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Cantor, Georg – closed interval

 Georg Cantor
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   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   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...

Carathéodory’s Theorem   See: outer measure.

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|;
• kl = |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

Cf. Charles Lutwidge Dodgson.

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.

Cauchy sequence   A sequence (x1, x2, x3, ... ) of elements of a metric space X with metric d(x, y) is Cauchy if for any e greater than zero there is some natural number N such that

In other words, in a Cauchy sequence, the elements eventually become “arbitrarily close together.” If the metric space X is closed, this condition is equivalent to the sequence being convergent.

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.

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 interval   An interval of the real number line (or any other totally ordered set) which includes its endpoints. An interval containing only one of its endpoints is called half-open.
Cf. open interval.

Cantor, Georg – closed interval

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