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

282 MATHEMATICAL PHILOSOPHY
of F be the numbers in the row; let the terms of "'s
range R be the row's integers. You see that 4 is an Si
limit of V and that ~ is not. Why not? Notice that 2
is not an Si limit of F under D3 nor a limit of V under
D2. If a term be a limit of a F under D3 or D2, must it
be a limit under D4? If, in the foregoing example, we
suppose the R to include 1 (besides the integers), will the
new V have an S limit? Why not? If we strike out
4 and 4 and suppose R to coincide with the new F (of the
new Si), will V have an S1 limit? Why not?
A C D
FIG. 28.
For other examples consider the following. Let S2
be a sequence having for its F the points of the line
segment AD, including A and D. Let us take predecessorsuccessor to mean, as before, left-right; let the R of V
be composed of A and the other points preceding B; you
see that all the points in R and the point B but no other
F points are S2 limits of F. Notice that the same would
be true if we supposed R to include B. Suppose the F
of S2 to be composed of B, B's predecessors, C and C's
successors; and, as before, let the terms of R be B's
predecessors; you readily see that the S2 limits of V are
B, B's predecessors and C; that all of these except C are
S2 limits of V under D3 and limits of J under D2; and
that, under D4, C is not an S2 limit if R include B (as well
as its predecessors). Why not? Next suppose the F of
S2 to consist of all the points of AD except B and C, and
let the terms of R be the midpoint A' of AB, the midpoint
of A'B, and so on, the midpoint B' of BC, the midpoint of
B'C, and so on, the midpoint C' of CD, the midpoint of
C'D, and so so; then show that D is the only S2 limit of