Proof-theoretic strengths of weak theories for positive inductive definitions

TL;DR

This paper analyzes the proof-theoretic strength of weak theories related to positive inductive definitions, showing how certain axioms and formulas influence the strength and boundaries of these theories.

Contribution

It calibrates the proof-theoretic ordinals of weak fragments of ID₁ and distinguishes between predicatively reducible and impredicative parts based on formula complexity.

Findings

The lightface Π₁¹-Comprehension axiom is proof-theoretically strong over RCA₀*.

Conjunctions of negative and positive formulas are weak in transfinite induction.

Disjunctions are strong, establishing a boundary between predicative and impredicative fragments.

Abstract

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