Minimal models of minimal theories
type: Item
instance of: scholarly article
title: Minimal models of minimal theories
zbMATH Open document ID: 0801.03027
author: Koichiro Ikeda
published in: Tsukuba Journal of Mathematics
publication date: 1994-12-01T00:00:00Z
review text: The author proves that a first-order theory, \(T\), has \(2^{\aleph_ 0}\) minimal models, if it is stable and trivial, and if it has a minimal and non-prime model in which each element has Lascar rank \(\leq 1\). This result is valid for any language in contrast to Marcus's and Toffalori's works that consider the language with one unary function [cf. \textit{L. Marcus}, Isr. J. Math. 18, 117-131 (1974; Zbl 0298.02059), and \textit{C. Toffalori}, Ill. J. Math. 35, No. 1, 1-26 (1991; Zbl 0696.03018), respectively]. The author shows that their example satisfies the assumptions of the above theorem as well as \textit{G. Fuhrken's} [Arch. Math. Logik Grundlagenforsch. 9, 3-11 (1966; Zbl 0168.007)]. On the other hand, \textit{S. Shelah's} do not, as all minimal models there have Lascar rank 2 [cf. J. Symb. Log. 43, 475-480 (1978; Zbl 0389.03014)].
Mathematics Subject Classification ID: 03C50
zbMATH DE Number: 591460
zbMATH Keywords: minimal models
zbMATH Keywords: Lascar rank
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496162276
OpenAlex ID: W1923095552
DOI: 10.21099/TKBJM/1496162276
description: scientific article; zbMATH DE number 591460