Totality problem is not computably enumerable
= Totality problem is not computably enumerable
If all indices of total computable functions could be enumerated as $e_0,e_1,\ldots$, then
$$
g(n)=\varphi_{e_n}(n)+1
$$
would be total and computable. Some $e_k$ would index $g$, giving $g(k)=g(k)+1$. Thus the totality problem is not computably enumerable.