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33 relations: Cardinal number, Church–Kleene ordinal, Class (set theory), Cofinality, Continuous function, Countable set, Equinumerosity, First uncountable ordinal, Georg Cantor, Greatest and least elements, Hilbert's paradox of the Grand Hotel, Infimum and supremum, Isolated point, John Horton Conway, Limit cardinal, Limit point, Maxima and minima, Natural number, Order topology, Order type, Ordinal arithmetic, Ordinal notation, Ordinal number, Recursively enumerable set, Richard K. Guy, Set (mathematics), Set theory, Successor ordinal, Transfinite induction, Union (set theory), Veblen function, Von Neumann cardinal assignment, Well-order.
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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Church–Kleene ordinal
In mathematics, the Church–Kleene ordinal, \omega^_1, named after Alonzo Church and S. C. Kleene, is a large countable ordinal.
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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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Cofinality
In mathematics, especially in order theory, the cofinality cf(A) of a partially ordered set A is the least of the cardinalities of the cofinal subsets of A. This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member.
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Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
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Countable set
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
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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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First uncountable ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number that, considered as a set, is uncountable.
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Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor (– January 6, 1918) was a German mathematician.
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Greatest and least elements
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if Hence, the greatest element of S is an upper bound of S that is contained within this subset.
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Hilbert's paradox of the Grand Hotel
Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets.
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Infimum and supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.
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Isolated point
In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S but there exists a neighborhood of x which does not contain any other points of S. This is equivalent to saying that the singleton is an open set in the topological space S (considered as a subspace of X).
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John Horton Conway
John Horton Conway FRS (born 26 December 1937) is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.
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Limit cardinal
In mathematics, limit cardinals are certain cardinal numbers.
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Limit point
In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself.
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Maxima and minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).
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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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Order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set.
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Order type
In mathematics, especially in set theory, two ordered sets X,Y are said to have the same order type just when they are order isomorphic, that is, when there exists a bijection (each element matches exactly one in the other set) f: X → Y such that both f and its inverse are strictly increasing (order preserving i.e. the matching elements are also in the correct order).
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Ordinal arithmetic
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation.
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Ordinal notation
In mathematical logic and set theory, an ordinal notation is a partial function from the set of all finite sequences of symbols from a finite alphabet to a countable set of ordinals, and a Gödel numbering is a function from the set of well-formed formulae (a well-formed formula is a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers.
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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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Recursively enumerable set
In computability theory, traditionally called recursion theory, a set S of natural numbers is called recursively enumerable, computably enumerable, semidecidable, provable or Turing-recognizable if.
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Richard K. Guy
Richard Kenneth Guy (born 30 September 1916) is a British mathematician, professor emeritus in the Department of Mathematics at the University of Calgary.
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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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Successor ordinal
In set theory, the successor of an ordinal number α is the smallest ordinal number greater than α.
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Transfinite induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.
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Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
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Veblen function
In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in.
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Von Neumann cardinal assignment
The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers.
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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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References
[1] https://en.wikipedia.org/wiki/Limit_ordinal
