Mathematical Logic

Mathematical logic is the subfield of mathematics that applies formal, symbolic methods to the study of reasoning and proof itself. It establishes rigorous frameworks, such as propositional and predicate logic, to analyze the validity of arguments, the relationship between mathematical theories and the structures they describe (model theory), and the inherent limits of what can be proven or computed (as famously explored by Gödel's incompleteness theorems). By treating logic as a mathematical object, this discipline provides the essential tools for investigating the very foundations of mathematics, ensuring its consistency and defining the boundaries of formal systems.

1. Introduction to Mathematical Logic
   1. The Nature of Formal Reasoning
      1. Definition of Logic
      2. Formal vs. Informal Reasoning
      3. Role of Symbolic Representation
      4. Deductive vs. Inductive Reasoning
      5. Mathematical vs. Philosophical Logic
   2. Historical Development
      1. Ancient Logic
         1. Aristotle's Syllogistic Logic
         2. Stoic Logic
      2. Medieval Logic
         1. Scholastic Contributions
         2. William of Ockham
      3. Early Modern Logic
         1. Leibniz and the Calculus Ratiocinator
         2. Port-Royal Logic
      4. Foundations of Modern Logic
         1. Boole and Algebraic Logic
         2. De Morgan's Laws
         3. Frege's Begriffsschrift
         4. Peano's Notation
         5. Russell's Paradox
         6. Principia Mathematica
         7. The Hilbert Program
         8. Gödel's Impact
   3. Distinction between Syntax and Semantics
      1. Definition of Syntax
      2. Definition of Semantics
      3. Examples Illustrating the Distinction
      4. Importance in Logical Systems
   4. Metalogic and Object Language
      1. Object Language vs. Metalanguage
      2. Use-Mention Distinction
      3. Expressing Properties of Formal Systems
      4. Examples of Metalogical Statements
   5. Basic Concepts
      1. Formal Languages
      2. Interpretations
      3. Logical Systems
      4. Proof and Truth