On predicate provability logics and binumerations of fragments of Peano arithmetic

Archive for Mathematical Logic 52 (7-8):871-880 (2013)
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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–Gödel 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 α(x) of a recursive theory T, let PL α(T) be the predicate provability logic of T defined by α(x). We prove that for any natural numbers i, j such that 0 < i < j, there exists a ∑1 binumeration α(x) of some recursive axiomatization of I∑ i such that ${{\sf PL}_\alpha({\rm I \Sigma}_i) \nsupseteq \bigcap_{\beta(x)}{\sf PL}_\beta({\rm I \Sigma}_j)}$ and ${{\sf PL}_\alpha({\rm I \Sigma}_i) \nsubseteq \bigcup_{\beta(x)}{\sf PL}_\beta({\rm I \Sigma}_j)}$ , where β(x) ranges over all ∑1 binumerations of recursive axiomatizations of I∑ j

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Citations of this work

An Escape From Vardanyan’s Theorem.Ana de Almeida Borges & Joost J. Joosten - 2023 - Journal of Symbolic Logic 88 (4):1613-1638.

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References found in this work

The Logic of Provability.George Boolos - 1993 - Cambridge and New York: Cambridge University Press.
Arithmetization of Metamathematics in a General Setting.Solomon Feferman - 1960 - Journal of Symbolic Logic 31 (2):269-270.
The predicate modal logic of provability.Franco Montagna - 1984 - Notre Dame Journal of Formal Logic 25 (2):179-189.
Metamathematics of First-Order Arithmetic.P. Hájek & P. Pudlák - 2000 - Studia Logica 64 (3):429-430.

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