Let $W_x$ be the domain of the partial computable function $\varphi_x$. I know that the set
$$\operatorname{Fin} = \{x : W_x\text{ is finite}\}$$
is not computable (as a corollary of Rice's theorem, for instance). My question is,
Is the set $\operatorname{Fin}$ recursively enumerable?
that is, is its semi-characteristic function computable? I think it's not, intuitively, there's no way to accept elements of this set since in this case we would need to compute $\varphi_x(n)$ for all $n$ and see if this eventually comes undefined for large enough $n$'s, but this requires infinitely many computations of course.
Best Answer
There is a general method to answer this kind of question, which Rogers called the Tarski-Kuratowski computation in his book. You first write down a definition of the set in the arithmetical hierarchy, in which all the leading quantifiers explicitly shown in prenex form and in which the central "matrix" is decidable: $$ e \in \text{Fin} \Leftrightarrow (\exists k)(\forall i)(\forall s) [ \phi_{e,s}(i)\mathord{\downarrow} \Rightarrow i < k]$$ Here as usual $\phi_{e,s}(i)\downarrow$ says that $\phi_e(i)$ halts in no more than $s$ steps, which is a decidable question.
From this, we see that the set Fin is $\Sigma^0_2$. That means you should next try to show the set is $\Sigma^0_2$ hard, at which point you will know the set is $\Sigma^0_2$ complete. In particular, that would mean the set is not r.e. because the r.e. sets are exactly the $\Sigma^0_1$ sets, and a $\Sigma^0_2$ hard set cannot be $\Sigma^0_1$.
In practice, most sets that arise in practice will be complete for the level of the arithmetical hierarchy that the natural prenex form of the set suggests ($\Sigma^0_2$, in this case). That is why this method is worth knowing.
The thing that remains is showing that the set Fin is $\Sigma^0_2$ hard. To do that, suppose you have a $\Sigma^0_2$ formula $\psi(x) \equiv (\exists n)(\forall m)\phi(n,m,x)$. Given $x$, we need to make a program $e = e(x)$ that will be in Fin if and only if $\psi(x)$ holds. That is not too hard to do, using the fact that $\phi(n,m,x)$ is decidable from $n$, $m$, and $x$. I'll leave it to you.
It also helps to know some examples of particular sets at particular levels of the hierarchy:
Once you know these, you can use them to show other sets are at particular levels of the arithmetical hierarchy.