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

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## Yu.I. Manin and Boris Zilber:

## Yu.I. Manin and Boris Zilber: *A Course in Mathematical Logic (2nd Edition)*

Published $\text {2010}$, **Springer Verlag**

- ISBN 978-1441906144.

### Subject Matter

### 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

- I Introduction to Formal Languages

- 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

- II Truth and Deducibility

- 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

- III The Continuum Problem and 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?

- IV The Continuum Problem and Constructible Sets

- 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

- V Recursive Functions and Church's Thesis

- 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

- VI Diophantine Sets and Algorithmic Undecidability

- 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

- VII Gödel's Incompleteness Theorem

- 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

- VIII Recursive Groups

- 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

- IX Constructive Universe and Computation

- 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

- X Model Theory

- Suggestions for Further Reading
- Index