Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. Admissible set theory is a major source of interaction between model theory, recursion theory and set theory, and plays an important role in definability theory. In this volume, the seventh publication in the Perspectives in Logic series, Jon Barwise presents the basic facts about admissible sets and admissible ordinals in a way that makes them accessible to logic students and specialists alike. It fills the artificial gap between model theory and recursion theory and covers everything the logician should know about admissible sets.
Author(s): Jon Barwise
Series: Perspectives in Logic 7
Publisher: Cambridge University Press
Year: 2017
Language: English
Pages: 410
Table of Contents......Page 12
Introduction......Page 16
Part A. The Basic Theory......Page 20
1. The Role of Urelements......Page 22
2. The Axioms of KPU......Page 24
3. Elementary Parts of Set Theory in KPU......Page 26
4. Some Derivable Forms of Separation and Replacement......Page 29
5. Adding Defined Symbols to KPU......Page 33
6. Definition by Σ Recursion......Page 39
7. The Collapsing Lemma......Page 45
8. Persistent and Absolute Predicates......Page 48
9. Additional Axioms......Page 53
1. The Definition of Admissible Set and Admissible Ordinal......Page 57
2. Hereditarily Finite Sets......Page 61
3. Sets of Hereditary Cardinality Less Than a Cardinal k......Page 67
4. Inner Models: the Method of Interpretations......Page 69
5. Constructible Sets with Urelements; HYPm Defined......Page 72
6. Operations for Generating the Constructible Sets......Page 77
7. First Order Definability and Substitutable Functions......Page 84
8. The Truncation Lemma......Page 87
9. The Levy Absoluteness Principle......Page 91
1. Formalizing Syntax and Semantics in KPU......Page 93
2. Consistency Properties......Page 99
3. M-Logic and the Omitting Types Theorem......Page 102
4. A Weak Completeness Theorem for Countable Fragments......Page 107
5. Completeness and Compactness for Countable Admissible Fragments......Page 110
6. The Interpolation Theorem......Page 118
7. Definable Well-Orderings......Page 120
8. Another Look at Consistency Properties......Page 124
1. On Set Existence......Page 128
2. Defining Π11 and Σ11 Predicates......Page 131
3. Π11 and Δ11 on Countable Structures......Page 137
4. Perfect Set Results......Page 142
5. Recursively Saturated Structures......Page 152
6. Countable M-Admissible Ordinals......Page 159
7. Representability in M-Logic......Page 161
PartB. The Absolute Theory......Page 166
1. Satisfaction and Parametrization......Page 168
2. The Second Recursion Theorem for KPU......Page 171
3. Recursion Along Well-founded Relations......Page 173
4. Recursively Listed Admissible Sets......Page 179
5. Notation Systems and Projections of Recursion Theory......Page 183
6. Ordinal Recursion Theory: Projectible and Recursively Inaccessible Ordinals......Page 188
7. Ordinal Recursion Theory: Stability......Page 192
8. Shoenfield's Absoluteness Lemma and the First Stable Ordinal......Page 204
1. Inductive Definitions as Monotonic Operators......Page 212
2. Σ Inductive Definitions on Admissible Sets......Page 220
3. First Order Positive Inductive Definitions and IHYPm......Page 226
4. Coding IHFm on M......Page 235
5. Inductive Relations on Structures with Pairing......Page 245
6. Recursive Open Games......Page 257
Part C. Towards a General Theory......Page 270
1. Some Definitions and Examples......Page 272
2. A Weak Completeness Theorem for Arbitrary Fragments......Page 277
3. Pinning Down Ordinals: the General Case......Page 285
4. Indiscernibles and upward Lowenheim-Skolem Theorems......Page 291
5. Partially Isomorphic Structures......Page 307
6. Scott Sentences and their Approximations......Page 312
7. Scott Sentences and Admissible Sets......Page 318
1. The König Infinity Lemma......Page 326
2. Strict Π11 Predicates: Preliminaries......Page 330
3. König Principles on Countable Admissible Sets......Page 336
4. König Principles K1 and K2 on Arbitrary Admissible Sets......Page 341
5. König's Lemma and Nerode's Theorem: a Digression......Page 349
6. Implicit Ordinals on Arbitrary Admissible Sets......Page 354
7. Trees and Σ1 Compact Sets of Cofinality ω......Page 358
8. Σ1 Compact Sets of Cofinality Greater than ω......Page 367
9. Weakly Compact Cardinals......Page 371
1. Compactness Arguments over Standard Models of Set Theory......Page 380
2. The Admissible Cover and its Properties......Page 381
3. An Interpretation of KPU in KP......Page 387
4. Compactness Arguments over Nonstandard Models of Set Theory......Page 393
References......Page 394
Index of Notation......Page 401
Subject Index......Page 403
