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

Yu.I. Manin: A Course in Mathematical Logic

Published $\text {1977}$, Springer Verlag

ISBN 0-387-90243-0

Subject Matter

• Mathematical Logic

Further Editions

This book also appears in the following edition:

• 2010: A Course in Mathematical Logic for Mathematicians, 2nd ed. (with Boris Zilber)

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

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

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