Computable Structure Theory: Within the Arithmetic

Antonio Montalbán

doi:10.1017/9781108525749

In mathematics, we know there are some concepts - objects, constructions, structures, proofs - that are more complex and difficult to describe than others. Computable structure theory quantifies and studies the complexity of mathematical structures, structures such as graphs, groups, and orderings. Written by a contemporary expert in the subject, this is the first full monograph on computable structure theory in 20 years. Aimed at graduate students and researchers in mathematical logic, it brings new results of the author together with many older results that were previously scattered across the literature and presents them all in a coherent framework, making it easier for the reader to learn the main results and techniques in the area for application in their own research. This volume focuses on countable structures whose complexity can be measured within arithmetic; a forthcoming second volume will study structures beyond arithmetic.

‘This exceptionally well-written book is the first modern monograph on computable structure theory in 20 years … The author succeeds in bringing together new and old results and presenting them in a coherent framework, making it easy for the reader to learn the main results and techniques in the area for application in their own research.’

Alexandra Andreeva Soskova Source: MathSciNet

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Computable Structure Theory

Within the Arithmetic

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Contents

Frontmatter
pp i-iv

Dedication
pp v-vi

Contents
pp vii-viii

Preface
pp ix-xii

Notation and Conventions
pp xiii-xxii

Chapter 1 - Structures
pp 1-10

Chapter 2 - Relations
pp 11-32

Chapter 3 - Existentially-Atomic Models
pp 33-50

Chapter 4 - Generic Presentations
pp 51-62

Chapter 5 - Degree Spectra
pp 63-74

Chapter 6 - Comparing Structures and Classes of Structures
pp 75-96

Chapter 7 - Finite-Injury Constructions
pp 97-114

Chapter 8 - Computable Categoricity
pp 115-136

Chapter 9 - The Jump of A Structure
pp 137-148

Chapter 10 - Σ-Small Classes
pp 149-176

Bibliography
pp 177-186

Index
pp 187-190

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Within the Arithmetic

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