"Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects."
Discusses the formal axioms underlying set theory, as well as the Zermelo–Fraenkel (ZF) axioms.
Sets: Definition and notation of sets, set operations such as union, intersection, and complement, subset relations, power sets, and the empty set.
Relations: Binary relations on sets, equivalence relations, partial orders, functions, and bijections.
Axioms of Set Theory: Zermelo-Fraenkel axioms, the axiom of foundation, and the axiom of choice.
Cardinality: Countable and uncountable sets, cardinal numbers, Cantor's diagonalization argument, and the continuum hypothesis.
Ordinals: Transfinite induction, ordinals, and the von Neumann ordinal representation.
Well-orderings and the Axiom of Choice: Zorn's lemma, the well-ordering theorem, and the independence of the Axiom of Choice.
Extensions of Set Theory: Large cardinals, inner models, and forcing.
Applications of Set Theory: Model theory, topology, mathematical logic, and number theory.
Paradoxes and unresolved problems: Russell's paradox, the liar paradox, the Banach-Tarski paradox, and the incompleteness theorem.
Zermelo-Fraenkel Set Theory (ZF): The most widely accepted and studied axiomatic set theory that defines a rigorous foundation for mathematical analysis. It includes the Axiom of Extension, the Axiom of Regularity, and the Axiom of Replacement.
Zermelo-Fraenkel-Choice Set Theory (ZFC): A set theory developed by adding the Axiom of Choice to ZF set theory. This theory establishes a more extended and flexible foundation for mathematical analysis, and most mathematicians use ZFC as their standard set theory.
Von Neumann-Bernays-Gödel Set Theory (NBG): A set theory developed by adding the Axiom of Class to Zermelo-Fraenkel Set Theory. This theory has more expressive power than ZF but is less widely used than ZFC.
Kripke-Platek Set Theory (KP): A weaker set theory than ZF developed to prove the consistency of mathematics. It includes a finiteness axiom that restricts the size of the sets.
Morse-Kelley Set Theory (MK): A set theory that is a slight modification of NBG, adding a stronger version of the Axiom of Infinity. MK is often used in category theory and to study geometric objects.
New Foundation Set Theory (NF): A set theory that aims to avoid Russell's paradox by adopting a new formulation for set membership. However, the lack of simplicity and elegance of this theory prevents it from being widely used.
Positive Set Theory (PST): A set theory in which sets are specified by their positive properties, rather than by their negations, making it possible to avoid some of the contradictions present in other set theories.
"The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s."
"Georg Cantor is commonly considered the founder of set theory."
"The non-formalized systems investigated during this early stage go under the name of naive set theory."
"The discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox)..."
"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."
"Set theory also provides the framework to develop a mathematical theory of infinity."
"It 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."
"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."
"Set theory is the branch of mathematical logic that studies sets..."
"The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s."
"Georg Cantor is commonly considered the founder of set theory."
"After the discovery of paradoxes within naive set theory...various axiomatic systems were proposed in the early twentieth century..."
"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."
"It has various applications in computer science, philosophy, and formal semantics."
"Its foundational appeal, together with its paradoxes, its implications for the concept of infinity, and its multiple applications..."
"Contemporary research into set theory covers a vast array of topics..."