 The class of \(\Delta^1_1\)-definable subsets of \(\mathbb{N}\) is also related to Kleene’s original study of the class of hyperarithmetical sets, customarily denoted \(\textrm{HYP}\). The definition of \(\textrm{HYP}\) depends on that of a system of constructive ordinal notations known as \(\mathcal{O} = \langle O, &lt;_O \rangle\) which Kleene had introduced in (1938). (It was also in the context of defining \(\mathcal{O}\) in which he proved the  Recursion Theorem 3.5—see  Rogers 1987, ch. 11.7 and Y. Moschovakis 2010.) \(\textrm{HYP}\) can be informally characterized as the class of sets of natural numbers \(A\) such that \(A \leq_T \emptyset^{(\alpha)}\) where \(\alpha\) is an ordinal which receives a notation \(e \in O\)—i.e., \(A \in \textrm{HYP}\) just in case it is computable from a transfinite iteration the of Turing jump up to the first non-recursive ordinal  \(\omega^{ck}_1\).[30]  Kleene’s original result was as  follows:[31] 