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

VARIABLES AND LIMITS 257
if the amounts of t's k-difference from the terms in R be
each more than null? If R be a finite class, the answer
is evidently no. If R be an infinite class, the answer
depends on k. Here we must make an important distinction. The difference-kind k may be such that, given any
two amounts of it, there are at most only a finite number
of intermediate amounts. Such a kind may be called a
discrete difference-kind. An example is the differencekind we have or should have in mind when, confining our
attention to zero and the positive integers, we talk of the
< differences " (strictly, the amounts of difference) found
by subtraction; of such amounts, the smallest is zero,
the next smallest is I, the next 2, and so on, and there are
no other amounts of the kind of difference we are here
dealing with. If we were talking of " differences,"amounts of difference-of (say) rational fractions, we
should have in mind a different kind of difference. As
in the foregoing example, so if k be any discrete differencekind, there is, as you readily see, an amount of the
k-difference next greater than the null of it. On the other
hand, a difference-kind may be such that, given any two
amounts of it, there is one amount (and hence infinitely
many amounts) intermediate to the given ones. Such a
kind may be called a compact or dense difference-kind. An
example is the difference-kind involved when, confining
our attention to the rational numbers, we say the amount
of difference of this fraction and that is so-and-so. It is
perfectly clear that no amount of dense difference-kind is
next greater than the null amount of it.
Let us now return to our query. If the k-difference
be discrete, the answer is negative, even though R be
infinite. For let d be the smallest amount of k-difference
except null; then no term t' of R is in the k-neighborhood