An Introduction to Gödel's Theorems

Preface; 1. What Gödel's theorems say; 2. Functions and enumerations; 3. Effective computability; 4. Effectively axiomatized theories; 5. Capturing numerical properties; 6. The truths of arithmetic; 7. Sufficiently strong arithmetics; 8. Interlude: taking stock; 9. Induction; 10. Two formalized arithmetics; 11. What Q can prove; 12. Io, an arithmetic with induction; 13. First-order Peano arithmetic; 14. Primitive recursive functions; 15. LA can express every p.r. function; 16. Capturing functions; 17. Q is p.r. adequate; 18. Interlude: a very little about Principia; 19. The arithmetization of syntax; 20. Arithmetization in more detail; 21. PA is incomplete; 22. Gödel's First Theorem; 23. Interlude: about the First Theorem; 24. The Diagonalization Lemma; 25. Rosser's proof; 26. Broadening the scope; 27. Tarski's Theorem; 28. Speed-up; 29. Second-order arithmetics; 30. Interlude: incompleteness and Isaacson's thesis; 31. Gödel's Second Theorem for PA; 32. On the 'unprovability of consistency'; 33. Generalizing the Second Theorem; 34. Löb's Theorem and other matters; 35. Deriving the derivability conditions; 36. 'The best and most general version'; 37. Interlude: the Second Theorem, Hilbert, minds and machines; 38. mi-Recursive functions; 39. Q is recursively adequate; 40. Undecidability and incompleteness; 41. Turing machines; 42. Turing machines and recursiveness; 43. Halting and incompleteness; 44. The Church-Turing thesis; 45. Proving the thesis?; 46. Looking back.

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