Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

514

SERIES

[PART V

a 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 S;P, where P is a serial
relation, and S is a one-many relation whose converse domain contains the
field of P. 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 P is transitive, we have
Ppo =P. * = P I C'rP.
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 P, the relation P, (defined in *121'02)
becomes P-_.2, 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, P, = A.
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 P is the series in which all the others are contained, and
a, 3, y,... are the fields of the contained series, the contained series themselves are P a, P /3, P C,.... 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.