Subsystems of Second Order Arithmetic / Edition 2

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Overview

Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are examined to prove particular theorems in core mathematical areas such as algebra, analysis, and topology, focusing on the language of second order arithmetic, the weakest language rich enough to express and develop the bulk of mathematics.

In many cases, if a mathematical theorem is proved from appropriately weak set existence axioms, then the axioms will be logically equivalent to the theorem. Furthermore, only a few specific set existence axioms arise repeatedly in this context, which in turn correspond to classical foundational programs. This is the theme of reverse mathematics, which dominates the first half of the book. The second part focuses on models of these and other subsystems of second order arithmetic. Additional results are presented in an appendix.

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Product Details

  • ISBN-13: 9780521150149
  • Publisher: Cambridge University Press
  • Publication date: 1/28/2010
  • Series: Perspectives in Logic Series
  • Edition description: New Edition
  • Edition number: 2
  • Pages: 464
  • Product dimensions: 6.10 (w) x 9.10 (h) x 1.10 (d)

Table of Contents

Preface
Acknowledgements
I Introduction 1
Pt. A Development of Mathematics Within Subsystems of Z[subscript 2]
II Recursive Comprehension 63
III Arithmetical Comprehension 105
IV Weak Konig's Lemma 127
V Arithmetical Transfinite Recursion 167
VI [Pi][subscript 1][superscript 1] Comprehension 217
Pt. B Models of Subsystems of Z[subscript 2]
VII [beta]-Models 245
VIII [omega]-Models 313
IX Non-[omega]-Models 363
X Additional Results 395
Bibliography 413
Index 425
List of Tables 445
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