Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

MORE ABOUT LIMITS 271
terms of the field, the terms of the sequence (a class of
couples) being the couples in the class; and he will
ordinarily have you understand that a number on the left
of another in the row is a predecessor of the latter and that
the latter is a successor of the former. This usual row
method of indicating sequences has obvious advantages,itt is mechanical, spatial, visual, diagrammatic,-but it
has to be used with care if confusion and error are to be
avoided, for, as you already see and will further see, it
disguises some of the nicer logicalities involved. For
example, the mathemetician may indicate the sequence
P' (instead of P) by writing the foregoing row of numbers;
in this case, a number to be a predecessor of another
must be on the right of the latter instead of on the left of
it; you see that the notion left-right (or its like) is not
that of predecessor-successor; the former is spatial and
sensuous, the latter logical and supersensuous; in the case
of P, 3 is the predecessor of 4, not because 3 is on the left
of 4 in the row, but simply because 3P4; and in the case
of P', 4 is the predecessor of 3, not because the former is
on the latter's right, but because 4P'3; again P begins at
I, not because I begins the row of numbers, for, you see,
P' ends at I, despite the fact that the row begins at I.
Keeping such precautions in mind, we may often very
conveniently employ the row method of indicating or
representing sequences.
We are at length almost prepared for a certain new
definition of the term " limit "; but there remains to be
explained one further preliminary. It is a sfollows: If
xRy implies xR'y, the relation R is said to be included
in the relation R'; in other words, R is included in R' if
every couple (x, y) in the class of couples constituting R
is a couple in the class of couples constituting R'; if R