1 Mathematician’s World ......................... 1
1.1 Mathematical Structures....................... 2
1.2 Everything Is a Set.......................... 25
1.3 Antinomies of Set Theory ...................... 36
1.4 The Axiomatic Method ....................... 43
1.5 The Necessity of Using Abstract Concepts . . . . . . . . . . . . . 54
Main Points of the Chapter ........................ 64
2 Language,Logic and Computations .................. 65
2.1 The Language of Mathematics.................... 66
2.2 Truth and Models .......................... 80
2.3 Proofs ................................ 92
2.4 Programs and Computations.....................123
2.5 The Lambda Calculus ........................146
Main Points of the Chapter ........................155
3 Set Theory.................................157
3.1 The Axioms of Set Theory......................159
3.2 The Arithmetic of Infinity......................176
3.3 What Is the Largest Number? ....................196
3.4 Controversial Axioms ........................215
3.5 Alternative Set-Theoretical Foundations . . . . . . . . . . . . . . 231
Main Points of the Chapter ........................253
4 Proofs of Impossibility..........................255
4.1 Impossibility Proofs in Geometry and Algebra . . . . . . . . . . . 256
4.2 The Incompleteness Theorems ...................272
4.3 Algorithmically Unsolvable Problems. . . . . . . . . . . . . . . . 300
4.4 Concrete Independence .......................319
4.5 The Independent Sentences of Set Theory. . . . . . . . . . . . . . 340
Main Points of the Chapter ........................364
5 The Complexity of Computations....................365
5.1 What Is Complexity? ........................366
5.2 Randomness, Interaction and Cryptography . . . . . . . . . . . . . 410
5.3 Parallel Computations........................437
5.4 Quantum Computations .......................448
5.5 Descriptional Complexity ......................479
Main Points of the Chapter ........................493
6 Proof Complexity.............................495
6.1 Proof Theory.............................496
6.2 Theories and Complexity Classes..................523
6.3 Propositional Proofs.........................540
6.4 Feasible Incompleteness.......................562
Main Points of the Chapter ........................580
7 Consistency,Truth and Existence....................583
7.1 Consistency and Existence......................584
7.2 The Attributes of Reality ......................609
7.3 Finitism and Physical Reality ....................646
Main Points of the Chapter ........................664
Bibliographical Remarks ...........................667
References .
· · · · · · (收起)