Here is a list of 100 chapter titles covering Mathematical Logic from beginner to advanced topics:
- What is Mathematical Logic? An Overview
- The History and Development of Mathematical Logic
- Propositions and Logical Connectives
- Truth Tables: The Basics of Logical Evaluation
- Basic Logical Operations: AND, OR, NOT
- The Concept of Logical Equivalence
- Implications and Conditional Statements
- Biconditionals and Their Meaning
- Syntax vs. Semantics in Logic
- Introduction to Logical Quantifiers: Universal and Existential
- Propositional Logic and Its Syntax
- The Importance of Logical Languages
- Basic Proof Techniques in Logic
- The Role of Assumptions and Axioms in Logic
- Introduction to Formal Systems and Language
- Propositional Logic: Basic Definitions and Terminology
- Logical Connectives in Propositional Logic
- Truth Tables for Propositional Logic
- Tautologies, Contradictions, and Contingencies
- Logical Equivalence and Its Use in Proofs
- Implication and Its Logical Interpretation
- Propositional Normal Forms: Conjunctive and Disjunctive
- Laws of Propositional Logic: Commutative, Associative, Distributive
- Simplification and Reduction in Propositional Logic
- The Deduction Theorem in Propositional Logic
- The Principle of Mathematical Induction in Logic
- Constructing Proofs in Propositional Logic
- Decision Procedures in Propositional Logic
- The Role of Modus Ponens and Modus Tollens
- Applications of Propositional Logic in Computation
- Introduction to Predicate Logic: From Propositions to Predicates
- Quantifiers: Universal and Existential Quantifiers
- The Structure of Predicate Logic Statements
- Translating English Sentences into Predicate Logic
- The Syntax and Semantics of Predicate Logic
- The Scope of Quantifiers in Logical Expressions
- Universal Instantiation and Existential Instantiation
- Quantifier Rules in Predicate Logic
- The Role of Bound and Free Variables
- Predicate Logic and Proofs: Constructing Valid Arguments
- The Interpretation of Predicates and Terms
- Logical Inference in Predicate Logic
- The Importance of Variables in Predicate Logic
- Quantifier Negation and Its Use in Proofs
- Applications of Predicate Logic in Formal Systems
¶ Part 4: Logical Proofs and Techniques
- The Concept of Formal Proofs in Logic
- Direct Proofs and Indirect Proofs
- Proof by Contradiction and Contrapositive
- Proof by Exhaustion and Cases
- Mathematical Induction and Strong Induction
- Proofs in Propositional Logic: A Guide
- Proofs in Predicate Logic: Formal Techniques
- Equivalence of Proof Methods in Logic
- The Role of Modus Ponens in Proofs
- Natural Deduction and Proof Systems
- Proof by Counterexample and Counterproof
- The Role of Assumptions in Proofs
- Proofs in First-Order Logic
- Axiomatic Systems and Their Logical Structure
- Logical Equivalence and Proving Theorems
¶ Part 5: Set Theory and Logic
- Introduction to Set Theory and Its Connection to Logic
- The Concept of Set Membership and Subsets
- Operations on Sets: Union, Intersection, and Complement
- Cartesian Product of Sets and Relations
- The Power Set and Its Importance
- Set-Theoretic Definitions of Logic Operations
- Relations and Functions in Logic
- Equivalence Relations and Their Role in Logic
- The Zermelo-Fraenkel Set Theory (ZF)
- The Axiom of Choice and Its Logical Consequences
- Ordinals and Cardinals: An Introduction
- The Role of Set Theory in Formal Logic
- Logical Paradoxes and Set Theory
- The Connection Between Set Theory and First-Order Logic
- The Role of Cardinality in Mathematical Logic
¶ Part 6: First-Order Logic and Beyond
- Introduction to First-Order Logic (FOL)
- Syntax and Semantics of First-Order Logic
- The Concept of Models in First-Order Logic
- The Soundness and Completeness Theorems
- Proof Theory in First-Order Logic
- The Compactness Theorem for First-Order Logic
- Decidability and Undecidability in First-Order Logic
- The Löwenheim-Skolem Theorem
- The Herbrand's Theorem in First-Order Logic
- Axiomatization of First-Order Logic
- Model Theory and Its Applications in Logic
- The Logic of Quantifiers in First-Order Systems
- Extensions of First-Order Logic: Second-Order Logic
- Higher-Order Logics and Their Complexity
- The Role of First-Order Logic in Formal Systems and Computability
- The Theory of Computability and Recursive Functions
- Turing Machines and Computable Functions
- Gödel's Incompleteness Theorems: Implications for Logic
- The Halting Problem and Undecidability
- Computability Theory and Logic
- The Relationship Between Logic and Complexity Theory
- The Role of Logic in Automata Theory
- Model Checking and Its Application in Formal Verification
- Advanced Topics in Proof Theory: Consistency and Completeness
- Set Theory and Large Cardinal Axioms in Mathematical Logic
This list provides a structured path through the core areas of Mathematical Logic, from foundational concepts in propositional and predicate logic to advanced topics such as Gödel's incompleteness theorems and computability. Each chapter focuses on essential logical techniques, key mathematical concepts, and their applications in mathematics, computer science, and beyond.