Models of Peano arithmetic as modules over initial segments
type: Item
instance of: scholarly article
title: Models of Peano arithmetic as modules over initial segments
zbMATH Open document ID: 1088.03035
author: Nobuya Suzuki
published in: Tsukuba Journal of Mathematics
publication date: 2006-03-28T00:00:00Z
review text: \textit{A. Tsuboi} and \textit{M. Murakami} proved in an unpublished preprint [see also their paper ``Expanding the additive reduct of a model of Peano arithmetic'', Math. Log. Q. 49, 363--368 (2003; Zbl 1026.03024)] that for every countable model \(M\) of PA there are a model \(N\) and its initial segment \(I\) such that the additive reducts of \(M\) and \(N\) coincide and, for all \(a\in M\), \(a\times^M a=a\times^N a\) iff \(a\in I\). The author proves a variant of this result which covers a large class of initial segments \(I\), but with a weaker conclusion. Let \(I\) be an initial segment of a model \(M\). If \(I\) is closed under \(+\) and \(\times\), then \(M\) can be considered in a natural way as an \(I\)-module over the field of fractions \(Q(I)\) (all models of arithmetic here are considered as rings). The main result is that if \(I\) is closed under \(+\), \(\times\), and the factorial function \(x\mapsto x!\), then there is a model \(N\), such that \(M\) and \(N\) coincide as \(I\)-modules, but \(\times^M\not=\times^N\).
Mathematics Subject Classification ID: 03C62
Mathematics Subject Classification ID: 03H15
zbMATH DE Number: 5015593
zbMATH Keywords: Peano arithmetic
zbMATH Keywords: additive reduct
zbMATH Keywords: multiplicative reduct
zbMATH Keywords: nonstandard model
reviewed by: Roman Kossak
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496164891
OpenAlex ID: W1161063
DOI: 10.21099/TKBJM/1496164891
description: scientific article; zbMATH DE number 5015593