The de Jongh property for Basic Arithmetic

Archive for Mathematical Logic 53 (7-8):881-895 (2014)

Abstract

We prove that Basic Arithmetic, BA, has the de Jongh property, i.e., for any propositional formula A built up of atoms p1,..., pn, BPC⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vdash}$$\end{document}A if and only if for all arithmetical sentences B1,..., Bn, BA⊢\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vdash}$$\end{document}A. The technique used in our proof can easily be applied to some known extensions of BA.

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References found in this work

The Interpretability Logic of Peano Arithmetic.Alessandro Berarducci - 1990 - Journal of Symbolic Logic 55 (3):1059-1089.
Models of Peano Arithmetic.Richard Kaye - 1991 - Clarendon Press.
Basic Propositional Calculus I.Mohammad Ardeshir & Wim Ruitenburg - 1998 - Mathematical Logic Quarterly 44 (3):317-343.
Basic Predicate Calculus.Wim Ruitenburg - 1998 - Notre Dame Journal of Formal Logic 39 (1):18-46.

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Citations of this work

The Σ 1 -Provability Logic of HA.Mohammad Ardeshir & Mojtaba Mojtahedi - 2018 - Annals of Pure and Applied Logic 169 (10):997-1043.
Logics of Intuitionistic Kripke-Platek Set Theory.Rosalie Iemhoff & Robert Passmann - 2021 - Annals of Pure and Applied Logic 172 (10):103014.

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