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Set Theory

Set Theory is the mathematical study of sets, which are collections of objects. It is a fundamental theory that provides a basis for various other areas of mathematics.
Sub-categories:

Axiomatic Set Theory explores the foundations of set theory through various axiomatic systems, like Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC).

Cardinal Numbers deals with the size of sets, focusing on understanding and comparing the cardinality of infinite sets.

Ordinal Numbers extends cardinal numbers by describing the order type of well-ordered sets, assigning a unique ordinal to each.

Naive Set Theory covers the basic concepts of set theory developed in a non-rigorous way, without formal axioms.

Cantor's Theorem is a fundamental result established by Georg Cantor concerning the hierarchy of infinite cardinalities.

Continuum Hypothesis concerns the possible sizes of infinite sets, questioning the existence of set sizes between integers and real numbers.

Descriptive Set Theory studies sets of real numbers with topological and measure-theoretic properties, focusing on complexity.

Fuzzy Set Theory generalizes classical set theory to handle the concept of partial truth, with degrees of membership in a set.

Set Operations covers the fundamental operations on sets, such as union, intersection, difference, and symmetric difference.

Venn Diagrams are diagrams that show all possible logical relations between a finite collection of different sets.

Forcing is a technique for proving consistency results in set theory, often used to demonstrate independence of axioms.

Transfinite Induction is an extension of mathematical induction to well-ordered sets, often used in ordinal number theory.

Set-Theoretic Topology is concerned with the foundations of topology based on set-theory principles, studying topological spaces.

Zermelo-Fraenkel Set Theory provides a foundational system for set theory, used by mathematicians to provide a formal framework.

Model Theory of Set Theory applies the concepts of model theory to understand and investigate models of various set-theoretic axioms and frameworks.

Large Cardinals explore higher infinities in sets, extending the traditional hierarchy of cardinal numbers to reach stronger axioms.