What is o-minimality?
type: Item
instance of: scholarly article
title: What is o-minimality?
zbMATH Open document ID: 1155.03021
author: Harvey M. Friedman
published in: Annals of Pure and Applied Logic
publication date: 2008-12-05T00:00:00Z
review text: Let \(\mathbb R\) denote the set of real numbers and let \(W\) be a set of functions, each from some \(\mathbb R^n\) to some \(\mathbb R^m\). Friedman proves that \((\mathbb R,<, 0,1,+,-,\times, W)\) is o-minimal if and only if there exists a least \(V\) such that \(W\subseteq V\) and: \(V\) contains all polynomial functions; it is is closed under compositions; for all \(f,g\) in \(V\), if \(f\) and \(g\) are of the same arity, then \(h(x)=(f(x),g(x))\) is in \(V\); if \(f:\mathbb R^{n+m}\to \mathbb R\) is in \(V\) then there is a \(g\) in \(V\) such that for all \(x\), if \(f(x,y)=0\) for some \(y\), then \(f(x,g(x))=0\); if \(f:\mathbb R\to \mathbb R\) in \(V\) is bounded, then \(f\) has a limit at infinity. The result is then generalized to arbitrary ordered rings \(R\), but to this end the last condition above has to be replaced by: for all \(f: R\to R\) in \(V\) there exist \(x_1<\dots <x_k\in R\), \(k\geq 1\), such that on each of the intervals \((-\infty, x_1), (x_1,x_2),\dots, (x_k,\infty)\), \(f\) is either constant, strictly increasing, or strictly decreasing.
reviewed by: Roman Kossak
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 12J15
Mathematics Subject Classification ID: 26E99
zbMATH DE Number: 5378326
zbMATH Keywords: o-minimality
zbMATH Keywords: ordered rings
zbMATH Keywords: monotonicity theorem
zbMATH Keywords: zero selection
MaRDI profile type: Publication
full work available at URL: j.apal.2008.06.007
OpenAlex ID: W2017552350
cites work: Real closed rings. II. Model theory
cites work: Definable Sets in Ordered Structures. I
cites work: Q4392286
cites work: Geometric categories and o-minimal structures
cites work: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
DOI: 10.1016/J.APAL.2008.06.007
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description: scientific article; zbMATH DE number 5378326