Subsets of models of arithmetic
type: Item
instance of: scholarly article
title: Subsets of models of arithmetic
zbMATH Open document ID: 0786.03027
DOI: 10.1007/BF01270396
published in: Archive for Mathematical Logic
publication date: 1994-01-13T00:00:00Z
review text: The paper contains various results concerning end extensions of countable models of PA. It is shown that a subset \(X\) of a countable model \(M\models\text{PA}\) can be coded in an elementary end extension of \(M\) iff \((M,X)\) satisfies a certain strong collection scheme. Examples are constructed to show that the scheme is weaker than full induction, but is not implied by \(\Sigma_ n\) induction scheme for \((M,X)\), for any \(n\). Other results include a proof that if \(X\subseteq M\) is coded in some elementary end extension of \(M\), then it is also coded in some minimal elementary end extension of \(M\), and a proof that every countable model \(M\) of PA has an elementary end extension in which \(M\) is a semiregular, but not regular, cut.
Mathematics Subject Classification ID: 03C62
zbMATH DE Number: 484794
zbMATH Keywords: elementary extensions
zbMATH Keywords: end extensions of countable models of PA
zbMATH Keywords: strong collection scheme
author: Roman Kossak
author: Jeff Paris
reviewed by: Roman Kossak
MaRDI profile type: Publication
cites work: Q3682484
cites work: Indicators, recursive saturation and expandability
cites work: On Extensions of Models of Strong Fragments of Arithmetic
full work available at URL: https://doi.org/10.1007/bf01270396
OpenAlex ID: W1999948456
description: scientific article; zbMATH DE number 484794