Can we find a formula defining a recursively enumerable set?

The definition of a recursive enumerable set is that it is the domain of some partial recursive function.

There is a recursive primitive function $\psi$, such that $\psi(n,t,x)=0$ if and only if $\phi_n(x)$ (the recursive function $n$ on entry $x$) halts in less than $t$ steps, else $\psi(n,t,x)=1$. Any recursive primitive function can be defined by a $\Delta_0$-formula. Hence

$$\phi_n(x)\mbox{ halts } \Leftrightarrow \exists t\; \psi(n,t,x)=0$$

The existence of $\psi$ is a consequence of Kleene T-predicate