Partially conservative extensions of arithmetic

Abstract:

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