 It can then be shown using the  Hierarchy Theorem 3.13  that neither \(T\) nor \(O\) is \(\Sigma^1_1\)-definable. These results provide the basis for an inductive analysis of the structure of \(\Delta^1_1\)- and \(\Pi^1_1\)-definable sets in terms of constructive ordinals which builds on  Theorem 3.15  (see Rogers 1987, ch. 16.4).