Contributions to model theory of metric structures

Tellez, Hernando

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https://hdl.handle.net/2142/16109 Copy

Description

Title

Contributions to model theory of metric structures

Author(s)

Tellez, Hernando

Issue Date

2010-05-19T18:36:29Z

Director of Research (if dissertation) or Advisor (if thesis)

Henson, C. Ward

Doctoral Committee Chair(s)

Solecki, Slawomir

Committee Member(s)

• Henson, C. Ward
• van den Dries, Lou
• Rosendal, Christian

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Continuous logic
• Metric structures
• Model theory
• Perturbations
• Banach spaces
• Algebraic closure

Abstract

Two Banach spaces X and Y are said to be almost isometric if for every λ > 1 there exists a λ-isomorphism f : X → Y . That is, a linear surjective map such that 1/λ ∥x∥ ≤ ∥f (x)∥ ≤ λ ∥x∥ for every x ∈ X . In this thesis we prove a Ryll-Nardzewski-style characterization of ω-categoricity up to almost isometry for Banach spaces using the concept of perturbations of metric structures and tools developed by Ben Yaacov ([6] and [5]). To this end we construct a single-sorted signature Lc for the study of the model theory of Banach spaces in the setting of continuous first order logic, we give an explicit axiomatization for the class of Lc -structures that come from unit balls of Banach spaces and we construct a perturbation system that is adequate for the study of almost isometric Banach spaces. Additionally, we study the algebraic closure construction for metric structures in the setting of continuous first order logic. We give several characterizations of algebraicity, and we prove basic properties analogous to ones that the algebraic closure satisfes in classical first order logic.

Graduation Semester

2010-05

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http://hdl.handle.net/2142/16109

Copyright and License Information

Copyright 2010 Hernando Tellez

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