Envelopes, indicators and conservativeness

摘要

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)~-.