Class (set Theory)
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as Von Neumann–Bernays–Gödel set theory, axiomatize the notion of "class", e.g., as entities that are not members of another entity.
Every set is a class, no matter which foundation is chosen. A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems.
Outside set theory, the word "class" is sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology. Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept.
Other articles related to "set theory, sets":
... ZF set theorydoes not formalize the notion of classes ... instead be described in the metalanguage,as equivalence classes of logical formulas ... For example,if is a structure interpreting ZF,then the metalanguage expression is interpreted in by the collection of all the elements from the domain of that is,all the setsin ...
Famous quotes containing the word class:
“I read, with a kind of hopeless envy, histories and legends of people of our craft who do not write for money. It must be a pleasant experience to be able to cultivate so delicate a class of motives for the privilege of doing ones best to express ones thoughts to people who care for them. Personally, I have yet to breathe the ether of such a transcendent sphere. I am proud to say that I have always been a working woman, and always had to be ...”
—Elizabeth Stuart Phelps (18441911)