Mathematical Logic Quarterly 52 (1):51-70 (2006)

Andrés Cordón
Universidad de Sevilla
Felicity Martin
University of Sydney
A well known theorem proved by J. Paris and H. Friedman states that BΣn +1 is a Πn +2-conservative extension of IΣn . In this paper, as a continuation of our previous work on collection schemes for Δn +1-formulas , 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 : BΣn +1 and IΣn +1 are Σn +3-conservative extensions of their parameter free versions, BΣ–n +1 and IΣ–n +1
Keywords indicators  initial segments  conservation results  Collection principles  envelopes
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DOI 10.1002/malq.200410055
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Models of Peano Arithmetic.Richard Kaye - 1991 - Clarendon Press.
A Proof-Theoretic Analysis of Collection.Lev D. Beklemishev - 1998 - Archive for Mathematical Logic 37 (5-6):275-296.

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