Envelopes, indicators and conservativeness

A well known theorem proved (independently) by J. Paris and H. Friedman states that B Σ n +1 (the fragment of Arithmetic given by the collection scheme restricted to Σ n +1 ‐formulas) is a Π n +2 ‐conservative extension of I Σ n (the fragment given by the induction scheme restricted to Σ n ‐formulas). In this paper, as a continuation of our previous work on collection schemes for Δ n +1 ( T )‐formulas (see [4]), we study a general version of this theorem and characterize theories T such that T + B Σ n +1 is a Π n +2 ‐conservative extension of T . We prove that this conservativeness property is equivalent to a model‐theoretic property relating Π n ‐envelopes and Π n ‐indicators for T . The analysis of Σ n +1 ‐collection we develop here is also applied to Σ n +1 ‐induction using Parsons' conservativeness theorem instead of Friedman‐Paris' theorem. As a corollary, our work provides new model‐theoretic proofs of two theorems of R. Kaye, J. Paris and C. Dimitracopoulos (see [8]): B Σ n +1 and I Σ n +1 are Σ n +3 ‐conservative extensions of their parameter free versions, B Σ – n +1 and I Σ – n +1 . (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)