Volume 1 / Part 5 — SUMMARY OF PART V.

A RELATION is said to be serial, or to generate a series, when it possesses three different properties, namely (1) being contained in diversity, (2) transitiveness, (3) connexity, i.e. the property that the relation or its converse holds between any two different members of its field. Thus is a serial relation if (1) , (2) , (3). The third characteristic, that of connexity, may be written more shortly i.e. , using the notation of *97; and this, in virtue of *97·23, is equivalent to

In virtue of *50·47, the first two characteristics are equivalent to When , we say that is "asymmetrical." Thus serial relations are such as are asymmetrical, transitive, and connected.

It might be thought that a serial relation need not be contained in diversity, since we commonly speak of series in which there are repetitions, i.e. in which an earlier term is identical with a later term. Thus, e.g. would be called a series of letters, although the letters and recur. But in all such cases, there is some means (in the above case, position in space) by which one occurrence of a given term is distinguished from another occurrence, and this will be found to mean that there is some other series (in the above case, the series of positions in a line) free from repetitions, with which our pseudo-series has a one-many correlation. Thus, in the above instance, we have a series of nine positions, which we may call which form a true series without repetitions; we have a one-many relation, that of occupying these positions, by means of which we distinguish occurrences of , the first occurrence being a as the correlate of 1, the second being as the correlate of 4. All series in which there are repetitions (which we may call pseudo-series) are thus obtained by correlation with true series, i.e, with series in which there is no repetition. That is to say, a pseudo-series has as its generating relation a relation of the form , where is a serial relation, and is a one-many relation whose converse domain contains the field of . Thus what we may call self-subsistent series must be series without repetitions, i.e. series whose generating relations are contained in diversity.

For our purposes, there is no use in distinguishing a series from its generating relation. A series is not a class, since it has a definite order, while a class has no order, but is capable of many orders (unless it contains only one term or none). The generating relation determines the order, and also the class of terms ordered, since this class is the field of the generating relation. Hence the generating relation completely determines the series, and may, for all mathematical purposes, be taken to be the series.

When is transitive, we have Hence all the propositions of Part II, Section E become greatly simplified when applied to series.

Also, since the field of a connected relation consists of a single family, a series has one first term or none, and one last term or none.

In the case of a serial relation , the relation (defined in *121·02) becomes , i.e. the relation "immediately preceding." In a discrete series, the terms in general immediately precede other terms. A compact series, on the contrary, is defined as one in which there are terms between any two: in such a series, .

It very frequently occurs that we wish to consider the relations of various series which are all contained in some one series; for example, we may wish to consider various series of real numbers, all arranged in order of magnitude. In such a case, if is the series in which all the others are contained, and , , , ... are the fields of the contained series, the contained series themselves are , , , .... Thus when series are given as contained in a given series, they are completely determined by their fields.

In what follows, Section A deals with the elementary properties of series, including maximum and minimum points, sequent points and limits.

Section B will deal with the theory of segments and kindred topics; in this section we shall define "Dedekindian" series, and shall prove the important proposition that the series of segments of a series is always Dedekindian, i.e. that every class of segments has either a maximum or a limit.

Section C, which stands outside the main developments of the book, is concerned with convergence and the limits of functions and the definition of a continuous function. Its purpose is to show how these notions can be expressed, and many of their properties established, in a much more general way than is usually done, and without assuming that the arguments or values of the functions concerned are either numerical or numerically measurable.

Section D will deal with "well-ordered" series, i.e. series in which every class containing members of the field has a first term. The properties of well-ordered series are many and important; most of them depend upon the fact that an extended variety of mathematical induction is possible in dealing with well-ordered series. The term "ordinal number" is confined by usage to the relation-number of a well-ordered series; ordinal numbers will also be considered in our fourth section.

Section E will deal with finite and infinite. We shall show that the distinction between "inductive" and "non-reflexive" does not arise in well-ordered series.

Section F will deal with "compact" series, i.e. series in which there is a term between any two, i.e. in which . In particular we shall consider "rational" series (i.e. series like the series of rationals in order of magnitude) and continuous series (i.e. series like the series of real numbers in order of magnitude). Our treatment of this subject will follow Cantor closely.