 Solution

Fix an effective enumeration $({\phi }_e)$ of the unary partial computable functions. Suppose the setwere computably enumerable, say as $e_0,e_1,\dots$. Thenwould be a total computable function. Hence $g={\phi }_{e_k}$ for some $k$, buta contradiction. Thus the set of Gödel numbers of total computable functions is not recursively axiomatizable; this is the totality problem is not computably enumerable argument.

Let $T$ be a recursively axiomatized theory of arithmetic. For each program index $e$, fix an arithmetical sentenceexpressing that the computation with index $e$ halts on every input. Enumerate all formal $T$-proofs and output $e$ whenever a proof ending in $\mathrm{Tot}(e)$ appears. This enumerates exactly the Gödel numbers of the computable functions that $T$ proves total, namely the provably total computable functions, so that set is recursively axiomatizable.

Assume now that $T$ is sound. Repeat every discovered index indefinitely, obtaining an effective infinite list $e_0,e_1,\dots$ of the functions whose totality $T$ proves. DefineSoundness makes every ${\phi }_{e_n}$ genuinely total, so $f_T$ is total and computable. If $T$ proved its totality, an index $e$ for $f_T$ would occur in the list, say $e_k=e$, and thenwhich is impossible. Thus $f_T$ is a total computable function whose totality is not provable in $T$.