内容简介

The two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability. Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs. Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory.

作品目录

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 .