 Proposition 3.8  may be proved by directly showing that for each partial recursive function \(\phi_e(\vec{x})\) it is possible to construct a corresponding \(\mathcal{L}_a\)-formula \(\varphi(\vec{x})\) whose logical structure mimics the steps in the definition of the former.  This can be achieved by formalizing primitive recursion using an arithmetically definable coding of finite sequences and expressing minimization using an unbounded existential quantifier (see, e.g., Kaye 1991, ch. 3). But it is also possible to obtain  Proposition 3.8  in a uniform manner by showing that there is a so-called universal formula for \(\Sigma^0_1\). In order specify such a formula, first note that it is possible to effectively enumerate all \(\Delta^0_0\)-formulas with \(k+1\) free variables as \(\psi^{k+1}_0(x,\vec{y}), \psi^{k+1}_1(x,\vec{y}), \ldots\) and then define a corresponding enumeration of \(\Sigma^0_1\)-formulas as \(\varphi^k_0(\vec{y}) = \exists x \psi_0(x,\vec{y}),\) \(\varphi^k_1(\vec{y}) = \exists x \psi_1(x,\vec{y}),\)…. We then have the following: