Involution.— If with each a of A a b of B be associated, any a and the associate b will be a pair. The same b may enter two or more pairs. The assemblage of all the pairs result ing from any such definite association is called a covering of A with B, and is denoted by f (A). A different covering results if with any a there be associated a b not associated with it before. The assemblage of all possible coverings of A with B is denoted by (BIA); then (BIA)= If(A) I. The power Y of (BIA) depends only on the powers a and P of A and B; hence the definition: aP=y. It may be proved that, if a, Y denote any three powers, aP •aY = ail aY•13Y=(a--/3)Y and It is an interesting fact that by means of the foregoing definitions of power, and addition, multiplication and involution of powers, the definition and the fundamental properties of the ordinary (finite) cardinals 1, 2, 3, . . . , . can be rigorously deduced.
The Smallest Transfinite Cardinal.— The cardinal number of the assemblage v of finite cardinals is denoted by it., alef-null.
Symbolically, N. I . The transfinite num ber s., has the properties: *.+ 1= s.; K. > whereµ is any finite cardinal; st. < a, where a is any transfinite cardinal different from N.; Ns + No; v • • P=Ite, where v is any finite cardinal; K.• st.v=--ts.; etc.
It is one of the interesting facts met with in the doctrine of transfinite assemblages that the cardinal number of the points of space or other continuum is precisely 2 K.
Simply Ordered Assemblages, Order types.— A is simply ordered when and only when its elements a are so disposed that of every pair a,, a, of them, one, as a., precedes, i. e., has lower rank and the other, as a., comes after, i.e., has higher rank, and of every triplet al, a,, a., ai is lower than a., if is lower than a, and a, is lower than as. To say symbolically that a. is lower in rank than a. and that a, is higher than a,, we write either a, i a, or a, f a,. A simply ordered assemblage that is further so arranged that it has an ele ment of lowest rank, a first element, and that every part of it has a first element, is said to be well-ordered. For example, the assemblage of rational fractions greater than zero and less than one, if arranged in natural order, so that the larger the fraction the higher its rank, is simply ordered but not well-ordered. The same assemblage can, however, be well ordered, thus: 4, 1, 1, 4, 1, 1, 1, 1, . . . , where the scheme is that El shall have lower rank q.
than 6 when p. + q, is less than p, + qs, and 92 if P. + q, = P3 + q., then the fraction having the smaller number for its numerator shall have the lower rank.
If A be a simply ordered assemblage, the new assemblage obtained by abstracting from the character of the elements of A is called the order-type of A and is denoted by A. Ob viously, A is simply ordered. If A and B are simply ordered, and if their elements can be paired in one-to-one fashion so that the rank relation of every two elements a, and a, of A shall be the same as the rank relation of their correspondents b, and b, in B, then A and B are said to be similar, and to be depictable on one another. These definitions of similar and depictable, it is noteworthy, are more restricted than that above given. The similarity of two similar simply ordered assemblages A and B is expressed by writing A If A is simply ordered, A^' A, and if B and C are simply ordered, and if Al`:C and B2iC, then B. It is plain, too, that either of the relations, A = B, implies the other. The definition of similarity given here is analogous to that of sameness of power in an earlier paragraph and can be rendered precise in the same man ner.
To every order-type, or ordinal number, corresponds a power, or cardinal number.
Thus to A corresponds A. The distinction of ordinal and cardinal is of no importance for finite assemblages, but is absolutely indis pensable in the doctrine of transfinites. All order-types corresponding to a finite cardinal a are similar, but those corresponding to a transfinite cardinal present a countless variety and are said to constitute a type-class [ a . To every transfinite cardinal corresponds such a type-class. Any type-class is itself an assem blage, namely, of order-types, and as such has its own cardinal number, which may be shown to be greater than that of each of the order types involved.
Addition and Multiplication of Order If A and B are simply ordered, it will be understood that in their union (A, B) the elements of A have the same rank relation as in A, that the like is true of B, and that every a is of lower rank than every b. Hence (A, B) is simply ordered. If A' and B' are simply ordered and if AEI' and B'" B', then (A, B)I (A', B'). Hence the order-type of (A, B) depends only on a =A and 13=B. Hence the definition of addition: a (A, B). Here a is the augend and ,8 the addend. If a, a, y be any three types, a + (i9 (a + 8) y; i.e., addition of ordinals is ciative; but, unlike cardinals, ordinals do not in general obey the commutative law. For example, if u= E where E denotes e,, e,, . . . , . . . fe,le,+,], and if f be any new ele ment, then 1 + u does not equal w +1, for (f, E) and (E, f) are not similar the latter having a last element, while the former has not.