Kleene's O is a $\Pi\_1^1$ complete set that decides every hyperarithmetic statement. A Turing Machine that uses this set as an oracle to decide a hyperarithmetic question can only look at a finite segment of the oracle before making a decision. The possible questions are all of the form $n$ is or is not a notation for a recursive ordinal. Can every hyperarithmetic question be decided by a single notation for a sufficiently large recursive ordinal?



Hyperarithemtic statements decidable by induction up to a recursive ordinal  
Every $\Pi^1\_1$ set is many-one reducible to Kleene's $\mathcal{O}$. In particular, the universal $\Pi^1\_1$ set is many-one reducible to Kleene's $\mathcal{O}$. Therefore, every $\Pi^1\_1$ sentence (and in particular hyperarithmetical sentences) can be decided by making a single query to Kleene's $\mathcal{O}$.


