Proof-Theoretic Strengths of Weak Theories for Positive Inductive Definitions

Abstract

In this paper the lightface Π 1 1 -Comprehension axiom is shown to be proof-theoretically strong even over RCA * 0 , and we calibrate the prooftheoretic ordinals of weak fragments of the theory ID1 of positive inductive definitions over natural numbers. Conjunctions of negative and positive formulas in the transfinite induction axiom of ID1 are shown to be weak, and disjunctions are strong. Thus we draw a boundary line between predicatively reducible and impredicative fragments of ID1.