On the Theory of Classes

Abstract

Abstract Apart from separate individual objects, which we shall also call INDIVIDUALS for short, logic is concerned with CLASSES of objects; in everyday life as well as in mathematics, classes are more often referred to as SETS. Arithmetic, for instance, frequently deals with sets of numbers, and in geometry our interest lies not so much in single points as in sets of points (that is, in geometrical configurations). Now, classes of individuals are called CLASSES OF THE FIRST ORDER. Relatively more rarely in our investigations we come upon CLASSES OF THE SECOND ORDER, that is, upon classes which consist, not of individuals, but of classes of the first order. Sometimes even CLASSES OF THE THIRD, FOURTH, and HIGHER ORDERS have to be dealt with. Here we shall be concerned almost exclusively with classes of the first order, and only exceptionally-as in Section 26-we shall have to deal with classes of the second order; however, our considerations can be applied with practically no changes to classes of any order. In order to distinguish between individuals and classes (and also among classes of different orders), we employ as variables letters of different shape and belonging to different alphabets. It is customary to designate individual objects such as numbers, and classes of such objects, by lower-case and capital letters of the Latin alphabet, respectively. In elementary geometry the opposite notation is the accepted one, capital letters designating points and lower-case letters (of the Latin or the Greek alphabet) designating sets of points.