Iterated Inductive Definitions and Σ21-AC

Publisher Summary This chapter discusses the relations between Σ 1 k -AC and Π 1 k -CA, k ≥0 and between Σ 1 k -DC and iterated generalized inductive definitions. A Π 1 k formula has a block of k function quantifiers, starting with a universal one, followed by only number quantifiers and propositional combinations of atomic formulae. EA is composed of first-order arithmetic together with the schema of ordinary induction on all second-order formulae. A Π 1 0 formula is a Π 1 0 formula. Σ 1 k -AC is stronger than Π 1 k -CA, in the sense that Σ 1 k -AC can prove the existence of an ω-model of Π 1 k -CA. By using the completeness of hyperjump among Π 1 1 predicates, a straightforward argument can be given. When the ω-rule is added to the theories Σ 1 k -AC, analogous results may be obtained.