Constructing \(\omega\)-stable structures: Model completeness.
type: Item
instance of: scholarly article
title: Constructing \(\omega\)-stable structures: Model completeness.
zbMATH Open document ID: 1042.03026
published in: Annals of Pure and Applied Logic
publication date: 2004-03-14T00:00:00Z
review text: The authors extend work by \textit{K. Holland} [J. Symb. Log. 64, No. 3, 946--962 (1999; Zbl 0945.03045)] by showing that the projective plane of \textit{J. T. Baldwin} [Trans. Am. Math. Soc. 342, No. 2, 695--711 (1994; Zbl 0796.03041)] and the finite rank bicoloured fields of \textit{J. T. Baldwin} and \textit{K. Holland} [J. Symb. Log. 65, No. 1, 371--391 (2000; Zbl 0957.03044)] are model-complete, while the infinite rank bicoloured field of \textit{B. Poizat} [J. Symb. Log. 64, No. 3, 1339--1355 (1999; Zbl 0938.03058)] is not. More generally, the finite rank collaps of a strongly minimal set expanded by a random unary predicate is almost strongly minimal and model-complete provided the strongly minimal set has a dense set of exactly rank \(k\) formulas, where \(k\) is the rank of the collapse (and this excludes triviality for \(k>2\)). The proof in the second case makes use of Lindström's theorem that inductive theories categorical in an infinite power are model-complete; in the first case it uses a new (but old-style) characterization of almost strongly minimal model-complete theories.
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C10
Mathematics Subject Classification ID: 03C35
zbMATH DE Number: 2056209
zbMATH Keywords: Hrushovski amalgamation construction
zbMATH Keywords: almost strongly minimal theories
zbMATH Keywords: model-completeness
zbMATH Keywords: projective plane
zbMATH Keywords: bicoloured fields
zbMATH Keywords: \(\omega\)-stable models
author: John T. Baldwin
author: Kitty Holland
MaRDI profile type: Publication
full work available at URL: j.apal.2003.10.003
OpenAlex ID: W2088730863
cites work: Categoricity and generalized model completeness
cites work: An Almost Strongly Minimal Non-Desarguesian Projective Plane
cites work: Some Projective Planes of Lenz-Barlotti Class I
cites work: Constructing ω-stable structures: rank 2 fields
cites work: Constructing \(\omega\)-stable structures: Computing rank
cites work: Stable generic structures
cites work: Trivial, strongly minimal theories are model complete after naming constants
cites work: Model completeness of the new strongly minimal sets
cites work: A new strongly minimal set
cites work: Non Σn axiomatizable almost strongly minimal theories
cites work: Le carre de l'egalite
DOI: 10.1016/J.APAL.2003.10.003
Recommended article: Expansions of models of ω-stable theories
Recommended article: Q3464624
Recommended article: Decidable models of \(\omega\)-stable theories
Recommended article: Constructing \(\omega\)-stable structures: Computing rank
Recommended article: Remarks on structure theorems for \(\omega_ 1\)-saturated models
Recommended article: Q3118396
Recommended article: The Model Completion of the Class of ℒ-Structures
Recommended article: ITERATING THE COFINALITY- CONSTRUCTIBLE MODEL
Recommended article: Model-theoretic properties of \(\omega\)-automatic structures
Recommended article: Model completeness of o-minimal structures expanded by Dedekind cuts
description: scientific article; zbMATH DE number 2056209