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

280 MATHEMATICAL PHILOSOPHY
of (r3) and of (r). But (r) indicates that F contains a
host of I's-the " inserted " I's; but if these are in F,
they are S predecessors and successors, and we have ISi
contrary to the definition of a sequence. You see that
(r) neither is, nor property indicates, a sequence. (It is
of course possible to define the terms "sequence" and
"limit " so that a sequence having a limit may be such
that the variable in running towards the limit reaches it
one or more times.) Here is a good place to emphasize
the fact that the field of a sequence never contains two
identical terms. Why not? Because a field is a class,
and a class contains all and only the verifiers of some
propositional function, say, p(x); if xi be a verifier of
p(x), then Xi is a term or member of the class; it is
evident that as such a member, it occurs but once. We
do indeed often speak (unprecisely) as if such were not the
case; but when we speak of a and a' as being identical
members of a class, we mean that a and a' are two different symbols for one and the same member of the class
and we do not mean that the two symbols are themselves
members of the class.
Serial (Ordinal) Definition of the Term " Limit."-We
have now before us three definitions —rDi, D2, Da-of the
term. It is important to observe that each of them
essentially involves the notion of quantity; they involve it,
for they involve the notion of the neighborhood of a term,
and this notion is quantitative; a given neighborhood
has a size; another one is larger or smaller; neighborhoods are among the things differing from one another in
respect of magnitude-quartity is of their essence. We
should not fail to observe, too, that, while the three
definitions thus agree in involving the notion of quantity, Da involves also the notion of a sequence or series,