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

~68 MATHEMATICAL PHILOSOPHY
sequence or linear order) is a relation R such that: (a) if
xRy, x and y are not the same; (b) if x and y are terms in
R's field, then xRy or else yRx; and (c) if xRy and yRz,
then xRz. In this discussion, let us use the shorter names,
sequence and series, for such an R, instead of the other
names. Evidently the above P is a specific instance of a
sequence or series. Consider. another instance, say P',
where P' is determined by the propositional function:
x is a positive integer greater than a positive integer y.
You see that P' is indeed a sequence. Notice that P and
P' are different sequences: for example, the couple (I, 2)
is a constituent of P but (2, i) is not, while (2, i) is a
constituent of P' but (I, 2) is not. Yet the field of P is
the same as the field of P'-namely, the class of positive
integers. You readily see that, if the field F of a given
sequence be infinite, there are infinitely many different
sequences having F for their field. It is plain that the
smallest class that can be the field of a sequence is a class
having two and only two members, say, a and b; even
in this case, there are two sequences having the field in
common; one of them consists of the couple (a, b), the
other of the couple (b, a). Let me, in passing, propose an
instructive little exercise. Given a class of three terms,
a, b and c, show that there are six sequences having the
class for field, that each sequence is a class of three couples,
and write down the couples for each case.
In our introductory study of sequences, or series, it is
desirable to learn something more of the subject's language; for as supersimians, we must chatter about the
subject, and as supersimian philosophers, we must try to
chatter intelligibly. If the relation R be a sequence we
say that the referents of R are predecessors-predecessors
for R, or R predecessors; that the relata of R are suc