Mathematical Logic A Course with Exercises

Contents of Part I

Notes from the translator

Notes to the reader

Introduction

5. Recursion theory

6. Formalization of arithmetic, Godel's theorems

7. Set theory

8. Som

Contents of Part I

xiv

Notes from the translator

xvii

Notes to the reader

xviii

Introduction

1

(6)

Recursion theory

7

(56)

Primitive recursive functions and sets

8

(10)

Some initial definitions

8

(2)

Examples and closure properties

10

(5)

Coding of sequences

15

(3)

Recursive functions

18

(7)

Ackerman's function

18

(4)

The μ-operator and the partial recursive functions

22

(3)

Turing machines

25

(15)

Description of Turing machines

25

(2)

T-computable functions

27

(5)

T-computable partial functions are recursive

32

(4)

Universal Turing machines

36

(4)

Recursively enumerable sets

40

(14)

Recursive and recursively enumerable sets

40

(4)

The halting problem

44

(2)

The smn theorem

46

(4)

The fixed point theorems

50

(4)

Exercises for Chapter 5

54

(9)

Formalization of arithmetic, Godel's theorems

63

(45)

Peano's axioms

64

(9)

The axioms

64

(5)

The ordering on the integers

69

(4)

Representable functions

73

(5)

Arithmetization of syntax

78

(10)

The coding of formulas

78

(5)

The coding of proofs

83

(5)

Incompleteness and undecidability theorems

88

(12)

Undecidability of arithmetic and predicate calculus

88

(2)

Godel's incompleteness theorems

90

(10)

Exercises for Chapter 6

100

(8)

Set theory

108

(71)

The theories Z and ZF

109

(11)

The axioms

109

(7)

Ordered pairs, relations, and maps

116

(4)

Ordinal numbers and integers

120

(14)

Well-ordered sets

120

(2)

The ordinals

122

(7)

Operations on ordinal numbers

129

(4)

The integers

133

(1)

Inductive proofs and definitions

134

(6)

Induction

134

(4)

The axiom of choice

138

(2)

Cardinality

140

(17)

Cardinal equivalence classes

140

(3)

Operations on cardinal equivalence classes

143

(3)

The finite cardinals

146

(2)

Countable sets

148

(3)

The cardinal numbers

151

(6)

The axiom of foundation and the reflection schemes

157

(14)

The axiom of foundation

157

(3)

Some relative consistency results

160

(4)

Inaccessible cardinals

164

(2)

The reflection scheme

166

(5)

Exercises for Chapter 7

171

(8)

Some model theory

179

(146)

Elementary substructures and extensions

180

(6)

Elementary substructures

180

(3)

The Tarski-Vaught test

183

(3)

Construction of elementary extensions

186

(7)

Elementary maps

186

(1)

The method of diagrams

187

(6)

The interpolation and definability theorems

193

(6)

Reduced products and ultraproducts

199

(6)

Preservation theorems

205

(12)

Preservation by substructures

205

(3)

Preservation by unions of chains

208

(3)

Preservation by reduced products

211

(6)

ℵ0-categorical theories

217

(12)

The omitting types theorem

217

(5)

ℵ0-categorical structures

222

(7)

Exercises for Chapter 8

229

(11)

Solutions to the exercises of Part II

Chapter 5

240

(20)

Chapter 6

260

(17)

Chapter 7

277

(21)

Chapter 8

298

(27)

Bibliography

325

(2)

Index

327

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Mathematical Logic A Course with Exercises Part II: Recursion Theory, Gödel's Theorems, Set Theory, Model Theory

Summary

Table of Contents

Supplemental Materials

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Mathematical Logic A Course with Exercises Part II: Recursion Theory, Gödel's Theorems, Set Theory, Model Theory

Summary

Table of Contents

Supplemental Materials

Digital License