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72 relations: Actual infinity, Atomic formula, Axiom of choice, Axiom of countable choice, Axiom of extensionality, Axiom of infinity, Axiom of regularity, Axiom schema, Axiom schema of replacement, Axiom schema of specification, Binary relation, Burali-Forti paradox, Cantor's paradox, Cardinal number, Class (set theory), Consistency, Dana Scott, Definite description, Domain of a function, Domain of discourse, Empty set, Equinumerosity, Existential quantification, First-order logic, Foundations of mathematics, Free variables and bound variables, Function (mathematics), George Boolos, Hierarchy (mathematics), Identity (mathematics), If and only if, Infinity, Injective function, Isomorphism, List of set theory topics, Mathematician, Mathematics, Mereology, Model theory, Morley's categoricity theorem, Morse–Kelley set theory, Naive set theory, Natural number, New Foundations, Number, Ontology, Ordinal number, Paradox, Peano axioms, Philosopher, ..., Philosophy of mathematics, Range (mathematics), Richard Milton Martin, Russell's paradox, S (set theory), Scott's trick, Set (mathematics), Set theory, Set-builder notation, Singleton (mathematics), Successor function, Transitive relation, Type theory, Uniqueness quantification, Universal quantification, Urelement, Von Neumann universe, Von Neumann–Bernays–Gödel set theory, Well-order, Willard Van Orman Quine, Zermelo set theory, Zermelo–Fraenkel set theory. Expand index (22 more) »
Actual infinity
In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects.
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Atomic formula
In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas.
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Axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.
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Axiom of countable choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function.
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Axiom of extensionality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory.
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Axiom of infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory.
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Axiom of regularity
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads: The axiom implies that no set is an element of itself, and that there is no infinite sequence (an) such that ai+1 is an element of ai for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true.
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Axiom schema
In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom.
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Axiom schema of replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set.
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Axiom schema of specification
In many popular versions of axiomatic set theory the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema.
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Binary relation
In mathematics, a binary relation on a set A is a set of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2.
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Burali-Forti paradox
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction.
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Cantor's paradox
In set theory, Cantor's paradox is a statement derivable from the theorem that there is no greatest cardinal number, so that the collection of "infinite sizes" is itself infinite.
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Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.
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Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.
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Consistency
In classical deductive logic, a consistent theory is one that does not contain a contradiction.
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Dana Scott
Dana Stewart Scott (born October 11, 1932) is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California.
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Definite description
A definite description is a denoting phrase in the form of "the X" where X is a noun-phrase or a singular common noun.
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Domain of a function
In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.
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Domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.
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Empty set
In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.
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Equinumerosity
In mathematics, two sets or classes A and B are equinumerous if there exists a one-to-one correspondence (a bijection) between them, i.e. if there exists a function from A to B such that for every element y of B there is exactly one element x of A with f(x).
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Existential quantification
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some".
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First-order logic
First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.
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Foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics.
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Free variables and bound variables
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place.
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Function (mathematics)
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
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George Boolos
George Stephen Boolos (September 4, 1940 – May 27, 1996) was an American philosopher and a mathematical logician who taught at the Massachusetts Institute of Technology.
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Hierarchy (mathematics)
In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set.
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Identity (mathematics)
In mathematics an identity is an equality relation A.
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If and only if
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
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Infinity
Infinity (symbol) is a concept describing something without any bound or larger than any natural number.
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Injective function
In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.
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Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
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List of set theory topics
This page is a list of articles related to set theory.
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Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Mereology
In philosophy and mathematical logic, mereology (from the Greek μέρος meros (root: μερε- mere-, "part") and the suffix -logy "study, discussion, science") is the study of parts and the wholes they form.
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Model theory
In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.
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Morley's categoricity theorem
In model theory, a branch of mathematical logic, a theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism.
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Morse–Kelley set theory
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first order axiomatic set theory that is closely related to von Neumann–Bernays–Gödel set theory (NBG).
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Naive set theory
Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
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Natural number
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
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New Foundations
In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica.
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Number
A number is a mathematical object used to count, measure and also label.
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Ontology
Ontology (introduced in 1606) is the philosophical study of the nature of being, becoming, existence, or reality, as well as the basic categories of being and their relations.
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Ordinal number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another.
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Paradox
A paradox is a statement that, despite apparently sound reasoning from true premises, leads to an apparently self-contradictory or logically unacceptable conclusion.
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Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano.
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Philosopher
A philosopher is someone who practices philosophy, which involves rational inquiry into areas that are outside either theology or science.
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Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics, and purports to provide a viewpoint of the nature and methodology of mathematics, and to understand the place of mathematics in people's lives.
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Range (mathematics)
In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage.
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Richard Milton Martin
Richard Milton Martin (1916, Cleveland, Ohio – 22 November 1985, Milton, Massachusetts) was an American logician and analytic philosopher.
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Russell's paradox
In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naïve set theory created by Georg Cantor led to a contradiction.
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S (set theory)
S is an axiomatic set theory set out by George Boolos in his article, Boolos (1989).
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Scott's trick
In set theory, Scott's trick is a method for giving a definition of equivalence classes for equivalence relations on a proper class (Jech 2003:65).
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Set (mathematics)
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
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Set theory
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.
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Set-builder notation
In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements or stating the properties that its members must satisfy.
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Singleton (mathematics)
In mathematics, a singleton, also known as a unit set, is a set with exactly one element.
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Successor function
In mathematics, the successor function or successor operation is a primitive recursive function S such that S(n).
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Transitive relation
In mathematics, a binary relation over a set is transitive if whenever an element is related to an element and is related to an element then is also related to.
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Type theory
In mathematics, logic, and computer science, a type theory is any of a class of formal systems, some of which can serve as alternatives to set theory as a foundation for all mathematics.
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Uniqueness quantification
In mathematics and logic, the phrase "there is one and only one" is used to indicate that exactly one object with a certain property exists.
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Universal quantification
In predicate logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all".
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Urelement
In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ur-, 'primordial') is an object (concrete or abstract) that is not a set, but that may be an element of a set.
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Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets.
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Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel set theory (ZFC).
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Well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.
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Willard Van Orman Quine
Willard Van Orman Quine (known to intimates as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century." From 1930 until his death 70 years later, Quine was continually affiliated with Harvard University in one way or another, first as a student, then as a professor of philosophy and a teacher of logic and set theory, and finally as a professor emeritus who published or revised several books in retirement.
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Zermelo set theory
Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory.
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Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.
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References
[1] https://en.wikipedia.org/wiki/Scott–Potter_set_theory
