Index: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Home > 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.

Some classes are sets, for instance the class of all integers that are even, but others are not, for instance the class of all ordinal numbers or the class of all sets. Classes that are not sets are called proper classes.

A proper class cannot be an element of a set or a class and is not subject to the Zermelo-Fraenkel axioms of set theory; thereby a number of paradoxes of naive set theory are avoided. Instead, these paradoxes become proofs that a certain class is proper. For example, Russell's paradox becomes a proof that the class of all sets is proper, and the Burali-Forti paradox becomes a proof that the class of all ordinal numbers is proper.

The standard Zermelo-Fraenkel set theory axioms do not talk about classes; classes exist only in the metalanguage as equivalence classes of logical formulas. Another approach is taken by the von Neumann-Bernays-Gödel axioms; classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. The proper classes, then, are those classes that are not elements of any other class.

Several objects in mathematics are too big for sets and need to be described with classes, for instance large categoriesCategory theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". See list of category theory topics for a breakdown of relevan or the class-field of surreal numberThe surreal numbers are an example of what is sometimes called a Field (with a capital F), meaning a proper class on which there is defined an addition, multiplication and multiplicative inverse which satisfy all of the axioms of a field except for the fas.

The word "class" is sometimes used synonymously with "set," most notably in the term " equivalence classIn mathematics, given a set X and an equivalence relation ~ on X the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a :[a] { x in X | x ~ a } The notion of equivalence classes is useful for constructing s." This usage dates from a historical period where classes and sets were not distinguished as they are in modern terminology. Many discussions of "classes" in the 19th centuryAlternative meaning: Nineteenth Century (periodical ( 18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801- 1900. Events The Little Ice Age ended and earlier are really referring to sets, or perhaps to a more ambiguous concept.

Category theoryCategory theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "generalized abstract nonsense". See list of category theory topics for a breakdown of relevan Set theory

Read more »

<Hide>

Topics: Class (set Theory), Classics, Classical Music Era, Class Envy...

ClassicsClassics particularly within the Western University tradition, when used as a singular noun, means the study of the language, literature, history, art, and other aspects of Greek and Roman culture during the time frame known as classical antiquity. As a p	Classical music eraThe Classical period in Western music occurred in the second half of the 18th century. Although the term classical music is used as a blanket term meaning all kinds of music in a certain tradition, it can also occasionally mean this particular era within	Class envyClass envy is a pejorative term sometimes used to describe criticisms of the rich and powerful by the poor and less powerful. Criticism of excessive wealth and the wealthy has been a staple of political discourse for generations and many believe that soci
Classical guitarA classical guitar also called a Spanish guitar is a musical instrument from the guitar family. The classical guitar is distinguished by a number of features. It is an acoustic instrument. The sound is amplified by a sound box. It has six strings. A few c	Classical mechanicsClassical mechanics is a model of the physics of forces acting upon bodies. It is often referred to as Newtonian mechanics after Newton and his laws of motion. Classical mechanics is subdivided into statics (which models objects at rest), kinematics (whic	Classical musicThis article is about the broad genre of classical music in the Western musical tradition. For the period of music in the 18th century see Classical music era, for articles on classical music of non-Western cultures, see: List of classical music tradition
Class	Classical element	Classical liberalism
Classical orders	Classical education	Class action
Class (computer science)	Classification of finite simple groups	Class (set theory)

Non User