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

258 MATHEMATICAL PHILOSOPHY
d of t for, if t' were in the neighborhood, then the amount
of k-difference of t and t' would be less than d, but there is
no such amount except null, and null is excluded by the
hypothesis of the query. On the other hand, if the
k-difference be dense, the answer is affirmative: there are
V's having the sort of limit required. For it is sufficient
that the range R be such that for some t not in R there is,
for any chosen k-neighborhood d of t, however small the
neighborhood, an R term in the neighborhood-that is,
an R term differing from t by more than null and less than
d; and the existence of such F's may be shown by letting
Y be the variable whose R has for its terms zero and all
ordinary fractions less than I; you note that i is a t not in
R; that in comparing the terms in question we employ
a dense difference-kind; that, however small a neighborhood of I be chosen, R has terms in the neighborhood;
and that I is, therefore, a limit of V. For another example
consider the variable x in (5) of our foregoing list of
variables; show that zero, which is not in the range, is a
limit of x. You will find it interesting and very instructive to examine all the variables of the cited list with a
view to ascertaining which of them have limits outside
their ranges and what the limits are.
The concept of a limit as defined by the definition D1
has, you see, some striking properties. JIt is, however,
too broad for certain highly useful purposes; for example,
it does not sufficiently discriminate variables among
themselves; according to it all variables (except null ones)
have limits, as we have seen, and every term in a variable'
range is a limit of the variable. It is obviously desirable
to classify variables with respect to the character or constitution of their ranges. Let us accordingly try a somewhat narrower definition D2 ofthe term limit.