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.
Peter Smith is Lecturer in Philosophy at the University of Cambridge. His books include Explaining Chaos (1998) and An Introduction to Formal Logic (2003), and he is a former editor of the journal Analysis.
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.