ASSEMBLAGES, General Theory of. The doctrine variously entitled Mengenlehre and Mannigfaltigkeitslehre by the Germans, Thiorie des ensembles by the French, and sometimes referred to in English as the theory or doctrine of manifolds or aggregates or by other analogous designations. Many of its ideas are at least as ancient as historical thought and have figured in important ways in logic, philosophy and mathematics steadily from the earliest times. On the other hand, many of the chief concepts involved in it, its char acteristic notions, and their organization into a distinct and self-supporting body of coherent doctrine, may be said to constitute the latest great mathematical creation. Indeed the ma jority of the founders and builders of the doc trine, including Georg Cantor as easily the first of them all, are still among the living. As mathematics is the most fundamental of the sciences, the theory of assemblages seems des tined to be regarded, if it be not already re garded, as the most fundamental branch of mathematics. Viewed in retrospect, it appears as an inevitable product of the modern critical i spirit. Already it is seen to underlie and inter penetrate both geometry and analysis. Its con nection with mathematical logic is most inti mate, often approximating identity with the latter; and even philosophy is surely, if but slowly, beginning to recognize in the theory of manifolds her own most inviting and promis ing field.
The Notions, Assemblage and Element — Roughly speaking, any collection of objects or things of whatever kind or kinds is an as semblage. Whitehead and Russell, however, have shown that certain limitations must be imposed on this great generality to avoid the most baffling paradoxes. Each object in the collection is .called an element of the assem blage. An assemblage, to be mathematically available, mast be defined, or, as usage has it. well-defined, (wohldefinirt, bien difini). An assemblage is defined when, by the logical prin ciple of the excluded middle, it can be regarded as intrinsically determined whether an arbi trarily given object is or is not an element of it. Means may or may not be known for mak ing the determination actual or extrinsic. Thus
if the elements of the assemblage be completely tabulated the determination can be actually ef fected by comparing the given object with the elements of the table. Again, if an assemblage, such as that of the endless sequence 1 2, 3, . . . of integers, be given by a definite law of formation of its elements, the law will gener ally enable one to determine actually whether any given object, as 5 or it or an apple or a sunset, is an element of the assemblage or not. But the possession of means the application of which is in our power is not essential to the notion of defined assemblage. A real * num ber is called algebraic or transcendental accord ing as it is or is not a root of an equation of the form, a.xx a having integral coefficients. Any real number, no matter what its origin or definition, is either algebraic or transcendental; it cannot be both and it cannot be neither. Hence the real alge braic numbers constitute a defined assemblage, and so, too, the transcendentals. Nevertheless no means is known for ascertaining in every case whether a given number is algebraic or not. It was really a great achievement when the trans cendental character of the long familiar clas sical numbers e and r was proved, for e by Hermite in 1874 and for r nine years later by Lindemann. Even the existence of non-alge braic numbers was unknown till it was proved by Lionville in 1851.
Depiction and Correspondence.—Assem blages will be denoted by large and elements by small letters. If, in any way, each element a of A is uniquely associated with an element b of B, A is said to be depicted on B. The b's so used are the pictures of the a's. If all the b's are thus made pictures, B is also depicted on A and the ds are pictures of the b's. If each a is a picture of a b and reciprocally, so that there is a one-to-one correspondence be tween them, the depiction is called similar. Ob viously an assemblage can be depicted either similarly or dissimilarly on itself, generally in more than one way, often in an endless variety of ways.