Constructibility (Perspectives in Mathematical Logic)

This Book Is Intended To Give A Fairly Comprehensive Account Of The Theory Of Constructible Sets At An Advanced Level. The Intended Reader Is A Graduate Mathe Matician With Some Knowledge Of Mathematical Logic. In Particular, We Assume Familiarity With The Notions Of Formal Languages, Axiomatic Theories In Formal Languages, Logical Deductions In Such Theories, And The Interpretation Oflanguages In Structures. Practically Any Introductory Text On Mathematical Logic Will Supply The Necessary Material. We Also Assume Some Familiarity With Zermelo-fraenkel Set Theory Up To The Development Or Ordinal And Cardinal Numbers. Any Number Of Texts Would Suffice Here, For Instance Devlin (1979) Or Levy (1979). The Book Is Not Intended To Provide A Complete Coverage Of The Many And Diverse Applications Of The Methods Of Constructibility Theory, Rather The Theory Itself. Such Applications As Are Given Are There To Motivate And To Exemplify The Theory. The Book Is Divided Into Two Parts. Part A (elementary Theory) Deals With The Classical Definition Of The La-hierarchy Of Constructible Sets. With Some Prun Ing, This Part Could Be Used As The Basis Of A Graduate Course On Constructibility Theory. Part B (advanced Theory) Deals With The Fa-hierarchy And The Jensen Fine-structure Theory. Preface......Page 9 Contents......Page 11 A. Elementary Theory......Page 15 1. The Language of Set Theory......Page 17 2. The Zermelo-Fraenkel Axioms......Page 18 3. Elementary Theory of ZFC......Page 20 4. Ordinal Numbers......Page 26 5. Cardinal Numbers......Page 27 6. Closed Unbounded Sets......Page 34 7. The Collapsing Lemma......Page 36 8. Metamathematics of Set Theory......Page 38 9. The Language L_V......Page 45 10. Definability......Page 58 11. Kripke-Platek Set Theory. Admissible Sets......Page 62 II. The Constructible Universe......Page 70 1. Definition of the Constructible Universe......Page 71 2. The Constructible Hierachy. The Axiom of Constructibility......Page 77 3. The Axiom of Choice in L......Page 85 4. Constructibility and Relative Consistency Results......Page 91 5. The Condensation Lemma. GCH in L......Page 92 6. Σ_n Skolem Functions......Page 99 7. Admissable Ordinals......Page 109 Exercises......Page 114 1. The Souslin Problem. ω_1-Trees. Aronszajn Trees......Page 122 2. The Kurepa Hypothesis......Page 132 3. Some Related Combinatorial Principles......Page 136 Exercises......Page 147 1. κ^+ Trees......Page 151 2. κ^+ Souslin Trees......Page 152 3. κ^+ Kurepa Trees......Page 163 4. Fine Structure......Page 166 5. Square_κ......Page 172 Exercises......Page 181 1. Review of Large Cardinals......Page 183 2. L-Indiscernables and 0^#......Page 190 3. Definability of 0^#......Page 199 4. 0^# and Elementary Embeddings......Page 202 5. The Covering Lemma......Page 210 Exercises......Page 232 B. Advanced Theory......Page 237 1. Rudimentary Functions......Page 239 2. The Jensen Hierachy of Constructible Sets......Page 265 3. The Σ_1 Skolem Functions......Page 272 4. The Σ_n Projectum......Page 280 5. Standard Codes......Page 288 6. Global Square......Page 298 Exercises......Page 312 1. Weakly Compact Cardinals and κ-Souslin Trees......Page 317 2. Ineffable Cardinals and κ-Kurepa Trees......Page 326 3. Generalised Kurepa Families and Diamond^+_κ,λ......Page 333 Exercises......Page 343 1. Cardinal Transfer Theorems......Page 346 2. Gap-1 Morasses......Page 352 3. Gap-2 Cardinal Transfer Theorem......Page 373 4. Simplified Morasses......Page 383 Exercises......Page 392 1. Silver Machines......Page 397 2. Square......Page 405 Exercises......Page 422 Remarks and Historical Notes......Page 423 Bibliography......Page 429 Glossary of Notation......Page 433 Index......Page 436