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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
We define a functional interpretation of KPω using Howard’s primitive recursive tree functionals of finite
type and associated terms. We prove that the Σ-ordinal of KPω is the least ordinal not given by a closed term of the ground type of the
trees (the Bachmann-Howard ordinal). We also extend KPω to a second-order theory with
Δ1-comprehension and strict-
reflection and show that the Σ-ordinal of this
theory is still the Bachmann-Howard ordinal. It is also argued that the
second-order theory is Σ1-conservative over
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