Book:Yu.I. Manin/A Course in Mathematical Logic for Mathematicians/Second Edition

Subject Matter

 * Mathematical Logic

Contents

 * 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 Beginner's 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
 * Diression: 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 $\mathrm L_2$ 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 Gödel'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 Gödel'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 Shor's Factoring Algorithm
 * 10 Kolmogorov Complexity and Growth of Recursive Functions


 * IV MODEL THEORY


 * X Model Theory
 * 1 Languages and Structure
 * 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