Mathematical Logic and Formalized Theories

I The Sentential Logic 1.1. Introduction 1.2. Sentential Connectives 1.3. The Sentential Logic P. Symbols and Formulas 1.4. Tautologies 1.5. Axiom Schemata of P. Rules of Inference and Theorems 1.6. Metamathematical Properties of PII The First-Order Predicate Logic: I 2.1. The First-Order Predicate Logic F1. Symbols, Quantifiers and Formulas 2.2. Interpretations. Truth and Validity 2.3. Axiom Schemata of F1. Rules of Inference and Theorems. Consistency of F1 2.4. The Deduction TheoremIII The First-Order Predicate Logic: II 3.1. Elementary Theories 3.2. Completeness Theorems 3.3. Further Corollaries. Decision Problem 3.4. The First-Order Predicate Logic with Identity 3.5. The First-Order Predicate Logic with Identity and Operation SymbolsIV The Second-Order Predicate Logic. Theory of Definition 4.1. Introduction 4.2. The Second-Order Predicate Logic F2 4.3. Second-Order Theories 4.4. Theory of DefinitionV The Natural Numbers 5.1. Introduction 5.2. Elementary Arithmetic: The Theory N 5.3. The Metamathematics of N 5.4. Second-Order Arithmetic: The Theory N2 5.5. The Metamathematics of N2VI The Real Numbers 6.1. The Theory R 6.2. The Metamathematics of R and of Elementary Algebra 6.3. Second-Order Real Number Theory: The Theory R2 6.4. The Metamathematics of R2VII Axiomatic Set Theory 7.1. Paradoxes 7.2. The Zermelo-Fraenkel Axioms 7.3. The Axiom of Choice 7.4. The Metamathematics of ZF 7.5. Strengthened Forms of ZFVIII Incompleteness. Undecidability 8.1. Introduction 8.2. Recursive Functions and Relations. Representability 8.3. Arithmetization 8.4. Gödel's First Incompleteness Theorem 8.5. Gödel's Second Incompleteness Theorem 8.6. Tarski's Theorem 8.7. Decision Problem. Church's Thesis. Recursively Enumerable Sets 8.8. UndecidabilityAuthor IndexSubject Index