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In this article the lightface
-Comprehension axiom is shown to be proof-theoretically strong even over
, and we calibrate the proof-theoretic ordinals of weak fragments of the theory
of positive inductive definitions over natural numbers. Conjunctions of negative and positive formulas in the transfinite induction axiom of
are shown to be weak, and disjunctions are strong. Thus we draw a boundary line between predicatively reducible and impredicative fragments of
In this paper we introduce a recursive notation system O(Π3) of ordinals. An element of the notation system is called an ordinal diagram. The system is designed for proof theoretic study of theories of Π3-reflection. We show that for each α < Ω in O(Π3) a set theory KP Π3 for Π3-reflection proves that the initial segment of O(Π3) determined by α is a well ordering. Proof theoretic study for such theories will be reported in [ 4].