"In particular, Georg Cantor is commonly considered the founder of set theory."
Set theory is the branch of mathematics that deals with sets and their properties. It is used in various areas of mathematics and logic.
Propositional logic: The study of logical relations between propositions.
Predicate logic: The study of logical relations between predicates.
Quantifiers: The terms that express the quantity of a statement.
Functions and Relations: The study of sets and their relation in a mathematical context.
Sets: A well-defined collection of objects.
Operations on sets: Union, intersection, complement, and other operations.
Cardinality and countability: The size of sets.
Axiomatic set theory: The foundational branch of set theory based on axioms.
Zermelo-Fraenkel set theory: One of the most widely used versions of axiomatic set theory.
Axiom of choice: A controversial axiom in set theory which allows the construction of sets without specifying their elements.
Transfinite induction: A method of proving statements about infinite sets.
Continuum hypothesis: A statement that relates the size of sets to the concept of infinity.
Model theory: The study of mathematical structures and their interpretations.
Topology: The study of the properties of spaces that are preserved under continuous transformations.
Mathematical logic: The study of mathematical reasoning and its formalization.
Zermelo-Fraenkel Set Theory: Zermelo-Fraenkel Set Theory is a foundational theory in mathematics that provides a formal framework for describing sets, their properties, and the operations that can be performed on them.
Von Neumann-Bernays-Gödel Set Theory: Von Neumann-Bernays-Gödel Set Theory (NBG) is an extension of set theory that allows for a foundational exploration of sets and classes, incorporating a class comprehension axiom and providing a framework for treating proper classes as well as sets.
Quine Set Theory: Quine Set Theory is a mathematical framework proposed by Willard Van Orman Quine that seeks to provide a foundation for mathematics by treating sets as the only fundamental objects of the theory.
Morse-Kelley Set Theory: Morse-Kelley Set Theory is an extension of Zermelo-Fraenkel Set Theory that includes proper classes and strengthens the axioms for set existence and set comprehension.
New Foundation Set Theory: New Foundation Set Theory is a mathematical approach that addresses the philosophical problems of traditional set theories, focusing on the constructive and predicative nature of sets.
Tarski-Grothendieck Set Theory: Tarski-Grothendieck Set Theory is a foundational theory that aims to provide a rigorous and axiomatic framework for set theory, addressing questions related to the existence and structure of sets while extending Zermelo-Fraenkel Set Theory with stronger axioms.
Aczel's Constructive Set Theory (CZF): Aczel's Constructive Set Theory (CZF) is a theory that aims to construct sets in a constructive and intuitionistic framework, accommodating both classical and anti-classical notions of sets.
IZF Set Theory: IZF Set Theory, based on intuitionistic logic and the Zermelo-Fraenkel axioms, aims to provide a foundational system for mathematics while rejecting the law of excluded middle, focusing on constructivity and intuition.
NFU Set Theory: NFU Set Theory is a version of set theory that resolves the paradoxes of set theory by allowing sets to be members of other sets while avoiding Russell's paradox.
Non-well-founded Set Theory: Non-well-founded Set Theory is a branch of set theory that allows for the existence of circular or self-referential sets, challenging the traditional notion of membership.
Fuzzy Set Theory: Fuzzy Set Theory is a mathematical framework that allows for the representation and manipulation of degrees of membership and uncertainty in sets, challenging the traditional binary logic of classical set theory.
Multisets: Multisets, also known as bags, are mathematical structures that allow for the repetition of elements, challenging the traditional notion of sets where each element appears only once.
Rough Set Theory: Rough set theory is a mathematical approach that deals with the approximate classification and analysis of imprecise or uncertain data.
Fuzzy-Rough Set Theory: Fuzzy-Rough Set Theory combines fuzzy set theory and rough set theory to handle vagueness and uncertainty in knowledge representation and decision-making processes.
Quantum Set Theory: Quantum Set Theory is a philosophical and mathematical framework that explores the nature of sets using the principles and concepts of quantum mechanics, allowing for more flexible and probabilistic descriptions of sets.
Topos Theory: Topos Theory is a branch of Mathematics that explores the notion of structure and logical reasoning from a philosophical perspective, providing a framework for studying categories and their internal logic.
Modal Logic: Modal Logic is a branch of logic that deals with the study of necessity, possibility, and other modal concepts.
Homotopy Type Theory: Homotopy Type Theory is a branch of mathematics that explores the connections between logic, type theory, and algebraic topology, aiming to apply topological concepts to enhance our understanding of types and proofs.
Algebraic Set Theory: Algebraic Set Theory is a branch of set theory that explores the formal relationship between sets and algebraic structures, such as lattices and boolean algebras.
Category Theory: Category Theory is a branch of mathematics that abstracts away the specific details of mathematical structures to focus on the relationships and interactions between them.
"The non-formalized systems investigated during this early stage go under the name of naive set theory."
"After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox)..."
"Various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied."
"Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice."
"Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science (such as in the theory of relational algebra), philosophy and formal semantics."
"Its foundational appeal, together with its paradoxes, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for logicians and philosophers of mathematics."
"The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s."
"Set theory, as a branch of mathematics, is mostly concerned with those (sets) that are relevant to mathematics as a whole."
"Sets can be informally described as collections of objects."
"Set theory is commonly employed as a foundational system for the whole of mathematics..."
"Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals."
"In particular, Georg Cantor is commonly considered the founder of set theory."
"After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century..."
"Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science (such as in the theory of relational algebra)..."
"Various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied."
"Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity..."
"Its foundational appeal, together with its paradoxes, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for logicians and philosophers of mathematics."
"Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice."
"Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals."