Subsystems of Second Order Arithmetic

Through a series of case studies, this volume examines these axioms to prove particular theorems in core mathematical areas.

Stephen G. Simpson (Author)

9780521150149, Cambridge University Press

Paperback, published 18 February 2010

464 pages
23.4 x 15.6 x 2.4 cm, 0.65 kg

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.

List of tables
Preface
Acknowledgements
1. Introduction
Part I. Development of Mathematics within Subsystems of Z2: 2. Recursive comprehension
3. Arithmetical comprehension
4. Weak König's lemma
5. Arithmetical transfinite recursion
6. ?11 comprehension
Part II. Models of Subsystems of Z2: 7. ?-models
8. ?-models
9. Non-?-models
Part III. Appendix: 10. Additional results
Bibliography
Index.

Subject Areas: Mathematical logic [PBCD]