)\}\).[29]  For instance \(x &lt; y \vee x = y\) defines the less-than-or-equal relation \(\leq\), \(\exists y(y + y = x)\) defines the even numbers, and 