When is a serial relation, we can greatly simplify the above definition of a limit. We can, in that case, define first the "boundary" of a class , i.e. its limits or maximum, and then proceed to distinguish the case where the boundary is the limit from the case where it is a maximum. For this purpose it is best to use the notion of "segment." We will speak of the "segment of defined by a class " as all those terms that have the relation to some one or more of the members of . This will be a segment in the sense defined [Pg 98] in Chapter VII.; indeed, every segment in the sense there defined is the segment defined by some class . If is serial, the segment defined by consists of all the terms that precede some term or other of . If has a maximum, the segment will be all the predecessors of the maximum. But if has no maximum, every member of precedes some other member of , and the whole of is therefore included in the segment defined by . Take, for example, the class consisting of the fractions i.e. of all fractions of the form for different finite values of . This series of fractions has no maximum, and it is clear that the segment which it defines (in the whole series of fractions in order of magnitude) is the class of all proper fractions. Or, again, consider the prime numbers, considered as a selection from the cardinals (finite and infinite) in order of magnitude. In this case the segment defined consists of all finite integers. Assuming that is serial, the "boundary" of a class will be the term (if it exists) whose predecessors are the segment defined by . A "maximum" of is a boundary which is a member of . An "upper limit" of is a boundary which is not a member of . If a class has no boundary, it has neither maximum nor limit. This is the case of an "irrational" Dedekind cut, or of what is called a "gap." Thus the "upper limit" of a set of terms with respect to a series is that term (if it exists) which comes after all the 's, but is such that every earlier term comes before some of the 's. We may define all the "upper limiting-points" of a set of terms as all those that are the upper limits of sets of terms chosen out of . We shall, of course, have to distinguish upper limiting-points from lower limiting-points. If we consider, for example, the series of ordinal numbers: [Pg 99] the upper limiting-points of the field of this series are those that have no immediate predecessors, i.e. The upper limiting-points of the field of this new series will be