Next from the simply ordered assemblages A and B, form the assemblage S by replacing each b by an assemblage Abp A. It is easily seen that the order-type of S depends only on a =A and 8=B. Hence the definition of mul tiplication ; a R = S. Here a is multiplicand and /3 is multiplier. It is readily proved, in respect to three types a, 8, y, that ( a iy) y a (4? 7) and that a (1/ ±)) a + a- y. That is, multiplication of ordinals like that of cardinals is associative and distributive. But in general ordinals do not obey, while car dinals always obey, the commutative law. The reader can easily convince himself that, for example, 2 2. Gr. is as u +4).2 ± 6).3 -1- . . . . and such order-types as , etc., are defined by analogous chains of additions.
of Denote by R the rational numbers less than 1 and greater than zero, taken in natural order. Let ft = R. Obviously a belongs to the type-class Ito, for we have seen that R is denumerable. More over, R is dense (contains an element between every two elements) and has no element of lowest rank and none of highest. By these three properties, R is completely characterized; that means that if A is simply ordered, dense, de numerable, and has neither lowest nor highest element, A and R are similar, and ri =A. It follows that e +e=e, 1=n, (1 +77) 77 (1+9 +I)* but, e +1 * 1 -Fe, and, thoughe-1-1-1-e=re,e+e-1-eiire,ife>1.
of Linear
De note by 9 the order-type of the linear con tinuum X= 1 xl, where 0 < x < 1, and where X is disposed in natural order, i.e., so that if x and x' be any two elements of X, x l x', when and only when x
Zermelo's In Zermelo pointed out that a certain principle, which amounts to the statement that given an as semblage A of assemblages B, there exists an assemblage C containing Just one member from every B, was being tacitly employed through out the theory of assemblages. Zermelo con sidered this axiomatic, but as soon as it was rendered explicit, most of the mathematicians who had formerly unconsciously employed it disowned it, and set themselves the task of freeing the theory of assemblages from all taint of it. In general, this work has been very successful, but there are a group of outstanding propositions which have not been proved except on the basis of Zermelo's axiom, some of which actually imply it. Among there are (1) that of two unequal transfinite cardinal numbers, one is always the greater; (2) that every assemblage can be well-ordered; (3) that every number is either finite or infinite (in the senses already defined). The con scientious mathematician, when he employs this principle, will always make clear the fact of this employment explicit. See also ALGE BRA, DEFINITIONS AND FUNDAMENTAL CON CEPTS.
Bettazi, delle grandezze' (1891) , Bolzano, 'Paradoxien des Unendlichen' (1850) ; du Bois-Reymond, 'Die allgemeine Funktionlehre' (1882) ; Bore), (Lecons sur la theorie des fonctions' (1898) ; Cantor, G., 'Grundlagen einer allgemeinen Mannigfaltigkeitslehre' (1883) and many ar ticles in Mathematische Annalen and Acta Mathematica; Dedekind, 'Was sind and was sollen die Zahlen?); Hobson, J., 'Theory of the Functions of a Real Variable' (Cam bridge 1907) ; Russell, B. A. W., 'Prin ciples of Mathematics' (Cambridge 1903) ; Schonflies, A., (Beitrage zur Theorie der Punktmengen' (in Mathematische Annalen, Leipzig 1906) ; Veronese, 'Fondamenti di geometria' (1891) ; Whitehead, A. N., and Russell, B. A. W. Mathematica' (Cambridge 1910- ) ; Young, W. H. and G. C., 'The Theory of Sets of Points' (Cambridge 1906) ' Zermelo, article in Vol. LIX, matische Annalen (1904).