Envelopes, indicators and conservativeness

A well known theorem proved (independently) by J. Paris and H. Friedman states that BΣn+1 (the fragment of Arithmetic given by the collection scheme restricted to Σn+1–formulas) is a Πn+2–conservative extension of IΣn (the fragment given by the induction scheme restricted to Σn–formulas). In this paper, as a continuation of our previous work on collection schemes for ∆n+1(T)–formulas (see [4]), we study a general version of this theorem and characterize theories, T, such that T + BΣn+1 is a Πn+2–conservative extension of T. We prove that this conservativeness property is equivalent to a model–theoretic property relating Πn–envelopes and Πn–indicators for T. The analysis of Σn+1–collection we develop here is also applied to Σn+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]): BΣn+1 and IΣn+1 are Σn+3–conservative extensions of their parameter free versions, BΣ − n+1 and IΣ − n+1.