Number theory as studied by the logician is the subject matter of the book. This first volume can stand on its own as a somewhat unorthodox introduction to mathematical logic for undergraduates, dealing with the usual introductory material: recursion theory, first-order logic, completeness, incompleteness, and undecidability. In addition, its second chapter contains the most complete logical discussion of Diophantine Decision Problems available anywhere, taking the reader right up to the frontiers of research (yet remaining accessible to the undergraduate). The first and third chapters also offer greater depth and breadth in logico-arithmetical matters than can be found in existing logic texts. Each chapter contains numerous exercises, historical and other comments aimed at developing the student's perspective on the subject, and a partially annotated bibliography.
The 2-volume introductory textbook on logical number theory - or, number theory as studied by the logician - consists of this first volume that can stand on its own as a somewhat unorthodox introduction to mathematical logic for undergraduates, and a second volume to follow soon with more sophisticated content and treatment but still accessible to students having read volume I or had a standard introductory course. The author is well known beyond his own field of specialisation for his expository talent and his lucid and enjoyable writing style.
I. Arithmetic Encoding.- 1. Polynomials.- 2. Sums of Powers.- 3. The Cantor Pairing function.- 4. The Fueter-Pólya Theorem, I.- *5. The Fueter-Pólya Theorem, II.- 6. The Chinese Remainder Theorem.- 7. The ?-Function and Other Encoding Schemes.- 8. Primitive Recursion.- *9. Ackermann Functions.- 10. Arithmetic Relations.- 11. Computability.- 12. Elementary Recursion Theory.- 13. The Arithmetic Hierarchy.- 14. Reading List.- II. Diophantine Encoding.- 1. Diophantine Equations; Some Background.- 2. Initial Results; The Davis-Putnam-Robinson Theorem.- 3. The Pell Equation, I.- 4. The Pell Equation, II.- 5. The Diophantine Nature of R.E. Relations.- 6. Applications.- 7. Forms.- *8. Binomial Coëfficients.- *9. A Direct Proof of the Davis-Putnam-Robinson Theorem.- *10. The 3-Variable Exponential Diophantine Result.- 11. Reading List.- III. Weak Formal Theories of Arithmetic.- 1. Ignorabimus?.- 2. Formal Language and Logic.- 3. The Completeness Theorem.- 4. Presburger-Skolem Arithmetic; The Theory of Addition.- *5. Skolem Arithmetic; The Theory of Multiplication.- 6. Theories with + and ?; Incompleteness and Undecidability.- 7. Semi-Repiesentability of Functions.- 8. Further Undecidability Results.- 9. Reading List.- Index of Names.- Index of Subjects.