An Introduction to Gödel’s Theorems

In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter?  Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic.

• An ideal textbook for philosophy and mathematics students taking a first course in mathematical logic • Rare coverage of a number of different proofs of the core incompleteness theorem • Includes a companion website with exercises

Preface; 1. What Gödel\'s Theorems say; 2. Decidability and enumerability; 3. Axiomatized formal theories; 4. Capturing numerical properties; 5. The truths of arithmetic; 6. Sufficiently strong arithmetics; 7. Interlude: taking stock; 8. Two formalized arithmetics; 9. What Q can prove; 10. First-order Peano Arithmetic; 11. Primitive recursive functions; 12. Capturing funtions; 13. Q is p.r. adequate; 14. Interlude: a very little about Principia; 15. The arithmetization of syntax; 16. PA is incomplete; 17. Gödel\'s First Theorem; 18. Interlude: about the First Theorem; 19. Strengthening the First Theorem; 20. The Diagonalization Lemma; 21. Using the Diagonalization Lemma; 22. Second-order arithmetics; 23. Interlude: incompleteness and Isaacson’s conjecture; 24. Gödel\'s Second Theorem for PA; 25. The derivability conditions; 26. Deriving the derivability conditions; 27. Reflections; 28. Interlude: about the Second Theorem; 29. Recursive functions; 30. Undecidability and incompleteness; 31. Turing machines; 32. Turing machines and recursiveness; 33. Halting problems; 34. The Church-Turing Thesis; 35. Proving the Thesis?; 36. Looking back.

\'Smith has written a wonderful book giving a clear and compelling presentation of Gödel’s Theorems and their implications. His style is both precise and engaging at the same time. The clarity of the writing is impressive, and there is a pleasing coverage of historical and philosophical topics. An Introduction to Gödel’s Theorems will work very well either as a textbook or as an introduction for any reader who wants a thorough understanding of some of the central ideas at the intersection of philosophy, mathematics and computer science.\'

– Professor Christopher Leary, Department of Mathematics, SUNY Geneseo

\'Peter Smith has succeeded in writing an excellent introduction to Gödel\'s incompleteness theorems and related topics which is accessible without being superficial. Philosophers in particular will appreciate the discussions of the Church-Turing Thesis, mechanism, and the relevance of Gödel\'s results in the philosophy of mathematics. It is certain to become a standard text.\'

– Richard Zach, Department of Philosophy, University of Calgary