User:IssaRice/Computability and logic/Characteristic function of a semirecursive set

What if we try to define a characteristic function for a semirecursive relation $S$ as follows?

${\chi }_S(x,y)=\{\begin{array}{cc}1& S(x,y)\\ 0& \mathrm{\neg }S(x,y)\end{array}$

Actually we should distinguish between different levels of badness of ${\chi }_S$, and ask separately if it is well-defined, total, and recursive.

First, given any $x$ and $y$, exactly one of $S(x,y)$ or $\mathrm{\neg }S(x,y)$ holds. So it seems ${\chi }_S$ is well-defined. Also in each case, we assign some number to ${\chi }_S(x,y)$, so it is total.

Since $S$ is semirecursive, there is some recursive relation $R$ such that $S(x,y)⟺\mathrm{\exists }t(R(x,y,t))$. And since $R$ is recursive, it has a recursive total characteristic function ${\chi }_R$. Our goal, then, is to try to define ${\chi }_S$ in terms of ${\chi }_R$.

$g(x,y)=\mu t[{\chi }_{\mathrm{\neg }R}(x,y,t)=0](x,y)=\mathrm{M}\mathrm{n}[{\chi }_{\mathrm{\neg }R}](x,y)$

The problem is that $g$ is only a semicharacteristic function. And it seems there is no way to get a characteristic function for $S$ using the processes for defining recursive functions. (Can we prove this?)