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a+1A a+1C a+1N Amer analogue applied arbitrary finite types arithmetical hierarchy bas(a based on 29a bounded with respect cofinal completely defined contradicting 7a Corollary course-of-values recursion cursive dg(a+ dg(a+jH ffi+1A fin(a fin(b finite hierarchies following properties Formulas Hao Wang hence of order hyperarithmetical hierarchy hypothesis induction corresponding inductive definition jump operator Kleene Lemma 11 limit ordinals Math Mostowski natural numbers non-zero order relation ordinal notations otherwise pairs of postfices partial recursive extension partial recursive function partial recursive predicate primitive recursive function primitive recursive predicate proved recursion theorem recursive in H recursive index recursive join recursive ordinals reduced matrices Relatively recursive relativized with respect Remark 22 representing function resp Result reverse inequality sequel sequence of degrees Similarly strictly increasing sequences successor ordinals techniques of RF Theorem 15 Theorem 9 transfinite extensions well-ordered Xa+1 XXII XXIII XXXVIIa