A short note on essentially \(\Sigma_1\) sentences
type: Item
instance of: scholarly article
title: A short note on essentially \(\Sigma_1\) sentences
zbMATH Open document ID: 1270.03039
author: Duccio Pianigiani
author: Franco Montagna
published in: Logica Universalis
publication date: 2013-04-08T00:00:00Z
review text: A modal formula \(F\) is essentially \(\Sigma_1\) with respect to theory \(T\) if, under any arithmetical interpretations \(\ast\) into \(T,\) \(F^\ast\) is a \(\Sigma_1\) formula. \textit{A.~Visser} proved in [Ann.~Pure~App.~Logic 73, No. 1, 109--142 (1995; Zbl 0828.03008)] that a formula \(F\) of the provability logic GL (Gödel-Löb) is essentially \(\Sigma_1\) with respect to PA if and only if \(F\) is provably equivalent in GL to a disjunction of formulas of the form \(\square B\). In [Stud.\ Fuzziness Soft Comput.\ 24, 246--254 (1999; Zbl 0923.03025)], \textit{D.\ de Jongh} and \textit{D.\ Pianigiani} extended the result to system \(R\) of Guaspari-Solovay. \textit{E.\ Goris} and \textit{J.\ J.\ Joosten} classified in [Log. J. IGPL 16, No. 4, 371--412 (2008; Zbl 1162.03033); ibid. 20, No. 1, 1--21 (2012; Zbl 1252.03140)] all essentially \(\Sigma_1\) sentences of ILM (interpretability logic with Montagna principle) with respect to interpretations in essentially reflexive recursively enumerable arithmetical theories. In the paper under review, the authors show that a characterization of this kind can be obtained also for formulas of the interpretability logic ILP, with respect to any finitely axiomatizable \(\Sigma_1\)-sound extension of \(\mathrm{I}\Delta_0 + \mathrm{Supexp}.\) It is proved that the same characterization does not extend to \(\mathrm I\Delta_0+\mathrm{Exp}\) and a conjecture is formulated about essentially \(\Sigma_1\) ILP-formulas with respect to \(\mathrm I\Delta_0+\mathrm{Exp}.\)
reviewed by: Mladen Vuković
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03F45
Mathematics Subject Classification ID: 03F30
zbMATH DE Number: 6151685
zbMATH Keywords: essentially \(\Sigma_1\) formula
zbMATH Keywords: provability logic
zbMATH Keywords: first-order fragments of arithmetic
zbMATH Keywords: interpretability logic
MaRDI profile type: Publication
full work available at URL: s11787-012-0070-9
OpenAlex ID: W2014462294
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DOI: 10.1007/S11787-012-0070-9
description: scientific article; zbMATH DE number 6151685