Building Models by Games

This volume presents research by algebraists and model theorists in accessible form for advanced undergraduates or beginning graduate students studying algebra, logic, or model theory. It introduces a general method for building infinite mathematical structures and surveys applications in algebra and model theory. Starting with an overview of basic model theory, the text examines a variety of algebraic applications, including completeness for Magidor-Malitz quantifiers, Shelah's recent and sophisticated omitting types theorem for L(Q), and applications to Boolean algebras and models of arithmetic. More than 160 exercises range from elementary drills to research-related items. 1985 ed.

Introduction

v

Preliminaries

1

(16)

Pictures

1

(4)

Model theory

5

(12)

References

16

(1)

Games and Forcing

17

(18)

A way of building models

17

(6)

Games

23

(4)

Forcing

27

(8)

References

34

(1)

Existential Closure

35

(48)

Adjunction of elements

36

(11)

Existentially closed models

47

(12)

E.c. groups

59

(13)

Robinson forcing

72

(11)

References

80

(3)

Chaos or Regimentation

83

(49)

Mass production

84

(11)

Atomic models

95

(10)

Finite-generic models

105

(6)

E.c. nilpotent groups of class 2

111

(21)

References

130

(2)

Classical Languages

132

(38)

Classical omitting types

133

(11)

Set-theoretical interruption: unbounded subsets

144

(8)

Saturation

152

(18)

References

167

(3)

Proper Extensions

170

(41)

Largeness properties

171

(14)

Definable ultrapowers

185

(10)

Uncountable boolean algebras

195

(16)

References

209

(2)

Generalised Quantifiers

211

(39)

L(Q)

212

(11)

Omitting types in L(Q)

223

(12)

Magidor-Malitz quantifiers

235

(15)

References

248

(2)

L(Q) in Higher Cardinalities

250

(25)

There is a problem

251

(9)

Completeness and omitting types

260

(15)

References

273

(2)

List of types of forcing

275

(1)

List of open questions

276

(6)

Bibliography

282

(21)

Index

303

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Building Models by Games

Summary

Table of Contents

Supplemental Materials