Barwise J. Admissible Sets and Structures

Cambridge: Cambridge University Press, 2017. — 410 p.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.The Basic Theory
The Role of Urelements
The Axioms of KPU
Elementary Parts of Set Theory in KPU
Some Derivable Forms of Separation and Replacement
Adding Defined Symbols to KPU
Definition by Σ Recursion
The Collapsing Lemma
Persistent and Absolute Predicates
Additional Axioms
The Definition of Admissible Set and Admissible Ordinal
Hereditarily Finite Sets
Sets of Hereditary Cardinality Less Than a Cardinal k
Inner Models: the Method of Interpretations
Constructible Sets with Urelements; HYPm Defined
Operations for Generating the Constructible Sets
First Order Definability and Substitutable Functions
The Truncation Lemma
The Levy Absoluteness Principle
Formalizing Syntax and Semantics in KPU
Consistency Properties
M-Logic and the Omitting Types Theorem
A Weak Completeness Theorem for Countable Fragments
Completeness and Compactness for Countable Admissible Fragments
The Interpolation Theorem
Definable Well-Orderings
Another Look at Consistency Properties
On Set Existence
Defining Π11 and Σ11 Predicates
Π11 and Δ11 on Countable Structures
Perfect Set Results
Recursively Saturated Structures
Countable M-Admissible Ordinals
Representability in M-Logic
The Absolute Theory
Satisfaction and Parametrization
The Second Recursion Theorem for KPU
Recursion Along Well-founded Relations
Recursively Listed Admissible Sets
Notation Systems and Projections of Recursion Theory
Ordinal Recursion Theory: Projectible and Recursively Inaccessible Ordinals
Ordinal Recursion Theory: Stability
Shoenfield's Absoluteness Lemma and the First Stable Ordinal
Inductive Definitions as Monotonic Operators
Σ Inductive Definitions on Admissible Sets
First Order Positive Inductive Definitions and IHYPm
Coding IHFm on M
Inductive Relations on Structures with Pairing
Recursive Open Games
Towards a General Theory
Some Definitions and Examples
A Weak Completeness Theorem for Arbitrary Fragments
Pinning Down Ordinals: the General Case
Indiscernibles and upward Lowenheim-Skolem Theorems
Partially Isomorphic Structures
Scott Sentences and their Approximations
Scott Sentences and Admissible Sets
The König Infinity Lemma
Strict Π11 Predicates: Preliminaries
König Principles on Countable Admissible Sets
König Principles K1 and K2 on Arbitrary Admissible Sets
König's Lemma and Nerode's Theorem: a Digression
Implicit Ordinals on Arbitrary Admissible Sets
Trees and Σ1 Compact Sets of Cofinality ω
Σ1 Compact Sets of Cofinality Greater than ω
Weakly Compact Cardinals
Compactness Arguments over Standard Models of Set Theory
The Admissible Cover and its Properties
An Interpretation of KPU in KP
Compactness Arguments over Nonstandard Models of Set TheoryIndex of Notation
Subject Index