Envelopes, indicators and conservativeness
Andrés Cordón–Franco, A. Fernández‐Margarit, Francisco Martín · Mathematical logic quarterly · 2005
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 Δn +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. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)