Barwise: infinitary logic and admissible sets

0. Introduction 1 1. Background on infinitary logic 2 1.1. Expressive power of Lω1ω 2 1.2. The back-and-forth construction 3 1.3. The Scott isomorphism theorem 4 1.4. ω-logic 7 1.5. Familiar theorems 8 1.6. Failure of compactness 9 2. Background on admissible sets 10 2.1. ∆0 formulas and Σ-formulas in set theory 10 2.2. Axioms of KP 11 2.3. Examples of admissible sets 12 2.4. The admissible set L(ωCK 1 ) 12 3. Admissible fragments 14 3.1. Completeness and compactness 14 3.2. Computable structures via Barwise compactness 15 3.3. Other applications of Barwise compactness 16 4. Admissible sets over M 19 4.1. KP with urelements 19 4.2. Truncation lemma 20 4.3. Admissible sets above M 21 4.4. Inductive definitions 21 5. Saturation properties 23 5.1. Computable saturation 23 5.2. ΣA-saturation 26 6. Conclusion 27