Abstract
A rigorous introduction to mathematical logic covering propositional logic, first-order logic, and the fundamental metatheorems. The book provides a thorough treatment of syntax, semantics, completeness, and compactness for both propositional and first-order logic. It includes detailed proofs of the completeness theorem using Henkin’s method, the Löwenheim-Skolem theorem, and Gödel’s incompleteness theorems. The text is known for its mathematical precision and careful attention to foundational issues.
Outline
Chapter 1: Sentential Logic
- Syntax and semantics of propositional logic
- Truth tables and tautologies
- Logical consequence and equivalence
- Adequate sets of logical connectives
Chapter 2: First-Order Logic
- Predicate calculus syntax
- Structures and models
- Quantifiers and variable binding
- Satisfiability and validity
Chapter 3: Proof Systems
- Axiomatic systems for first-order logic
- Natural deduction
- The deduction theorem
- Soundness of proof systems
Chapter 4: The Completeness Theorem
- Henkin’s construction
- Maximal consistent sets
- Canonical models
- Gödel’s completeness theorem for first-order logic
Chapter 5: Applications of Completeness
- Compactness theorem
- Löwenheim-Skolem theorems
- Elementary equivalence
- Nonstandard models
Chapter 6: Incompleteness and Undecidability
- Arithmetization of syntax
- Gödel’s first incompleteness theorem
- Gödel’s second incompleteness theorem
- Undecidability of first-order logic