"In particular, Georg Cantor is commonly considered the founder of set theory."
Explores the relationship between set theory and logic, with a focus on constructions such as the Russell paradox and the axiom of choice.
Sets and elements: Understanding the basic concepts of sets and the elements they contain, including the empty set and universal set.
Subsets and superset: The relationship between subsets and superset, including proper subsets.
Union and intersection: Understanding operations of union and intersection between sets.
Complement and difference: Understanding complement and difference operations between sets.
Cartesian product: Understanding the Cartesian product operation, and the resulting ordered pairs.
Relations and functions: Understanding concepts of relations and functions, including injections, surjections, and bijections.
Equivalence relations: Understanding the properties of equivalence relations, and their application in partitioning sets.
Partial order relations: Understanding partial order relations, including anti-symmetry, reflexivity, and transitivity.
Well-ordering: Understanding well-ordering properties of sets, and their application in proofs of mathematical induction.
Axiomatic set theory: Understanding set theory in the context of axiomatic systems, including the Zermelo-Fraenkel axioms and the axiom of choice.
Propositional logic: Understanding propositional logic, including logical operators (AND, OR, NOT), truth tables, and duality.
Predicate logic: Understanding predicate logic, including quantifiers (existential and universal), and inference rules (modus ponens, modus tollens).
Formal proofs: Understanding the process of creating formal proofs of statements and the use of axioms and inference rules.
Model theory: Understanding the relationship between models and theories in mathematical logic.
Gödel's incompleteness theorems: Understanding limitations of formal systems, including Gödel's incompleteness theorems.
Zermelo-Fraenkel Set Theory: This is a standard type of Set Theory, which is based on a set of axioms that define the properties of sets. It is named after the mathematicians Ernst Zermelo and Abraham Fraenkel.
Axiomatic Set Theory: This is a type of Set Theory, which is based on a set of axioms that define the basic properties of sets. Axiomatic Set Theory is also called Formal Set Theory.
Intuitionistic Set Theory: This type of Set Theory is based on the idea that all mathematical constructions must be constructible by a procedure of finite computations.
Fuzzy Set Theory: This type of Set Theory is used to describe the uncertainty and imprecision of the objects in a set.
Non-Well-Founded Set Theory: This type of Set Theory allows sets that refer to themselves.
Alternative Set Theory: This type of Set Theory is based on the idea that the basic objects of set theory are not sets but functions, categories, or classes.
Modal Logic: This type of Logic is used to express assertions about necessity, possibility, and contingency.
Propositional Logic: This type of Logic focuses on the logical relationships between propositions.
Predicate Logic (First-Order Logic): This type of Logic extends Propositional Logic to allow quantification over objects or entities.
Higher-Order Logic: This type of Logic extends Predicate Logic to allow quantification over functions, predicates, and sets.
Temporal Logic: This type of Logic is used to reason about the progression of time.
Modal Temporal Logic: This type of Logic combines Modal Logic and Temporal 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."