## Tags - Part of: - Related: - Includes: - Additional: ## Main resources - - <iframe src="https://en.wikipedia.org/wiki/Set_theory" allow="fullscreen" allowfullscreen="" style="height:100%;width:100%; aspect-ratio: 16 / 5; "></iframe> ## Written by AI (may include factually incorrect information) - 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.