Partially conservative extensions of arithmetic

Abstract

Let T be a consistent r.e. extension of Peano arithmetic; Σ n 0 \Sigma _n^0 , Π n 0 \Pi _n^0 the usual quantifier-block classification of formulas of the language of arithmetic (bounded quantifiers counting “for free"); and Γ \Gamma , Γ ′ \Gamma ’ variables through the set of all classes Σ n 0 \Sigma _n^0 , Π n 0 \Pi _n^0 . The principal concern of this paper is the question: When can we find an independent sentence ϕ ∈ Γ \phi \, \in \,\Gamma which is Γ ′ \Gamma ’ -conservative in the following sense: Any sentence χ \chi in Γ ′ \Gamma ’ which is provable from T + ϕ T + \phi is already provable from T? (Additional embellishments: Ensure that ϕ \phi is not provably equivalent to a sentence in any class “simpler” than Γ \Gamma ; that ϕ \phi is not conservative for classes “more complicated” than Γ ′ \Gamma ’ .) The answer, roughly, is that one can find such a ϕ \phi , embellishments and all, unless Γ \Gamma and Γ ′ \Gamma ’ are so related that such a ϕ \phi obviously cannot exist. This theorem has applications to the theory of interpretations, since “ ϕ \phi is Γ \Gamma -conservative” is closely related to the property “ T + ϕ T + \phi is interpretable in T"-or to variants of it, depending on Γ \Gamma . Finally, we provide simple model theoretic characterizations of Γ \Gamma -conservativeness. Most results extend straightforwardly if extra symbols are added to the language of arithmetic, and most have analogs in the Levy hierarchy of set theoretic formulas (T then being an extension of ZF).