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

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

Published $1977$, Springer Verlag

ISBN 0-387-90243-0.

### 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
II Truth and deducibility
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
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
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?
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
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
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
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
Index