# Book:Yu.I. Manin/A Course in Mathematical Logic

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## Yu.I. Manin:

## Yu.I. Manin: *A Course in Mathematical Logic*

Published $\text {1977}$, **Springer Verlag**

- ISBN 0-387-90243-0.

### Subject Matter

### Further Editions

This book also appears in the following edition:

### Contents

- Part 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
- Digression: Kennings
- 6 Gödel'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

- 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 $\mathrm{L_2Real}$
- 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

- Part 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 graphs of the functorial and the binomial coefficients are Diophantine
- 8 Versal families
- 9 Kolmogorov complexity

- VI Diophantine sets and algorithmic undecidability

- Part 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

- Index