Table of Contents
72 relations: Absolute value, Aleph number, Area, Axiom of choice, Beth number, Bijection, Cantor's diagonal argument, Cantor's first set theory article, Cantor's paradox, Cantor's theorem, Cardinal number, Cardinality of the continuum, Class (set theory), Commensurability (mathematics), Continuum hypothesis, Countable set, Counting, Dedekind-infinite set, Disjoint sets, Element (mathematics), Equinumerosity, Equivalence class, Equivalence relation, Euclid's Elements, Finite set, Fraktur, Function (mathematics), Geometry, Georg Cantor, Giuseppe Peano, Gottlob Frege, Hilbert's paradox of the Grand Hotel, History of mathematics, Hypercube, Independence (mathematical logic), Infinite set, Injective function, Intersection (set theory), Interval (mathematics), Irrational number, Law of trichotomy, Length, Line segment, Mathematics, Mathematische Annalen, Modulo, Morse–Kelley set theory, Natural number, Number line, Order type, ... Expand index (22 more) »
- Basic concepts in infinite set theory
- Cardinal numbers
Absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign.
See Cardinality and Absolute value
Aleph number
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. Cardinality and aleph number are cardinal numbers.
See Cardinality and Aleph number
Area
Area is the measure of a region's size on a surface.
Axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty.
See Cardinality and Axiom of choice
Beth number
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written \beth_0, \beth_1, \beth_2, \beth_3, \dots, where \beth is the Hebrew letter beth. Cardinality and beth number are cardinal numbers.
See Cardinality and Beth number
Bijection
A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the first set (the domain) is mapped to exactly one element of the second set (the codomain).
Cantor's diagonal argument
Cantor's diagonal argument (among various similar namesthe diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof) is a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbersinformally, that there are sets which in some sense contain more elements than there are positive integers. Cardinality and Cantor's diagonal argument are cardinal numbers.
See Cardinality and Cantor's diagonal argument
Cantor's first set theory article
Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties.
See Cardinality and Cantor's first set theory article
Cantor's paradox
In set theory, Cantor's paradox states that there is no set of all cardinalities. Cardinality and Cantor's paradox are cardinal numbers.
See Cardinality and Cantor's paradox
Cantor's theorem
In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A, the set of all subsets of A, known as the power set of A, has a strictly greater cardinality than A itself. Cardinality and Cantor's theorem are cardinal numbers.
See Cardinality and Cantor's theorem
Cardinal number
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. Cardinality and cardinal number are cardinal numbers.
See Cardinality and Cardinal number
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. Cardinality and cardinality of the continuum are cardinal numbers.
See Cardinality and Cardinality of the continuum
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.
See Cardinality and Class (set theory)
Commensurability (mathematics)
In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio is a rational number; otherwise a and b are called incommensurable.
See Cardinality and Commensurability (mathematics)
Continuum hypothesis
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. Cardinality and continuum hypothesis are Basic concepts in infinite set theory and cardinal numbers.
See Cardinality and Continuum hypothesis
Countable set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Cardinality and countable set are Basic concepts in infinite set theory and cardinal numbers.
See Cardinality and Countable set
Counting
Counting is the process of determining the number of elements of a finite set of objects; that is, determining the size of a set.
Dedekind-infinite set
In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A. Explicitly, this means that there exists a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Cardinality and Dedekind-infinite set are Basic concepts in infinite set theory and cardinal numbers.
See Cardinality and Dedekind-infinite set
Disjoint sets
In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common.
See Cardinality and Disjoint sets
Element (mathematics)
In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set.
See Cardinality and Element (mathematics)
Equinumerosity
In mathematics, two sets or classes A and B are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, 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). Cardinality and Equinumerosity are Basic concepts in infinite set theory and cardinal numbers.
See Cardinality and Equinumerosity
Equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes.
See Cardinality and Equivalence class
Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
See Cardinality and Equivalence relation
Euclid's Elements
The Elements (Στοιχεῖα) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid 300 BC.
See Cardinality and Euclid's Elements
Finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Cardinality and finite set are cardinal numbers.
See Cardinality and Finite set
Fraktur
Fraktur is a calligraphic hand of the Latin alphabet and any of several blackletter typefaces derived from this hand.
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of.
See Cardinality and Function (mathematics)
Geometry
Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor (– 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics.
See Cardinality and Georg Cantor
Giuseppe Peano
Giuseppe Peano (27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist.
See Cardinality and Giuseppe Peano
Gottlob Frege
Friedrich Ludwig Gottlob Frege (8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician.
See Cardinality and Gottlob Frege
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.
See Cardinality and Hilbert's paradox of the Grand Hotel
History of mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past.
See Cardinality and History of mathematics
Hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square and a cube.
Independence (mathematical logic)
In mathematical logic, independence is the unprovability of a sentence from other sentences.
See Cardinality and Independence (mathematical logic)
Infinite set
In set theory, an infinite set is a set that is not a finite set. Cardinality and infinite set are Basic concepts in infinite set theory and cardinal numbers.
See Cardinality and Infinite set
Injective function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies.
