数学家用的数理逻辑教程 第2版 影印版 (德)Y. I.马宁(Yu.I.Manin) 著 生活 文轩网

   Preface to the Second Edition

Preface to the First Edition

Ⅰ PROVABILITY

Ⅰ Introduction to Formal Languages

1 General Information

2 First-Order Languages

Digression: Names

3 Beginners' Course in Translation

Digression: Syntax

Ⅱ Truth and Deducibility

1 Unique Reading Lemma

2 Interpretation: Truth, Definability

3 Syntactic Properties of Truth

Digression: Natural Logic

4 Deducibility

Digression: Proof

5 Tautologies and Boolean Algebras

Digression: Kennings

6 Godel's Completeness Theorem

7 Countable Models and Skolem's Paradox

8 Language Extensions

9 Undefinability of Truth: The Language SELF

10 Smullyan's Language of Arithmetic

11 Undefinability of Truth: Tarski's Theorem

Digression: Self-Reference

12 Quantum Logic

Appendix: The Von Neumann Universe

The Last Digression. Truth as Value and Duty: Lessons of Mathematics

Ⅲ The Continuum Problem and Forcing

1 The Problem: Results, Ideas

2 A Language of Real Analysis

3 The Continuum Hypothesis Is Not Deducible in L2 Real

4 Boolean-Valued Universes

5 The Axiom of Extensionality Is "True"

6 The Axioms of Pairing, Union, Power Set, and Regularity Are "True"

7 The Axioms of Infinity, Replacement, and Choice Are "True".

8 The Continuum Hypothesis Is "False" for Suitable B

9 Forcing

Ⅳ The Continuum Problem and Constructible Sets

1 Godel's Constructible Universe

2 Definability and Absoluteness

3 The Constructible Universe as a Model for Set Theory

4 The Generalized Continuum Hypothesis Is L-True

5 Constructibility Formula

6 Remarks on Formalization

7 What Is the Cardinality of the Continuum?

Ⅱ COMPUTABILITY

Ⅴ Recursive Functions and Church's Thesis

1 Introduction. Intuitive Computability

2 Partial Recursive Functions

3 Basic Examples of Recursiveness

4 Enumerable and Decidable Sets

5 Elements of Recursive Geometry

Ⅵ Diophantine Sets and Algorithmic Undecidability

1 The Basic Result

2 Plan of Proof

3 Enumerable Sets Are D-Sets

4 The Reduction

5 Construction of a Special Diophantine Set

6 The Graph of the Exponential Is Diophantine

7 The Factorial and Binomial Coefficient Graphs Are Diophantine

8 Versal Families

9 Kolmogorov Complexity

Ⅲ PROVABILITY AND COMPUTABILITY

Ⅶ Godel's Incompleteness Theorem

1 Arithmetic of Syntax

2 Incompleteness Principles

3 Nonenumerability of True Formulas

4 Syntactic Analysis

5 Enumerability of Deducible Formulas

6 The Arithmetical Hierarchy

7 Productivity of Arithmetical Truth

8 On the Length of Proofs

Ⅷ Recursive Groups

1 Basic Result and Its Corollaries

2 Free Products and HNN-Extensions

3 Embeddings in Groups with Two Generators

4 Benign Subgroups

5 Bounded Systems of Generators

6 End of the Proof

Ⅸ Constructive Universe and Computation

1 Introduction: A Categorical View of Computation

2 Expanding Constructive Universe: Generalities

3 Expanding Constructive Universe: Morphisms

4 Operads and PROPs

5 The World of Graphs as a Topological Language

6 Models of Computation and Complexity

7 Basics of Quantum Computation I: Quantum Entanglement

8 Selected Quantum Subroutines

9 Shot's Factoring Algorithm

10 Kolmogorov Complexity and Growth of Recursive Functions

Ⅳ MODEL THEORY

Ⅹ Model Theory

1 Languages and Structures

2 The Compactness Theorem

3 Basic Methods and Constructions

4 Completeness and Quantifier Elimination in Some Theories

5 Classification Theory

6 Geometric Stability Theory

7 Other Languages and Nonelementary Model Theory

Suggestions for Further Reading

Index