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

262 MATIHEMATICAL PHILOSOPHY
neighborhood, however small, of any given real number,
there are real numbers whose numerical difference from
the given number is more than zero. If we are to talk of
(S) as having or not having a limit, we must indicate
what we are going to mean by " difference " of geometric
series as such. This we may do as follows: let (ai, rl)
be a given pair of real numbers, and let d and d' be given
positive numbers; we agree to say that the neighborhood
d of ai, and the neighborhood d' of r1, together determine a
neighborhood (d, d') of the series al +airl +alrl2+..,
and that, if a2 be in the first neighborhood and r2 in the second, then and only then the series a2+a2r2+a2r22+ +.,
is in the third. Now, as we saw a moment ago, ai is a
limit of a, and ri of r; hence, the series ai +air1 +
aiir2+... is a limit of a+ar+ar2+..., that is, of
(S); hence, every series in the range of (S) is a limit of (S).
You will recall that in (i I) of our little list of variables,
p is a variable whose range is the class of all the true
propositions having a given propositional function p(x)
for their common matrix. Can we associate the notion
of limit with p? We can, as follows: Let F be a variable
whose range R is the class of verifiers of (x); denote the
range of p by R'; if xi be a given term in R, then the
proposition ()(xi) is a definite term in R'-to each R
term there thus corresponds an R' term, and conversely;
let k be a suitable difference-kind for the R terms; it will
evidently be a suitable difference-kind for the R' terms,
for, if and only if the amount of k-difference between the
R terms xi and x2 be null, the corresponding R' terms
4(xi), c(x2) are identical propositions-indistinguishable
with reference to k; we will regard the amount of k-differences between xl and X2 as the measure of k-difference
between o(xi) and O(x2); let t be a term (in R or not)