Structures associated with real closed fields and the axiom of choice
type: Item
instance of: scholarly article
title: Structures associated with real closed fields and the axiom of choice
zbMATH Open document ID: 1388.03036
author: Merlin Carl
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2016-10-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1402.6130
full work available at URL: https://projecteuclid.org/euclid.bbms/1473186514
reviewed by: J. M. Plotkin
Mathematics Subject Classification ID: 03C55
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03E30
Mathematics Subject Classification ID: 03E75
Mathematics Subject Classification ID: 12J20
Mathematics Subject Classification ID: 12L12
zbMATH DE Number: 6643012
zbMATH Keywords: axiom of choice
zbMATH Keywords: real closed fields
zbMATH Keywords: integer part
zbMATH Keywords: residue field section
zbMATH Keywords: value group section
MaRDI profile type: Publication
arXiv ID: 1402.6130
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review text: Let \(K\) be a real closed field (RCF). A discretely ordered subring \(I\) of \(K\) is an integer part of \(K\) if and only if 1 is the minimal positive element of \(I\) and for every \(x\in K\) there is an \(i\in I\) with \(i\leq x<i+1\). \textit{M. H. Mourgues} and \textit{J. P. Ressayre} [J. Symb. Log. 58, No. 2, 641--647 (1993; Zbl 0786.12005)] used a Zorn's lemma argument to show that every RCF has an integer part. They also used the same non-constructive methods to produce other structures associated with RCFs: namely, value group sections and residue field sections.NEWLINENEWLINEThe author of this article shows that the three structures mentioned above all require non-constructive methods. In particular, he constructs transitive models of ZF containing an RCF \(K\), but no integer part. He also constructs models containing RCFs without value group sections and residue field sections.NEWLINENEWLINEWith regard to integer parts, the author gives extra conditions on RCFs that guarantee the existence of integer parts in ZF. The author strengthens his results and obtains others by proving the following theorem: Let \(T\) be a countable, consistent o-minimal theory. Then there is a transitive model \(N\) of ZF such that \(N\) contains a model \(\hat{M}\models T\) with no unbounded discrete subset.
description: scientific article; zbMATH DE number 6643012