"In particular, Georg Cantor is commonly considered the founder of set theory."
A branch of mathematics that deals with the study of sets, which are collections of distinct objects.
Basic Concepts of Set Theory: Introduces the concept of sets and elements, subsets, and set operations such as union, intersection, and complement.
Axiomatic Set Theory: Discusses the formal axioms underlying set theory, as well as the Zermelo–Fraenkel (ZF) axioms.
Ordinal and Cardinal Numbers: Discusses the different types of numbers used in set theory: ordinal and cardinal numbers. Includes concepts like transfinite induction and arithmetic operations on transfinite numbers.
Relations and Functions: Relates the concepts of relations and functions to sets, including equivalence relations, partial orderings, and well-orderings.
Set Theory and Logic: Explores the relationship between set theory and logic, with a focus on constructions such as the Russell paradox and the axiom of choice.
Model Theory: Discusses the different models or interpretations of set theory, as well as their relation to each other.
Applications of Set Theory: Examines the use of set theory in other fields, including computer science, topology, and mathematical logic.
"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."