On predicate provability logics and binumerations of fragments of Peano arithmetic

Abstract

Solovay proved (Israel J Math 25(3---4):287---304, 1976) that the propositional provability logic of any ?2-sound recursively enumerable extension of PA is characterized by the propositional modal logic GL. By contrast, Montagna proved in (Notre Dame J Form Log 25(2):179---189, 1984) that predicate provability logics of Peano arithmetic and Bernays---Godel set theory are different. Moreover, Artemov proved in (Doklady Akademii Nauk SSSR 290(6):1289---1292, 1986) that the predicate provability logic of a theory essentially depends on the choice of a binumeration of the theory which is used to construct the provability predicate. In this paper, we compare predicate provability logics of I? n 's. For a binumeration ?(x) of a recursive theory T, let PL ?(T) be the predicate provability logic of T defined by ?(x). We prove that for any natural numbers i, j such that 0 < i < j, there exists a ?1 binumeration ?(x) of some recursive axiomatization of I? i such that $${{\sf PL}_\alpha({\rm I \Sigma}_i) \nsupseteq \bigcap_{\beta(x)}{\sf PL}_\beta({\rm I \Sigma}_j)}$$ PL ? ( I Σ i ) ? ? β ( x ) PL β ( I Σ j ) and $${{\sf PL}_\alpha({\rm I \Sigma}_i) \nsubseteq \bigcup_{\beta(x)}{\sf PL}_\beta({\rm I \Sigma}_j)}$$ PL ? ( I Σ i ) ? ? β ( x ) PL β ( I Σ j ) , where β(x) ranges over all ?1 binumerations of recursive axiomatizations of I? j .