According to http://en.wikipedia.org/wiki/Analytical_hierarchy
The set of all natural numbers which are indices of computable ordinals is a $\Pi^1_1$ set which is not $\Sigma^1_1$.
However, "the set of all natural number which are indices of computable ordinals" can be interpreted as
- the set of indices of recursive binary relations which well-order some subset of natural numbers;
- Kleene's $\cal O$.
To which interpretation the above result refers? Ideally the two interpretations are reducible to one another, and the result refers to both, but it's not obvious to me. And also, where can I read more about this result?
Thank you in advance.