Solution
= Solution
If $g_0$ were total recursive, its recursion-theorem fixed point $e$ would satisfy $W_e=W_j$ when $e\in\mathrm{Inf}$ and $W_e=W_i$ when $e\in\mathrm{Fin}$, a contradiction in either case. The same argument applies to $g_1$. A total recursive reduction $h$ with $n\in\mathrm{Inf}\iff h(n)\in\mathrm{Fin}$ would provide such a forbidden switch (and would collapse the known distinct arithmetical complexities), so none exists.