Volume 1 / Part 5 — SECTION A. GENERAL THEORY OF SERIES.

Summary of Section A.

In the present section, we shall be concerned with the properties common to all series. Such properties, for the most part, are very simple, and present no difficulties of any kind. Many of the properties of series do not require all the three characteristics by which serial relations are defined, but only one or two of these properties: we therefore begin with numbers in which, though the properties proved derive their chief importance from their applicability to series, the hypotheses are only that the relations in question have one or two of the properties of serial relations. Thence we proceed to the most elementary properties peculiar to series, and thence to the theory of minimum and maximum members of classes contained in a series, and of the successors and limits of classes. We then proceed to the correlation of a series with part of itself. The ground covered is familiar, and the difficulties encountered are less than in most previous sections.

It will be observed that where series are concerned, if is an existent class contained in , is correlative to (which is ): is "predecessors of some ," and is "successors of all 's." If is an existent class contained in , the whole of , with the exception of the last term of (if there is such a term), belongs to one or other of the classes , , of which the first wholly precedes the second. The division of into these two classes is the Dedekind "cut" defined by . But when only part of is contained in , we must replace by , since if has any member not belonging to . Again, if , we have . But what we want is the complement to , which in this case is null. Hence we must replace by : this is when , i.e. when . In any other event it is equal to . If is contained in and is not null, . Thus the Dedekind "cut" defined by a class , whether or not this class is contained in whole or part in , is always the two classes

Throughout the elementary propositions of this section, we have been careful to avoid stronger hypotheses than are required: we have not assumed to be serial, if our conclusion would follow (e.g.) from the hypothesis that is transitive and connected. It will be found that many properties of series depend upon the fact that, if , are two different terms of a series , then (*204·3). Here the implication requires that should be asymmetrical, i.e. that we should have , or . The implication requires that should be connected. Thus the hypothesis required is not that should be serial, but that should be connected and asymmetrical (*202·5).

Again, consider the proposition that if is a series, . This relation is the very useful relation "immediately preceding"; thus the above proposition is important, as is the further proposition that if is a series, is a one-one relation. It will be remembered that (by *121) "" means that consists of two terms. It was shown in *121·304·305 that if is contained in diversity, "" implies "" and is equivalent to the statement that and constitute the whole interval and are not identical. Also by *121·254, . It is evident that, if is contained in diversity, and , we cannot have , because there is no term other than and in the interval , and we cannot have or . Hence if , we have . Hence by what was said above (*121·305), if , we shall have . On the other hand, if is transitive, we have (*201·61). Combining these two facts, and remembering that if is transitive, (*201·18), we find that if is transitive and contained in diversity. We find further (*202·7) that if is connected, is one-one. Hence we need the full hypothesis that is a series in order to prove that is a one-one (*204·7). This is a good example of the way in which the various separate characteristics that make up the definition of series are relevant in proving the properties of series.