Totality problem

For an effective enumeration of computably enumerable languages $W_v$, the totality problem isIt is ${\Pi }_2$-complete: a ${\Pi }_2$ condition $w\in P⟺\forall xf(w,x)\downarrow$ maps effectively to a program whose domain is $\{x:f(w,x)\downarrow \}$.

If all indices of total computable functions could be enumerated as $e_0,e_1,\dots$, thenwould be total and computable. Some $e_k$ would index $g$, giving $g(k)=g(k)+1$. Thus the totality problem is not computably enumerable.