Envelopes, indicators and conservativeness
|Author||Cordón Franco, Andrés
Fernández Margarit, Alejandro
Lara Martín, Francisco Félix
|Department||Universidad de Sevilla. Departamento de Ciencias de la Computación e Inteligencia Artificial|
|Published in||Mathematical Logic Quaterly, 52 (1), 51-70.|
|Abstract||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 ...
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 ), 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 ): BΣn +1 and IΣn +1 are Σn +3‐conservative extensions of their parameter free versions, BΣ–n +1 and IΣ–n +1.
|Cite||Cordón Franco, A., Fernández Margarit, A. y Lara Martín, F.F. (2006). Envelopes, indicators and conservativeness. Mathematical Logic Quaterly, 52 (1), 51-70.|