Large Infinitary Languages (eBook)

Front Cover 1
Large Infinitary Languages: Model Theory 4
Copyright Page 5
Preface 8
Contents 11
Chapter 0. Preliminaries 18
§ 1. Set-theoretical background 18
§ 2. “Constructibly” defined inaccessible cardinals 28
§ 3. Filters and ultrafilters ultraproducts
§ 4. Measurable cardinals 49
§ 5. Partition calculus 59
Chapter 1. The Basic Notions of Infinitary Languages 76
§ 1. Elementary syntax and semantics of the language Lk. 76
§ 2. Examples 89
§ 3. Classification of formulas, model classes and maps 104
Chapter 2. Some Results From The Model Theory of L.. And L.1. 119
§ 1. Background on types and saturated structures 119
§ 2. Models generated by indiscernibles 130
§ 3. Some results from the model theory of L.1. 145
Chapter 3. General Results 175
§ 1. Incompactness of accessible cardinals 175
§ 2. Incompactness results for inaccessible cardinals 182
§ 3. Larger cardinals 189
§ 4.The downward Löwenheim-Skolem theorem 222
§ 5.The upward Lowenheim-Skolem theorem and generalizations Hanf numbers
Chapter 4. Infinitary Languages With Finite Quantifiers 247
§ 1. The Hanf number of L... and Morley numbers 247
§ 2. Lower bounds for Hanf and Morley numbers 251
§ 3. Omitting types upper bounds for Morley numbers
§ 4. Applications: non-definability of well-orderings in L., 273
§ 4. Applications: non-definability of well-orderings in Lk. 273
§ 5. Further Hanf number computations 280
Chapter 5. Languages with Infinite Quantifiers 299
§ 1. The Hanf number of languages with infinite quantifiers 299
§ 2. The language L.. for . strongly inaccessible 308
§ 3. Extension of partial isomorphisms 325
§ 4. Applications of the back and forth method 390
Appendix A. Induction For Well-Founded Relations and The Shepherdson–Mostowski Theorem 420
Appendix B.an L.1.1 Formula With no Prenex Form 424
I. Some absoluteness notions and results 424
II. Construction of the formula 428
Appendix C. Axiomatizability, Completeness and Definability Results 434
I. A review of (positive) completeness and definability results for infinitary logic 435
II. Scott’s undefinability theorem 440
Appendix D. Results on The Constructible Universe 448
Appendix E. Real-valued Measurable Cardinals and the Tree Property 452
References 455
Index 462
List of Symbols 474