Theory of Aggregates

AGGREGATES, THEORY OF. An aggregate may be defined, for the moment, as a collection of elements of some kind or other which is subjected to laws of operation and, on occasion, provided with a structure in such a way as to make it capable of being used to facilitate the arguments and calculations of mathe matical analysis. From the point of view of pure philosophy such collections are of considerable interest in themselves and there is an extensive subject, which may be called for purposes of reference the theory of classes, which has grown up around them. This theory involves, as might be expected, considerable difficul ties of a somewhat metaphysical nature and is in parts highly con troversial. It is, however, possible to isolate from the rest of the theory a part which is of much practical value and almost com pletely free from controversy. This is the theory of aggregates. It concerns classes considered not as entities to be subjected to logical scrutiny and philosophical analysis but simply as symbols of a certain kind which are used to facilitate mathematical cal culations in much the same way that numbers are used. The prog ress that has been made in some branches of mathematics since the introduction in the subject of the theory of aggregates is very striking.

Preliminary Notions.

It will be assumed that there is given a set of elements of some kind or other each of which can be dis tinguished from all the others and so may be indicated by an appropriately attached name. This set is described as the field of operations and the individual constituents which it contains are its elements. Specified elements are conveniently denoted by let ters a, b, c, ... and unspecified elements by letters x, y, z, ... . There are, naturally, a great many fields of operations available. It is supposed that a particular one is selected and the subsequent is then relative to this field. The theory of classes, of course, has to take into account all possible fields of operations. The theory of aggregates is at most concerned with the reali zation of a single field of operations which shall satisfy the con ditions which may at the time be imposed. The question of realiz ing the field of operations is almost the only part of the subject which is controversial.

Aggregates are, roughly speaking, collections of elements of the field of operations. This definition is, however, subject to the criticism that it is not very precise. Moreover, the idea of a collection involves certain metaphysical difficulties which, al though some of them have been known for several thousands of years, have not been completely resolved. An alternative defi nition has therefore been adopted by some authorities. Nothing is said about the nature of aggregates but instead their use is defined. Aggregates are taken to be symbols the sentences con taining which are interpreted by given conventions in such a way that the resulting formal laws are, up to a point, identical with those which would result from the application of the rough defi nition originally given. The requirements of mathematics are simply that the conventions used should be consistent. Experience seems to show that this is the case. The requirements of the mathematician are that the conventions should be easily applied. These are met by treating aggregates as though there were actual collections. Experience again shows that this procedure, if cer tain quite simple safeguards are employed, never leads to error. Specified aggregates may be denoted by letters A, B, C, ... and unspecified aggregates by letters X, Y, Z, ... .

The Relations E and co.—Between any element x of the field of operations and any given aggregate A one or other of two mutually exclusive relations must hold. Either x belongs to the aggregate, in a sense that will presently be made precise, or x is excluded by the aggregate. In the first case x e A is written and in the second x w A. The elements which belong to A are described as its members and of them A is said to be composed.

The aggregate of which the elements which do not belong to A are the members is the complement of A. An interchange of e and co plainly interchanges an aggregate and its complement.

Propositional Forms.

A sentence like "a is blue" expresses a property of the element a. On the other hand a form of words and symbols like "x is blue," where x denotes an unspecified ele ment of the field of operations, expresses nothing. It may be described as a propositional form. When the variable x which it contains is replaced by a name it becomes a proposition. Certain forms of words and symbols which have the appearance of propo sitional forms can be shown to be incapable of yielding propo sitions when treated in the above way. What results when the name is substituted is necessarily a meaningless array of words. The convention is therefore made that only propositional forms that are significant over the field of operations are taken into con sideration in the theory of aggregates associated with any given field of operations.

The Fundamental Convention.

It is assumed that to each significant propositional form there corresponds a definite ag gregate. This is described as the aggregate determined by the form. If the form is denoted by f (x) the corresponding aggre gate is denoted by [f (x)].

The main convention governing the use of aggregates is that sentences involving [f(x)] are to have the same meaning as cor responding sentences involving f (x). Thus, to take a rough ex ample, if A is the aggregate [x is blue] then "A is extensive" is taken to mean " `x is blue' is often true." More precise examples are: "a e [f(x)]" means "f(a) is true" and "a co [f(x)]" means "f (a) is false." It is possible to interchange the conventions in volving E and CO without causing any alteration in the formulae, provided that certain simple precautions are taken. In this way each formula can be made to yield two propositions. For this reason it is convenient, at any rate for mathematical purposes, to keep the notion of an aggregate distinct from that of a propo sitional form.

Null Aggregates.

An aggregate determined by a propositional form which never reduces to a true proposition, no matter what name is substituted for its variable, is described as null. A null aggregate is denoted by the symbol o. Null aggregates have formal properties closely resembling those of the number o. The complement of a null aggregate is the aggregate consisting of all the elements of the field of operations.

Arithmetical Theory.

Between any two aggregates oper ations similar to those of arithmetic may be defined. Thus:— The addition or multiplication of any number of aggregates presents no difficulty. In this respect the theory of aggregates is considerably easier than the theory of numbers. Operations in volving limits can also be defined. Calculations in the arithmetical theory are not much different from those in the arithmetic of numbers. It is often found that the results of extremely ele mentary calculations involving aggregates have an interpretation which sheds considerable light on comparatively difficult problems in other branches of mathematics.

Geometrical Theory.

A completely different but equally useful theory results when the aggregates considered are en dowed with a structure in some way. Between any two points of a given aggregate, for instance, a distance may be defined. Any aggregate which has been given a structure is described as a space. A considerable variety of spaces is known and their consideration provides much information as to the nature of the more familiar spaces, such as the Euclidean spaces.

Application to the Theory of Functions.

There are many problems in the theory of functions (q.v.) which have been solved either wholly or in part by means of the theory of aggregates. This is in measure due to the intrinsic power of the theory and also to the concise and expressive way in which the theory allows the formulation of ideas. A notable advance is due to the fact that numerical functions can be defined by means of a table of aggregates. In order to know a function (1) (x) it is sufficient to know the aggregates [0(x)>p] for only the rational values of p, whereas in the ordinary way it is necessary to know cp (x) for all values, irrational as well as rational of x.

See F. Hausdorff, Mengenlehre (1927, bibl.) ; C. J. de la Vallee Poussin, Integrales de Lebesque, fonctions d'ensemble, classes de Baire 0916). Hausdorff's work provides an unusually interesting general introduction to the subject. The reader already familiar with the theory of functions of a real variable will find some valuable applica tions of the theory of aggregates in the tract by de la Vallee Poussin.

(S. Po.)