Interpolation and the interpretability logic of PA
type: Item
instance of: scholarly article
title: Interpolation and the interpretability logic of PA
zbMATH Open document ID: 1114.03048
author: Evan Goris
published in: Notre Dame Journal of Formal Logic
publication date: 2007-02-15T00:00:00Z
review text: The interpretability logic is a nonstandard modal logic which describes the relation of relative interpretability between arithmetical theories like \(T+A\) and \(T+B,\) where \(A\) and \(B\) are formulas (for detailed definitions see, e.g., [\textit{A. Visser}, ``An overview of interpretability logic'', in: M. Kracht et al. (eds.), Advances in modal logic. Vol. 1. CSLI Lect. Notes 87, 307--359 (1998; Zbl 0915.03020)] or [\textit{D. de Jongh} and \textit{G. Japaridze}, ``The logic of provability'', in: S. R. Buss (ed.), Handbook of proof theory, 475--546 (1998; Zbl 0915.03019]). The modal system ILM is the interpretability logic of Peano arithmetic. This system does not have the interpolation property. For this reason the author considers an extensions of the system ILM by an additional operator for \(\Sigma_1\)-ness. An axiomatization for this extension is given (the system ILM(S)). The author proved modal and arithmetical completeness of the system ILM(S) in an earlier paper. The main result of the paper under review is that the system ILM(S) does not have the interpolation property.
reviewed by: Mladen Vuković
Mathematics Subject Classification ID: 03F45
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03F30
zbMATH DE Number: 5127267
zbMATH Keywords: provability logic
zbMATH Keywords: interpretability logic
zbMATH Keywords: interpolation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1153858645
OpenAlex ID: W1975821687
DOI: 10.1305/NDJFL/1153858645
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description: scientific article; zbMATH DE number 5127267