In this text for first-year graduate students, the authors provide an elementary exposition of some of the basic concepts of model theory--focusing particularly on the ultraproduct construction and the areas in which it is most useful. The book, which assumes only that its readers are acquainted with the rudiments of set theory, starts by developing the notions of Boolean algebra, propositional calculus, and predicate calculus.
Model theory proper begins in the fourth chapter, followed by an introduction to ultraproduct construction, which includes a detailed look at its theoretic properties. An overview of elementary equivalence provides algebraic descriptions of the elementary classes. Discussions of completeness follow, along with surveys of the work of Jónsson and of Morley and Vaught on homogeneous universal models, and the results of Keisler in connection with the notion of a saturated structure. Additional topics include classical results of Gödel and Skolem, and extensions of classical first-order logic in terms of generalized quantifiers and infinitary languages. Numerous exercises appear throughout the text.
Author(s): Bell J.L., Slomson A.B.
Edition: 3rd printing
Publisher: North-Holland
Year: 1969
Language: English
Pages: 322
City: Amsterdam
J.L.Bell, A.B.Slomson “Models and Ultraproducts: An Introduction” (1969) ......Page 1
Table of contents ......Page 5
Preface to the second printing ......Page 8
Introduction ......Page 9
Ch. 0. Prerequisites ......Page 12
1. Lattices ......Page 15
2. Filters ......Page 20
3. Ultrafilters ......Page 22
4. Homomorphisms and quotient algebras ......Page 25
5. Filters in topology ......Page 30
6. The Stone representation theorem ......Page 31
7. Atomless Boolean algebras and Cantor spaces ......Page 35
8. Historical and bibliographical remarks ......Page 39
1. The system SC ......Page 40
2. The Lindenbaum algebra ......Page 48
3. The completeness of propositional calculus ......Page 50
4. The compactness of propositional calculus ......Page 52
5. Historical and bibliographical remarks ......Page 57
1. The language L ......Page 58
2. Interpretations of L ......Page 62
3. An axiom system for predicate calculus ......Page 65
4. The Lindenbaum algebra of PC ......Page 69
5. The completeness of PC ......Page 70
6. Uncountable languages ......Page 75
7. Constants ......Page 76
8. Function symbols ......Page 77
9. Historical and bibliographical remarks ......Page 79
1. The basic notions ......Page 80
2. Unions of chains ......Page 87
3. Löwenheim-Skolem theorems ......Page 88
4. Hanf numbers ......Page 92
5. Historical and bibliographical remarks ......Page 94
1. The ultraproduct construction ......Page 95
2. Łoš’s theorem ......Page 97
3. Finite axiomatizability ......Page 100
4. The compactness theorem ......Page 110
5. The completeness theorem, the axiom of choice and the ultrafilter theorem ......Page 111
6. Historical and bibliographical remarks ......Page 114
1. More about ultrafilters ......Page 115
2. Set theoretic properties of ultraproducts ......Page 130
3. The cardinality of ultraproducts ......Page 133
4. Well-ordered ultraproducts ......Page 141
5. The Rabin-Keisler theorem ......Page 144
6. Historical and bibliographical remarks ......Page 147
1. The various sorts of elementary classes ......Page 148
2. Keisler’s ultrapower theorem ......Page 153
3. Characterizing elementary classes ......Page 159
4. Craig’s interpolation lemma ......Page 161
5. EC as a Boolean algebra ......Page 165
6. Historical and bibliographical remarks ......Page 167
1. Two basic lemmas ......Page 169
2. Ultralimits ......Page 173
3. Elementary classes and ultralimits ......Page 177
4. Ultralimits and Craig’s interpolation lemma ......Page 178
5. Historical and bibliographical remarks ......Page 182
1. Completeness ......Page 183
2. Model completeness ......Page 191
3. Universal and existential sentences ......Page 193
4. A test for model completeness ......Page 196
5. Applications of Robinson’s test ......Page 199
6. Historical and bibliographical remarks ......Page 208
1. The main definitions ......Page 209
2. The uniqueness of M-homogeneous, M-universal structures ......Page 212
3. The existence of M-homogeneous, M-universal structures ......Page 217
4. Full expansions ......Page 221
5. Historical and bibliographical remarks ......Page 224
1. Saturated models ......Page 225
2. Ultraproducts and saturated structures ......Page 230
3. More about saturated structures ......Page 232
4. Special relational structures ......Page 237
5. Historical and bibliographical remarks ......Page 240
1. Godel’s completeness proof ......Page 241
2. A non-standard model of arithmetic ......Page 244
3. MacDowell and Specker’s theorem ......Page 248
4. Cardinal-like orderings ......Page 252
5. Two-cardinal theorems ......Page 253
6. Vaught’s two-cardinal theorem ......Page 257
7. Chang’s two-cardinal theorem ......Page 261
8. Historical and bibliographical remarks ......Page 267
Ch. 13. Generalized quantifiers ......Page 268
1. Generalized quantifiers ......Page 269
2. The quantifiers Qα ......Page 270
3. The Chang quantifier ......Page 275
4. Fuhrken’s reduction technique ......Page 278
5. Löwenheim-Skolem and compactness theorems for LQ ......Page 283
6. Completeness theorem for LQ ......Page 287
7. Historical and bibliographical remarks ......Page 293
1. The language Lαβ ......Page 294
2. The compactness property; incompactness results for accessible cardinals ......Page 297
3. Incompactness results for inaccessible cardinals ......Page 301
4. Measurable cardinals ......Page 303
5. Keisler’s ultrapower proof ......Page 306
6. Measurable cardinals and the axiom of constructibility ......Page 308
7. Historical and bibliographical remarks ......Page 313
Some suggestions for further reading ......Page 316
Bibliography ......Page 317
Index of named theorems ......Page 325
General index ......Page 327
