Class (set theory)

with class is designated in particular today in mathematics, and in the set theory, a summary by quantities or other mathematical objects. Necessary condition for the definition of the class is a clear characteristic, which all objects of the class possess. Before the axiomatic formulationthe set theory the terms class and quantity were used inconsistently, and step also in such a way in the mathematical literature 19. Century up.

examples

1. the class <math> K< /math> all groups. One writes <math> G \ in K< /math> exactly if <math> G< /math> a group is.
2. The class<math> K< /math> all ordinal numbers
3. certain categories form
4. the class for classes <math> K< /math> the surrealen numbers. <math> K< /math> has all characteristics of a body to be except the characteristic, a quantity.
5. The class <math> K< /math> all quantities, the all class in such a way specified
6. the class of all einelementigen quantities.

Each quantity is also a class (or somewhat more formalistic: can as class be understood, i.e. as class of the elements, which them contain), but not all classes are quantities. Classes, which are not quantities, are called genuine classes or enormous amounts.

Informally one can say that a class is genuine,if it is “too large”. Like that for instance the class of all whole numbers a quantity is - infinitely largely however nevertheless manageable; the class of all groups however, as well as the class of all quantities, and therefore genuine classes are “too large”.

Classes are not subject to the axioms that Zermelo Fraenkel set theory. However also classes know /math by <the operations> math \< cap> and <math> \ cup< /math> to be linked with one another: <math> M \ in K \ cup L< /math>, exactly if <math> M \ in K< /math> or <math> M \ in L< /math>. Others from the set theory well-known operations, like for example the power quantity, may be transferred not at will to classes.In particular a class never contains a genuine class.

By the use of classes a set of paradoxes of the naive set theory are avoided. Instead each former paradox is a proof that a certain class is genuine. For example the Russell Antinomie is a proof that thoseAll class a genuine class is. The Burali Forti paradox is a proof for the authenticity of the class of all ordinal numbers.

The Zermelo Fraenkel set theory does not speak about classes. These exist only in a meta language as equivalence classes of logical formulas.

Another approach is of John von Neumann, Paul Bernays and briefly Gödel: Classes are the basic objects of their theory, and quantities are classes, which appear as element of another class. The genuine classes are then the classes, which are not element of another class.