The field of reals with a predicate for the real algebraic numbers and a predicate for the integer powers of two

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• 作者：Mohsen Khani
• 关键词：O ; minimality ; Dense pairs ; Integer powers of two ; 03C10
• 刊名：Archive for Mathematical Logic
• 出版年：2015
• 出版时间：November 2015
• 年：2015
• 卷：54
• 期：7-8
• 页码：885-898
• 全文大小：456 KB
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• 作者单位：Mohsen Khani (1)
  
  1. Albert-Ludwigs-Universitaet Freiburg, Abteilung f眉r Mathematische Logik, Eckerstr 1, Raum 305, 79104, Freiburg im Breisgau, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematical Logic and Foundations
  Mathematics
  Algebra
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0665

文摘

Given a theory T of a polynomially bounded o-minimal expansion R of \({\bar{\mathbb{R}} = \langle\mathbb{R}, +, ., 0, 1, < \rangle}\) with field of exponents \({\mathbb{Q}}\), we introduce a theory \({\mathbb{T}}\) whose models are expansions of dense pairs of models of T by a discrete multiplicative group. We prove that \({\mathbb{T}}\) is complete and admits quantifier elimination when predicates are added for certain existential formulas. In particular, if T = RCF then \({\mathbb{T}}\) axiomatises \({\langle\bar{\mathbb{R}}, \mathbb{R}_{alg}, 2^{\mathbb{Z}}\rangle}\), where \({\mathbb{R}_{alg}}\) denotes the real algebraic numbers. We describe types and definable sets in our models and prove that \({\mathbb{T}}\) is dependent. Keywords O-minimality Dense pairs Integer powers of two