Volume 1 / Part 5 — SECTION B. ON SECTIONS, SEGMENTS, STRETCHES, AND DERIVATIVES.

In this section, our chief topic will be sections and segments. This topic will occupy *211, *212 and *213, and *210 will consist of propositions whose chief utility lies in their application to segments. In *214, we shall consider Dedekindian series, which are intimately connected with segments, owing to the fact that one of the chief propositions in the subject is that the series of segments of a series is Dedekindian. In *215, we shall consider "stretches," which consist of any consecutive piece of a series, and are constituted by the product of an upper and lower section. Finally, in *216, we shall consider the derivative of a series, or of a class contained in a series: the former is the series of limit-points of the series, i.e. , the latter is the class of limits of existent sub-classes of , i.e. .

A class is called a section of when it is contained in , and contains all the predecessors of its members, i.e. is a section of if . Thus a section consists of all the field up to a certain point. It may consist of all the predecessors of , i.e. it may be of the form ; or again, it may consist of these together with , in which case it is of the form ; or again, it may be not definable by means of a single sequent or maximum, but be of the form , where is a class without a limit or maximum. The class of sections of is denoted by . A section of will be called an "upper section" of .

The idea of a segment is slightly less general than that of a section. We define a segment of as any class of the form , i.e. as any member of . Provided is transitive, segments are contained among sections. But even in a series sections are not, in general, contained among segments: if is a series, and if is a member of which has no immediate successor, will be a section but not a segment.

If a segment has a maximum, it must also have a sequent. Segments which have no maximum form a specially important class of segments: these are classes such that ; they form the class .

The properties of sections and segments considered as classes of classes are many and various: they are considered in *211. In *212, we pass to the consideration of the series of sections and segments. These series are and (cf. *170). The series of such segments as have no maximum is . We put It then appears that so that it is unnecessary to introduce a special notation for the series of sections.

Whenever is connected and transitive, turns out to be equivalent to logical inclusion combined with diversity (with the field limited to ). That is to say (*212·23), Hence it follows (*212·24) that We have also (*211·6·17) Hence it easily follows that whenever is connected, is a series. Similarly will be a series if is transitive and connected.

The fact of connection, which is required in order that or may be a series, results from In order to deal with such cases generally, we study, in a preliminary number (*210), the consequences to be deduced from the hypothesis We find that, with this hypothesis, putting if (*210·13), and thus in the same circumstances is a series (*210·14).

The interesting point about such series is their behaviour with regard to limits. Assuming that is not a unit class (so as to insure , if is any sub-class of , the logical product is the minimum of if it is a member of (*210·21), and the lower limit of if it is a member of but not of (*210·23). Similarly is the maximum of if it is a member of (*210·211), and the upper limit of if it is not a member of but is a member of (*210·231). Thus if is such that, whenever , we have , it follows that every sub-class of has either a maximum or a limit, i.e. the series is Dedekindian. Now each of the three classes , , verifies this condition, i.e. the sum of any sub-class of any one of these classes belongs to the class in question (*211·63·64·65). (This holds without any hypothesis as to .) Hence we arrive at the result that (i.e. the series of sections) is a Dedekindian series whenever is connected and is not null (*214·32), while (i.e. the series of segments) is a Dedekindian series whenever is transitive and connected and not null (*214·33), and (the series of segments having no maximum) is a Dedekindian series whenever it exists and is connected (*214·34). These propositions are important, and are the source of much of the utility of sections and segments.

For many purposes, especially in ordinal arithmetic, it is necessary to consider sections not as classes, but as series. That is to say, if is a member of , we want to deal with rather than with . The series of all such terms as might be supposed to be . But here a limitation is necessary owing to the fact that, if exists, and are both sections, and and are both , so that will be a relation which will have to itself. In order to avoid this, we first exclude from the sections to be considered, and thus put Then is the series of segments considered as series. Provided is a series, the relation holds between any two members and of its field when, and only when, . The subject of is considered in *213; the utility of the propositions of this number will not appear until we come to ordinal arithmetic.

The subject of Dedekindian relations is next considered (*214). We define a Dedekindian relation as one such that every class has either a maximum or a sequent. A Dedekindian series must have a first and a last term, since the first term must be the sequent of , and the last must be the maximum of the field. A Dedekindian series may be discrete, or compact (i.e. such that there is a term between any two, i.e. such that ), or partly one and partly the other. A finite series must be Dedekindian: a well-ordered series is Dedekindian if it has a last term. But the chief importance of the Dedekindian property is in connection with compact series. A compact Dedekindian series is said to possess "Dedekindian continuity"; such series have many important properties. They are a wider class than series possessing Cantorian continuity; these latter will be considered in Section F of this Part.