Admissible Sets and Structures

Description

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. Table of Contents Introduction Part A. The Basic Theory Chapter I. Admissible Set Theory 1. The Role of Urelements 2. The Axioms of KPU 3. Elementary Parts of Set Theory in KPU 4. Some Derivable Forms of Separation and Replacement 5. Adding Defined Symbols to KPU 6. Definition by Σ Recursion 7. The Collapsing Lemma 8. Persistent and Absolute Predicates 9. Additional Axioms Chapter II. Some Admissible Sets 1. The Definition of Admissible Set and Admissible Ordinal 2. Hereditarily Finite Sets 3. Sets of Hereditary Cardinality Less Than a Cardinal k 4. Inner Models: the Method of Interpretations 5. Constructible Sets with Urelements; HYPm Defined 6. Operations for Generating the Constructible Sets 7. First Order Definability and Substitutable Functions 8. The Truncation Lemma 9. The Levy Absoluteness Principle Chapter III. Countable Fragments of L∞ω 1. Formalizing Syntax and Semantics in KPU 2. Consistency Properties 3. M-Logic and the Omitting Types Theorem 4. A Weak Completeness Theorem for Countable Fragments 5. Completeness and Compactness for Countable Admissible Fragments 6. The Interpolation Theorem 7. Definable Well-Orderings 8. Another Look at Consistency Properties Chapter IV. Elementary Results on HYPm 1. On Set Existence 2. Defining Π11 and Σ11 Predicates 3. Π11 and Δ11 on Countable Structures 4. Perfect Set Results 5. Recursively Saturated Structures 6. Countable M-Admissible Ordinals 7. Representability in M-Logic PartB. The Absolute Theory Chapter V. The Recursion Theory of Σ1 Predicates on Admissible Sets 1. Satisfaction and Parametrization 2. The Second Recursion Theorem for KPU 3. Recursion Along Well-founded Relations 4. Recursively Listed Admissible Sets 5. Notation Systems and Projections of Recursion Theory 6. Ordinal Recursion Theory: Projectible and Recursively Inaccessible Ordinals 7. Ordinal Recursion Theory: Stability 8. Shoenfield's Absoluteness Lemma and the First Stable Ordinal Chapter VI. Inductive Definitions 1. Inductive Definitions as Monotonic Operators 2. Σ Inductive Definitions on Admissible Sets 3. First Order Positive Inductive Definitions and IHYPm 4. Coding IHFm on M 5. Inductive Relations on Structures with Pairing 6. Recursive Open Games Part C. Towards a General Theory Chapter VII. More about L∞ω 1. Some Definitions and Examples 2. A Weak Completeness Theorem for Arbitrary Fragments 3. Pinning Down Ordinals: the General Case 4. Indiscernibles and upward Lowenheim-Skolem Theorems 5. Partially Isomorphic Structures 6. Scott Sentences and their Approximations 7. Scott Sentences and Admissible Sets Chapter VIII. Strict Π11 Predicates and König Principles 1. The König Infinity Lemma 2. Strict Π11 Predicates: Preliminaries 3. König Principles on Countable Admissible Sets 4. König Principles K1 and K2 on Arbitrary Admissible Sets 5. König's Lemma and Nerode's Theorem: a Digression 6. Implicit Ordinals on Arbitrary Admissible Sets 7. Trees and Σ1 Compact Sets of Cofinality ω 8. Σ1 Compact Sets of Cofinality Greater than ω 9. Weakly Compact Cardinals Appendix. Nonstandard Compactness Arguments and the Admissible Cover 1. Compactness Arguments over Standard Models of Set Theory 2. The Admissible Cover and its Properties 3. An Interpretation of KPU in KP 4. Compactness Arguments over Nonstandard Models of Set Theory References Index of Notation Subject Index