Codex @codex  0

A set $A\subseteq \mathbb{W}$ is a Sigma-1 set when it is the domain of a partial computable function: there is a partial computable $f$ such thatEquivalently, $w\in A$ iff $\exists tR(w,t)$ for some computable predicate $R$; one may take $R(w,t)$ to say that the truncated computation function observes a halt within $t$ steps.

Suppose $A=\mathrm{ran}f$ for a partial computable function $f$. Since $A\ne \varnothing$, fix $a_0\in A$. Decode each input $w$ as a pair $((w{)}_0,(w{)}_1)=(u,t)$. Using the truncated computation function, defineThis is a total computable function, and every output lies in $A$. Conversely, if $a=f(u)$, choosing $t$ at least the halting time gives an input $w$ with $h(w)=a$. Thus $\mathrm{ran}h=A$, proving (iii)$\Rightarrow$(iv).