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Cantor, Georg – closed
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Georg Cantor
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Cantor, Georg
(born 1845)
German mathematician and founder of modern set theory. Cantor’s concern with the set-theoretic nature of the real numbers began while he was working on certain properties of trigonometric series during the early 1870’s. Inspired by the work of his friend Dedekind, Cantor proved in 1873 that the rational numbers are countable, and later the same year proved that the real numbers themselves are uncountable. These proofs introduced important methods, including the notion of a one-to-one correspondence and the method of diagonalization. Cantor’s results were counterintuitive and broke with established dogma about the nature of infinity. Consequently, he became a controversial figure, opposed by some mathematicians (e.g., Kronecker) who used their influence to stifle Cantor’s career as much as possible. This opposition exacerbated Cantor’s predisposition to severe depression, and he ultimately died, of a heart attack, in a sanitorium.
Related article: Cantor's Theorem
Related article: Cantor Set
Related MiniText: Infinity -- You Can't Get There From Here...
Cantor-Bendixson Theorem
Every closed subset of the real numbers is the disjoint union of a perfect set and a countable set.
Cf. derived set.
Cantor-Schröder-Bernstein Theorem
For any sets A and B, if there exists an injective (one-to-one) function from A into B and also an injective function from B into A, then there exists a surjective (onto) function from A onto B.
Cantor set
ARTICLE
A subset of the unit interval (on the real number line) which is a perfect set, nowhere dense, uncountable, and homeomorphic to the entire unit interval. See the article for a complete exposition.
Cantor’s Theorem
ARTICLE
Given any set, a new set may be constructed which is cardinally larger, i.e., whose size is represented by a larger cardinal than the size of the initial set. More specifically, for no set X is there a bijection of X onto its power set. This theorem establishes both that there are different sizes of infinity, and that there is no greatest such “size.” Cantor’s proof was the first use of a diagonalization argument to derive a contradiction.
Related MiniText: Infinity -- You Can't Get There From Here...
cardinal
A cardinal number represents the size of a set irrespective of the order or structure of its elements. If X is a set, its cardinality is generally indicated by |X|. Two sets A and B are said to have the same cardinality if there is a function from A into B that is bijective (i.e., one-to-one and onto).
A cardinal is an initial ordinal, i.e., an ordinal for which there does not exist a bijection onto any lesser ordinal. Thus, all finite ordinals (i.e., the natural numbers) are cardinals, but most transfinite ordinals are not cardinals.
If a is a cardinal, then we denote by a+ the least cardinal greater than a. A cardinal k is called a successor cardinal if k = a+ for some cardinal a. Otherwise k is called a limit cardinal. The heirarchy of transfinite cardinals is defined by recursion, and denoted using the Hebrew letter aleph:
where w is the first infinite ordinal.
Cf. cardinal arithmetic, inaccessible cardinal, regular.
Related MiniText: Infinity -- You Can't Get There From Here...
cardinal arithmetic
Let k and l denote cardinals and A and B denote sets such that k = |A| and l = |B|. Denote by BA the set of all functions from B to A, and by A × B the Cartesian product of A and B. The operations of cardinal arithmetic are then defined by- k + l = |A union B|;
- k • l = |A × B|;
- kl = |BA|
For finite cardinals, this corresponds to the operations of ordinary arithmetic, but is quite different in the case of transfinite cardinals. For instance, if k is not finite, then k + k = k.
Cf. ordinal arithmetic.
cardinality
Two sets X and Y have the same cardinality if there exists a bijection between them. Specifically, the cardinality of X may be understood as a canonical object meant to represent the proper class of sets to which X is bijective. If the axiom of choice is assumed, then the cardinality of X is the unique cardinal number having the same cardinality as X
Carroll, Lewis
See Charles Lutwidge Dodgson.
Carroll’s Paradox
ARTICLE
Paradox published by Lewis Carroll as “What the Tortoise Said to Achilles.” See the article for a full exposition.
Cf. Charles Lutwidge Dodgson.
Cartesian plane
The Cartesian product R2, represented graphically by two real number lines at right angles to one another, with the point (0,0) at the intersection.
Used for graphing functions from the set of real numbers to itself. The quadrants, numbered I - IV as shown, indicate the regions of the plane where the x and y axes are positive and negative. Compare: Argand plane.
Cartesian product
For any collection {Ai}, i = 1, 2, 3, ..., n, of sets, the Cartesian product
is the set of ordered n-tuples (a1,a2, ... ,an) with a1 an element of A1, a2 an element of A2, etc. The Cartesian product R2 of the set of real numbers is called the Cartesian plane, and in general n-dimensional real space is the Cartesian product Rn. The assertion that the Cartesian product of an infinite collection of non-empty sets is non-empty is equivalent to the axiom of choice.
CH
See: continuum hypothesis.
chain
If X is a partially ordered set, then a subset Y of X is called a chain if it is totally ordered, that is, if for any two elements a, b of Y, either a  b or b a.
Cf. antichain.
chain condition
For a an infinite cardinal, a partial order P is said to have the a-chain condition if every antichain in P has cardinality not greater than a. If a = w, this is called the countable chain condition, or “c.c.c.”
characteristic function
Given a subset E of a space X, the characteristic function cE is defined by cE(x) = 1 if x is in E, and cE(x) = 0 otherwise. All properties of sets and set operations may be expressed by means of characteristic functions.
choice, axiom of
See: axiom of choice.
circle
In a plane, the locus of all points equidistant from a given point, called the center. The general equation for a circle in the Cartesian plane is given by (x - h) 2 + (y - k) 2 = r 2, where r is the radius of the circle (distance from the center to the locus of points), and (h, k) are the coordinates of the center.
The interior of a circle is referred to as an open disk.
A circle is also a conic section; a special case of an ellipse in which the foci coincide.
Related article: Conics
circumference
Geometry: The distance around a circle in the plane, or around a great circle of a sphere.
Graph Theory: The circumference of a graph G is defined as the length of the longest cycle of G. The circumference is ususally denoted by c(G), and is undefined if G has no cycles.
class
See proper class.
closed
General: A set is closed under an operation if applying the operation to its elements returns only elements in the set. For example, the set of integers is closed under addition, since adding two integers always gives another integer, but it is not closed under division, since dividing two integers may result in a non-integer.
Geometry: A plane figure is closed if it consists of lines and/or curves that entirely enclose an area. Similarly, a figure in 3-dimensional space is called closed if it entirely encloses a volume.
See the following listings for other uses of the word “closed” in mathematics.
Topology: A set is topologically closed if it is not open.
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