Sequences in countable nonstandard models of the natural numbers
type: Item
instance of: scholarly article
title: Sequences in countable nonstandard models of the natural numbers
zbMATH Open document ID: 0675.03041
DOI: 10.1007/BF00370555
author: Steven Leth
published in: Studia Logica
publication date: 1989-01-01T00:00:00Z
review text: The author studies countable models of the set of natural numbers. Let \({\mathcal M}\) be such a model, \({\mathcal M}=<M,+,\cdot,0,1,A^ m>,\) where A is a unary predicate symbol whose interpretation in \({\mathbb{N}}\) is the standard sequence A. There is an order-preserving mapping q: \(M\to {\mathbb{Q}}\cap [0,1]\). Now the addition within the model induces a new ``addition'' on \({\mathbb{Q}}\cap [0,1]\). This addition function is continuous but it does not extend continuously to [0,1]. A sequence \(A\subseteq {\mathbb{N}}\) is associated with a subset q(A) of \({\mathbb{Q}}\cap [0,1]\). It is proved that the topological properties of the subset q(A) are independent of the way in which q is chosen. Relationships between A and q(A) are studied. The following result is typical: q(A) has at least one isolated point iff there is an \(n\in {\mathbb{N}}\) such that for all \(m\in {\mathbb{N}}\) there exists an interval of length m which contains exactly n elements of A. A mapping e: \(M\to [0,1]\) is introduced so that \(e(m)=e(n)\) iff m/n is in the monad of 1. The mapping e induces another addition \([+]\) on [0,1]\(\cap {\mathbb{Q}}\). The function \([+]\) can be extended continuously to [0,1].
Mathematics Subject Classification ID: 03H15
Mathematics Subject Classification ID: 11U10
zbMATH DE Number: 4104954
zbMATH Keywords: countable models
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4104954