Sigma_1^0 sets are exactly the computably enumerable sets

If $x\in A⟺\mathrm{\exists }sR(x,s)$ with $R$ primitive recursive, evaluate $R(x,s)$ for $s=0,1,\dots$ and accept on the first true value. Every primitive-recursive predicate is computable: its initial functions are computed directly and its finite composition and primitive-recursion derivation become composition and finite loops of machines, as in A partial function is partial recursive if and only if it is Turing computable. The search therefore recognizes exactly $A$.

Conversely, let $M$ recognize $A$. A recognizer may halt rejecting on negative inputs, so replace each rejecting halt of $M$ by a fixed loop, obtaining a machine $M^{\mathrm{\prime }}$ that halts exactly when $M$ accepts. Fix its code $e$. By Kleene's T-predicate and output function, $T(e,x,s)$ is a primitive-recursive predicate saying exactly that $s$ codes a complete halting history of $M^{\mathrm{\prime }}$ on $x$. Thus $x\in A⟺\mathrm{\exists }sT(e,x,s)$, a ${\mathrm{\Sigma }}_1^0$ definition.