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Mathematical logic is a subfield of mathematics exploring applications of formal logic to mathematics. It deals with mathematical properties of formal logical systems. Here's a comprehensive list of the various branches and topics within mathematical logic:

1. Set Theory ¶

Axiomatic Set Theory (ZFC, Von Neumann-Bernays-Gödel, etc.)
Cardinal and Ordinal Numbers
Infinite Combinatorics
Large Cardinals and Independence Results
Constructible Universe (L) and Inner Models

2. Model theory ¶

Structures and Interpretations
Elementary Equivalence and Embeddings
Quantifier Elimination and Model Completeness
Types and Omitting Types Theorem
Stability Theory and Classification Theory
Nonstandard Models and Ultraproducts

3. Proof Theory¶

Structural Proof Theory
Natural Deduction and Sequent Calculus
Cut-Elimination and Normalization
Arithmetic Hierarchy and Reverse Mathematics
Transfinite Induction and Ordinal Analysis
Constructive Logic and Type Theory

4. Computability Theory (Recursion Theory)¶

Turing Machines and Church-Turing Thesis
Recursive and Recursively Enumerable Sets
Degrees of Unsolvability
Algorithmic Randomness
Computable Functions and Effective Operations
Complexity Classes and Reducibility

5. Formal Arithmetic and Number Theory¶

Peano Arithmetic and Gödel's Incompleteness Theorems
Presburger Arithmetic and Diophantine Equations
Nonstandard Models of Arithmetic
Finite Model Theory
Arithmetic Complexity and Cryptography

6. Logic and Foundations of Mathematics¶

Philosophy of Mathematics
Foundations of Set Theory
Axiomatic Systems and Consistency Proofs
Constructivism and Intuitionism
Platonism vs. Formalism

7. Higher-Order Logic¶

Second-Order and Higher-Order Logics
Type Theories and Lambda Calculus
Higher-Order Arithmetic and Set Theory

8. Descriptive Set Theory¶

Polish Spaces and Borel Sets
Analytic and Projective Sets
Determinacy Axioms
Descriptive Complexity

9. Non-Classical Logics¶

Modal Logic and Possible World Semantics
Temporal Logic and Temporal Structures
Intuitionistic Logic and Heyting Algebras
Fuzzy Logic and Many-Valued Logics

10. Categorical Logic¶

Functorial Semantics
Topos Theory
Cartesian Closed Categories and Locally Cartesian Closed Categories
Internal Languages of Categories

11. Algebraic Logic¶

Boolean Algebras and Stone Spaces
Cylindric Algebras and Relation Algebras
Algebraic Semantics of Logical Systems

12. Applications of Logic to Computer Science¶

Formal Verification and Model Checking
Automated Theorem Proving
Logic Programming and Lambda Calculus
Database Theory and Query Languages

13. Quantum Logic¶

Logical Foundations of Quantum Mechanics
Quantum Computation and Information

14. Decision Theory and Game Theory¶

Logical Models of Decision Making
Game Semantics

Mathematical logic is a deep and broad field, bridging mathematics and philosophy, and it has profound implications in computer science, especially in areas like algorithms, computation, and information theory. It continues to evolve, addressing both foundational questions about mathematics and practical issues in computing and information sciences.

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