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Set theory is a fundamental branch of mathematical logic that studies sets, which are collections of objects. It's a foundational system for mathematics, underpinning the formal structure of mathematical reasoning. Here's a comprehensive list of various branches and subfields within set theory:

1. Naive Set Theory¶

Basic Concepts and Operations on Sets
Venn Diagrams
Cardinality of Sets
Subsets, Power Sets
Union, Intersection, and Difference of Sets

2. Axiomatic Set Theory¶

Zermelo-Fraenkel Set Theory (ZF)
Axiom of Choice and its Equivalents (ZFC)
Von Neumann–Bernays–Gödel Set Theory (NBG)
Morse–Kelley Set Theory (MK)
Constructible Universe (L)

3. Ordinal and Cardinal Numbers¶

Ordinal Arithmetic
Cardinal Arithmetic
Aleph Numbers
Beth Numbers
Cofinality and Regular Cardinals

4. Combinatorial Set Theory¶

Infinite Combinatorics
Ramsey Theory
Partition Calculus
Singular Cardinals Hypothesis

5. Descriptive Set Theory¶

Polish Spaces
Borel Sets
Analytic Sets
Projective Sets
Perfect Set Property
Descriptive Complexity

6. Large Cardinals and Independence¶

Measurable Cardinals
Inaccessible Cardinals
Weakly and Strongly Compact Cardinals
Woodin Cardinals
Consistency and Independence Proofs

7. Forcing and Independence Proofs¶

Forcing Method
Suslin's Hypothesis
Martin's Axiom
Independence of the Continuum Hypothesis
Easton's Theorem

8. Model Theory in Set Theory¶

Inner Models
Outer Models
Ultraproducts and Ultrapowers
Constructibility and Core Models

9. Infinitary Combinatorics¶

Infinite Graphs
Trees and Kurepa's Hypothesis
Club and Stationary Sets
Diamond Principle

10. Generalized Set Theory¶

Fuzzy Set Theory
Rough Set Theory
Intuitionistic Set Theory

11. Set-Theoretic Topology¶

Topological Spaces from a Set-Theoretic Perspective
Martin's Axiom and Its Applications in Topology
Continuum Theory

12. Set Theory and Foundations of Mathematics¶

Role of Set Theory in the Foundations
Set-Theoretic Reductionism
Set-Theoretic Realism and Platonism

13. Transfinite Recursion and Induction¶

Recursive Definitions on Ordinals
Transfinite Sequences
Applications in Ordinal Analysis

14. Higher Infinite Sets¶

Hyperreal Numbers in Nonstandard Analysis
Large Cardinal Axioms
Extendible and Supercompact Cardinals

15. Applications of Set Theory¶

Applications in Algebra and Analysis
Set-Theoretic Methods in Logic
Set Theory in Computer Science

Set theory is not only a core part of mathematical logic but also an essential tool in various areas of mathematics. Its concepts and techniques are widely used in the study of algebra, topology, real analysis, and beyond. The study of set theory ranges from very concrete and applicable aspects to highly abstract and theoretical ones.

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