See Cardinality and Injective function
Intersection (set theory)
In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A.
See Cardinality and Intersection (set theory)
Interval (mathematics)
In mathematics, a (real) interval is the set of all real numbers lying between two fixed endpoints with no "gaps".
See Cardinality and Interval (mathematics)
Irrational number
In mathematics, the irrational numbers (in- + rational) are all the real numbers that are not rational numbers.
See Cardinality and Irrational number
Law of trichotomy
In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.
See Cardinality and Law of trichotomy
Length
Length is a measure of distance.
Line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints.
See Cardinality and Line segment
Mathematics
Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
See Cardinality and Mathematics
Mathematische Annalen
Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann.
See Cardinality and Mathematische Annalen
Modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the modulus of the operation).
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).
See Cardinality and Morse–Kelley set theory
Natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0. Cardinality and natural number are cardinal numbers.
See Cardinality and Natural number
Number line
In elementary mathematics, a number line is a picture of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely.
See Cardinality and Number line
Order type
In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y such that both and its inverse are monotonic (preserving orders of elements).
See Cardinality and Order type
Ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
See Cardinality and Ordinal number
Parity (mathematics)
In mathematics, parity is the property of an integer of whether it is even or odd.
See Cardinality and Parity (mathematics)
Pigeonhole principle
In mathematics, the pigeonhole principle states that if items are put into containers, with, then at least one container must contain more than one item.
See Cardinality and Pigeonhole principle
Power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of, including the empty set and itself.
Quanta Magazine
Quanta Magazine is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science.
See Cardinality and Quanta Magazine
Real number
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.
See Cardinality and Real number
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic.
See Cardinality and Richard Dedekind
Schröder–Bernstein theorem
In set theory, the Schröder–Bernstein theorem states that, if there exist injective functions and between the sets and, then there exists a bijective function. Cardinality and Schröder–Bernstein theorem are cardinal numbers.
See Cardinality and Schröder–Bernstein theorem
Set (mathematics)
In mathematics, a set is a collection of different things; these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.
See Cardinality and Set (mathematics)
Set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.
See Cardinality and Set theory
Space-filling curve
In mathematical analysis, a space-filling curve is a curve whose range reaches every point in a higher dimensional region, typically the unit square (or more generally an n-dimensional unit hypercube).
See Cardinality and Space-filling curve
St. Lawrence University
St.
See Cardinality and St. Lawrence University
Subset
In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).
Surjective function
In mathematics, a surjective function (also known as surjection, or onto function) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that.
See Cardinality and Surjective function
Trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.
See Cardinality and Trigonometric functions
Uncountable set
In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. Cardinality and uncountable set are Basic concepts in infinite set theory and cardinal numbers.
See Cardinality and Uncountable set
Union (set theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.
See Cardinality and Union (set theory)
Vertical bar
The vertical bar,, is a glyph with various uses in mathematics, computing, and typography.
See Cardinality and Vertical bar
Von Neumann cardinal assignment
The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. Cardinality and von Neumann cardinal assignment are cardinal numbers.
See Cardinality and Von Neumann cardinal assignment
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–choice set theory (ZFC).
See Cardinality and Von Neumann–Bernays–Gödel set theory
Zeitschrift für Philosophie und philosophische Kritik
The Zeitschrift für Philosophie und philosophische Kritik (english "Journal of Philosophy and Philosophical Criticism") was an academic journal.
See Cardinality and Zeitschrift für Philosophie und philosophische Kritik
Zermelo–Fraenkel set theory
In set theory, 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.
See Cardinality and Zermelo–Fraenkel set theory
See also
Basic concepts in infinite set theory
- Cardinality
- Cocountability
- Cofiniteness
- Continuum hypothesis
- Countable set
- Dedekind-infinite set
- Equinumerosity
- Infinite set
- Limitation of size
- List of set identities and relations
- Set-theoretic definition of natural numbers
- Transfinite number
- Uncountable set
Cardinal numbers
- Θ (set theory)
- Aleph number
- Amorphous set
- Beth number
- Cantor's diagonal argument
- Cantor's paradox
- Cantor's theorem
- Cardinal and Ordinal Numbers
- Cardinal assignment
- Cardinal characteristic of the continuum
- Cardinal function
- Cardinal number
- Cardinality
- Cardinality of the continuum
- Cichoń's diagram
- Cofinality
- Continuum function
- Continuum hypothesis
- Countable set
- Dedekind-infinite set
- Easton's theorem
- Equinumerosity
- Finite set
- Gimel function
- Hartogs number
- Infinite set
- König's theorem (set theory)
- Large cardinals
- Limit cardinal
- Natural number
- Rathjen's psi function
- Regular cardinal
- Schröder–Bernstein theorem
- Second continuum hypothesis
- Singular cardinals hypothesis
- Strong partition cardinal
- Successor cardinal
- Suslin cardinal
- Tarski's theorem about choice
- Tav (number)
- Transfinite number
- Uncountable set
- Von Neumann cardinal assignment
- Weak continuum hypothesis
References
Also known as Cardinal comparison, Cardinalities, Finite cardinality, Number of elements, Set modulus, Set size.