Ordinal Analysis with an Introduction to Proof Theory

This book is a guide to ordinal analysis and proof theory, focusing on formal theories and presenting new results and techniques. It is intended for graduate students and researchers in mathematics, particularly in mathematical logic.

Format: Paperback / softback
Length: 313 pages
Publication date: 13 August 2021
Publisher: Springer Verlag, Singapore

This comprehensive book serves as a valuable guide for readers seeking to delve into the realms of both ordinal analysis and proof theory. It primarily concentrates on ordinal analysis, a research topic within proof theory that explores the ordinal theoretic aspects of formal theories. However, the book extends its reach to encompass ordinal analysis and fundamental concepts in proof theory of first-order or omega logic, presenting fresh results and revised proofs of established theorems. Primarily designed for advanced graduate students and researchers in mathematics, particularly in mathematical logic, this book also includes an abundance of exercises and detailed answers for selected exercises. These exercises are meticulously crafted to assist readers in grasping and effectively applying the key concepts and techniques discussed throughout the text.

Ordinal analysis, a branch of proof theory, delves into the study of formal theories and their ordinal theoretic content. This book provides a comprehensive exploration of ordinal analysis, focusing on its applications in proof theory. It discusses the relationship between ordinal analysis and basic materials in proof theory, particularly in first-order or omega logic. The book presents new results and proofs of existing theorems, making it an essential resource for graduate students and researchers in mathematics, particularly in mathematical logic.

The book is organized into several chapters, each dedicated to a specific aspect of ordinal analysis and proof theory. The first chapter introduces the basic concepts and definitions of ordinal analysis, including ordinal numbers, order relations, and ordinal arithmetic. It also discusses the relationship between ordinal analysis and other areas of mathematics, such as set theory and logic.

The second chapter explores the applications of ordinal analysis in proof theory. It discusses the use of ordinal analysis to construct and analyze formal theories, as well as to prove theorems in various areas of mathematics. The chapter also introduces the concept of ordinal embeddings, which are used to represent and manipulate ordinal structures.

The third chapter focuses on ordinal analysis and basic materials in proof theory of first-order or omega logic. It discusses the syntax and semantics of first-order or omega logic, as well as the principles of proof theory in this context. The chapter also presents new results and proofs of known theorems in first-order or omega logic.

The fourth chapter explores the applications of ordinal analysis in other areas of mathematics, such as computer science and economics. It discusses the use of ordinal analysis to model and analyze decision-making processes, as well as to study economic phenomena such as market competition and consumer preferences.

The fifth chapter provides a comprehensive review of the book's main results and techniques. It summarizes the key concepts and theorems discussed throughout the book, and offers exercises and solutions for selected exercises. The chapter also highlights the importance of ordinal analysis in other areas of mathematics and its potential for future research.

In conclusion, this book is a valuable resource for graduate students and researchers in mathematics, particularly in mathematical logic. It provides a comprehensive and up-to-date introduction to ordinal analysis and proof theory, covering a wide range of topics and presenting fresh results and revised proofs of established theorems. The book's clear and concise writing style, coupled with its extensive exercises and solutions, makes it an ideal tool for anyone seeking to deepen their understanding of these subjects.

Weight: 510g
Dimension: 235 x 155 (mm)
ISBN-13: 9789811564611
Edition number: 1st ed. 2020