1 (p1): Prologue.Prerequisites and notation1 (p1-1): 1 Sets2 (p1-2): 2 Functions4 (p1-3): 3 Relations and predicates4 (p1-4): 4 Logical notation5 (p1-5): 5 References7 (p2): 1 Computable functions7 (p2-1): 1 Algorithms，or effective procedures9 (p2-2): 2 The unlimited register machine16 (p2-3): 3 URM-computable functions22 (p2-4): 4 Decidable predicates and problems23 (p2-5): 5 Computability on other domains25 (p3): 2 Generating computable functions25 (p3-1): 1 The basic functions25 (p3-2): 2 Joining programs together29 (p3-3): 3 Substitution32 (p3-4): 4 Recursion42 (p3-5): 5 Minimalisation48 (p4): 3 Other approaches to computability：Church’s thesis48 (p4-1): 1 Other approaches to computability49 (p4-2): 2 Partial recursive functions（Godel-Kleene）51 (p4-3): 3 A digression：the primitive recursive functions52 (p4-4): 4 Turing-computability57 (p4-5): 5 Symbol manipulation systems of Post and Markov65 (p4-6): 6 Computability on domains other than N67 (p4-7): 7 Church’s thesis72 (p5): 4 Numbering computable functions72 (p5-1): 1 Numbering programs76 (p5-2): 2 Numbering computable functions79 (p5-3): 3 Discussion：the diagonal method81 (p5-4): 4 The s-m-n theorem85 (p6): 5 Universal programs85 (p6-1): 1 Universal functions and universal programs90 (p6-2): 2 Two applications of the universal program93 (p6-3): 3 Effective operations on computable functions95 (p6-4): Appendix.Computability of the function σn100 (p7): 6 Decidability，undecidability and partial decidability101 (p7-1): 1 Undecidable problems in computability106 (p7-2): 2 The word problem for groups107 (p7-3): 3 Diophantine equations108 (p7-4): 4 Sturm’s algorithm109 (p7-5): 5 Mathematical logic112 (p7-6): 6 Partially decidable predicates121 (p8): 7 Recursive and recursively enumerable sets121…