How can I construct/define arbitrary semi-computable (but not computable) sets?
Recall that a set A is semi-computable if it is domain of a computable function f. Recall also that a set A is computable if and only if both A and the complement set (Ac) are semi-computable.
In particular, I am looking for a semi-computable (but not computable) set A, such that A is a proper subset of E, where E is the set of even numbers. E = { 2x | x € N }
On
The answer is provided by @Mitchell Spector in comments.
Pick any semi-computable, but not computable set $K$. Then, a new set can be defined as $\{ 2x \mid x \in K\}$.
Such a set is a proper subset of $E$ because it consists only of even numbers, and it cannot be computable (but it is semi-computable), because if it was computable, then $K$ would be computable as well.
Let $\phi_0,\phi_1,\phi_2,\ldots$ be an enumeration of all $n$-ary partial recursive functions.
It is well-known that the set $K=\{x\mid x\in dom\,\phi_x\}$ is semi-computable. Moreover, the set $K\cap E$ is semi-computable which you can easily check.
Add: Suppose $K\cap E$ is decidable. Then the complement must be semi-decidable. Otherwise, $K$ would be decidable. Now change the encoding $\phi_n$ of the partial recursive function by exchanging even and odd.