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74 relations: Aleph, Algebraic number, American Mathematical Society, Axiom of choice, Axiom of countable choice, Bijection, Borel hierarchy, Cardinal number, Cardinality, Cardinality of the continuum, Coanalytic set, Cofinality, Composite number, Computable number, Consistency, Constructible number, Countable set, Cube (algebra), Definable real number, Divergent series, Easton's theorem, Epsilon numbers (mathematics), Equinumerosity, Extended real number line, Finite set, Fixed point (mathematics), Fixed-point lemma for normal functions, Forcing (mathematics), Fourth power, Function (mathematics), Georg Cantor, Georg Cantor's first set theory article, Group theory, Hebrew alphabet, Inaccessible cardinal, Infinite set, Integer, Kurt Gödel, Limit (mathematics), Limit ordinal, Mathematics, Monotype Imaging, Natural number, Omega, Order type, Ordinal number, Parity (mathematics), Paul Cohen, Perfect power, Polish space, ..., Prime number, Prime power, Rational number, Real line, Real number, Regular cardinal, Scott's trick, Sequence, Set theory, Sigma-algebra, Springer Science+Business Media, Square number, String (computer science), Subset, Successor cardinal, Successor ordinal, Total order, Transfinite induction, Transfinite number, Uncountable set, Vector space, Von Neumann cardinal assignment, Well-order, Zermelo–Fraenkel set theory. Expand index (24 more) »
Aleph
Aleph (or alef or alif) is the first letter of the Semitic abjads, including Phoenician 'Ālep 𐤀, Hebrew 'Ālef א, Aramaic Ālap 𐡀, Syriac ʾĀlap̄ ܐ, Arabic ا, Urdu ا, and Persian.
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Algebraic number
An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).
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American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.
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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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Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.
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Borel hierarchy
In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets.
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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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Cardinality
In mathematics, the cardinality of a set is a measure of the "number of elements of the set".
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Cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum.
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Coanalytic set
In the mathematical discipline of descriptive set theory, a coanalytic set is a set (typically a set of real numbers or more generally a subset of a Polish space) that is the complement of an analytic set (Kechris 1994:87).
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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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Composite number
A composite number is a positive integer that can be formed by multiplying together two smaller positive integers.
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Computable number
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm.
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Consistency
In classical deductive logic, a consistent theory is one that does not contain a contradiction.
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Constructible number
In geometry and algebra, a real number is constructible if and only if, given a line segment of unit length, a line segment of length || can be constructed with compass and straightedge in a finite number of steps.
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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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Cube (algebra)
In arithmetic and algebra, the cube of a number is its third power: the result of the number multiplied by itself twice: It is also the number multiplied by its square: This is also the volume formula for a geometric cube with sides of length, giving rise to the name.
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Definable real number
Informally, a definable real number is a real number that can be uniquely specified by its description.
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Divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
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Easton's theorem
In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets.
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Epsilon numbers (mathematics)
In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map.
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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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Extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system by adding two elements: and (read as positive infinity and negative infinity respectively).
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Finite set
In mathematics, a finite set is a set that has a finite number of elements.
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Fixed point (mathematics)
In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function.
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Fixed-point lemma for normal functions
The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points (Levy 1979: p. 117).
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Forcing (mathematics)
In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results.
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Fourth power
In arithmetic and algebra, the fourth power of a number n is the result of multiplying four instances of n together.
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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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Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor (– January 6, 1918) was a German mathematician.
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Georg Cantor's first set theory article
Georg Cantor's first set theory article was published in 1874 and contains the first theorems of transfinite set theory, which studies infinite sets and their properties.
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Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.
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Hebrew alphabet
The Hebrew alphabet (אָלֶף־בֵּית עִבְרִי), known variously by scholars as the Jewish script, square script and block script, is an abjad script used in the writing of the Hebrew language, also adapted as an alphabet script in the writing of other Jewish languages, most notably in Yiddish (lit. "Jewish" for Judeo-German), Djudío (lit. "Jewish" for Judeo-Spanish), and Judeo-Arabic.
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Inaccessible cardinal
In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic.
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Infinite set
In set theory, an infinite set is a set that is not a finite set.
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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Kurt Gödel
Kurt Friedrich Gödel (April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher.
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Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.
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Limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal.
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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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Monotype Imaging
Monotype Imaging Holdings, Inc. is a Delaware corporation based in Woburn, Massachusetts.
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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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Omega
Omega (capital: Ω, lowercase: ω; Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the 24th and last letter of the Greek alphabet.
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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 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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Parity (mathematics)
In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd.
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Paul Cohen
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician.
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Perfect power
In mathematics, a perfect power is a positive integer that can be expressed as an integer power of another positive integer.
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Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset.
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Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
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Prime power
In mathematics, a prime power is a positive integer power of a single prime number.
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Regular cardinal
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality.
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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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Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
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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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Sigma-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Square number
In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself.
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String (computer science)
In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable.
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Subset
In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.
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Successor cardinal
In the theory of cardinal numbers, we can define a successor operation similar to that in the ordinal numbers.
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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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Total order
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.
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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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Transfinite number
Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite.
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Uncountable set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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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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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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Redirects here:
Aleph 0, Aleph 1, Aleph Null, Aleph One, Aleph function, Aleph naught, Aleph notation, Aleph nought, Aleph null, Aleph numbers, Aleph one, Aleph zero, Aleph-0, Aleph-1, Aleph-Null, Aleph-Zero, Aleph-naught, Aleph-nought, Aleph-null, Aleph-one, Aleph-zero, AlephOne, Alpeh-omega, א0, ℵ, ℵ0, ℵ1, ℵ₀, ℵ₁.
References
[1] https://en.wikipedia.org/wiki/Aleph_number
