Class

In mathematics, a class refers to a collection of sets 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 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, such as Russells paradox, are avoided.

The standard Zermelo-Fraenkel set theory axioms do not talk about classes and classes are defined afterwards as equivalence classes of logical formulas. Another approach is taken by the von Neumann- Bernays-Gödel set theory: classes are the basic objects in this theory and sets are then defined to be those classes which are elements of other classes. The proper classes then are those classes that are not element of any other class.

Several objects in mathematics are too big for sets and need to be described with classes, for instance categories or the surreal numbers.

Classes vs. types

Philosophers sometimes distinguish classes from types and kinds. We can talk about the class of human beings, just as we can talk about the type (or natural kind), human being, or humanity. How, then, might classes be thought to differ from types? One might well think they are not actually different categories of being, but typically, while both are treated as abstract objects, classes are not usually treated as universals, whereas types usually are. Whether natural kinds ought to be considered universals is vexed; see natural kind.

There is, in any case, a difference in how we talk about types and kinds versus how we talk about classes. We say that Socrates is a token of a type, or an instance of the natural kind, human being. But notice that we say instead that Socrates is a member of the class of human beings. We would not say that Socrates is a "member" of the type or kind, human beings. He is a token (instance) of the type (kind). So the linguistic difference is: types (or kinds) have tokens (or instances); classes, on the other hand, have members.

The distinctions are admittedly none too clear, and one thing that philosophers do in thinking about these topics is to try to get clear on them.

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In object-oriented programming languages, a class refers to the description of a collection of objects with the same internal structure. A class specifies the data items each object of the class contains and the operations or methods that can be performed on each object belonging to the class. An object belonging to a class is also called an instance of the class.

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In politics and sociology, the word class is often used to refer to social class, i.e. a socio-economic stratum of society.