On two problems concerning end extensions
type: Item
instance of: scholarly article
title: On two problems concerning end extensions
zbMATH Open document ID: 1147.03017
author: Ch. Cornaros
author: Costas Dimitracopoulos
published in: Archive for Mathematical Logic
publication date: 2008-08-20T00:00:00Z
review text: The paper presents a comprehensive survey and some new results concerning two questions on end extensions of models of fragments of PA. The first, due to Paris, asks whether every countable model of \(\text{B}\Sigma_1\) has a proper end extension to a model of \(\text{I}\Delta_0\); the second, due to Clote, asks whether every countable model of \(\text{B}\Sigma_n\), \(n\geq 2\), has a proper \(\Sigma_n\)-elementary end extension to a model of \(\text{B}\Sigma_{n-1}\). Concerning the first question, \textit{A. Wilkie} and \textit{J. Paris} proved in [``On the existence of end extensions of models of bounded induction'', in: J. E. Fenstad et al. (eds.), Logic, methodology and philosophy of science VIII, Proc. 8th Int. Congr., Moscow/USSR 1987, Stud. Logic Found. Math. 126, 143--161 (1989; Zbl 0695.03019)] that the answer is affirmative if the model is \(\Delta_0\)-full. Cornaros and Dimitracopoulos analyze this notion and relate it to other notions of weak saturation. They introduce a notion of \(\Gamma\)-1-fullness, where \(\Gamma\) is a class of formulas, and they prove that if \(\Gamma\supseteq \text{I}\Delta_0\) is a recursive set of sentences and \(M\) is a countable model of \(\text{B}\Sigma_1+\Pi_1(\Gamma)\) and there is a sentence \(\lambda\) such that \(M\models\lnot\lambda\) and \(M\) is \((\Gamma+\lambda)\)-1-full, then there exists \(K\models\Gamma\) which is a proper \(\Sigma_1\)-elementary end extension of \(M\). They also prove that if there is a \(K\) as above and \(\Gamma\supseteq \text{I}\Sigma_1\), then \(M\) is \(\Gamma\)-1-full. Concerning Clote's question, Cornaros and Dimitracopoulos prove that for \(n\geq 2\) every countable model of \(\text{B}\Sigma_n\) has a proper \(\Sigma_n\)-elementary end extension to a model of parameter-free collection scheme for \(\Sigma_{n-1}\) formulas. They also modify a proof from the above mentioned paper of Wilkie and Paris to show that for every \(n\geq 2\) there exists a countable \(M\models \text{B}\Sigma_n\) …
reviewed by: Roman Kossak
Mathematics Subject Classification ID: 03C62
zbMATH DE Number: 5314396
zbMATH Keywords: end extension problem
zbMATH Keywords: fragments of arithmetic
zbMATH Keywords: weak saturation
zbMATH Keywords: overspill
zbMATH Keywords: survey
MaRDI profile type: Publication
full work available at URL: s00153-007-0055-1
OpenAlex ID: W2026277548
cites work: A note on the Mac Dowell-Specker theorem
cites work: Q4384146
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cites work: Q3931391
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cites work: On the scheme of induction for bounded arithmetic formulas
DOI: 10.1007/S00153-007-0055-1
description: scientific article; zbMATH DE number 5314396