A nontrivial Index Set cannot be computable.
Proof
Let be The Halting Problem.
For a fixed consider the following function
Note that is computable, as we can run
and if then it will diverge.
Otherwise, we run .
By The s-m-n Theorem we get a computable function
such that
If then .
If then
Let be an Index Set.
Fix some such that .
Then either or .
Case 1
If then find some .
We claim that is a reduction from to .
If then so .
If , then , so .
This proves so is not computable.
Case 2
If by similar arguments find
and prove that reduces to .
Then so is not computable.