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

276 MATHEMATICAL PHILOSOPHY
diffêrence.kind appropriate for comparing (as by subtraction) the values of real numbers. Let V be the
variable whose range is F. It is easy to see that i and -I
are both of them limits of F under D2 and that neither of
them is an S limit of Y under D3. For choose a neighborhood d of I, no matter how small; plainly there is an F
term (a positive number in the row, but not a negative
one) differing from I numerically by less than d, and
such a term is in the chosen neighborhood; accordingly
I is, by D2, a limit of F; but I is not an S limit of V for i
is not even a term of S's field. Like reasoning would show
that - is a limit of F under D2 but is not, under D3,
an S limit of F, where S is the sequence above indicated.
We have seen that, if two sequences S and S' have a
common field F and if F be a variable whose range R
is a part of F, a term t may be an S limit of V without
being an S' limit of F. This fact is so important that it
seems advisable to give it further exemplification. Let F
be the class of all the positive rational numbers and zero.
Consider the following sequences Si, S2, Sa, having F
for their common field.
Si is to be such that, if xSiy, x and y are in F and x
is less than y; and such that, if x and y be in F, then
xSiy if and only if x is less than y. We commonly say
that Si as defined arranges the terms of F in the order
of increasing magnitude.
To define S2 consider the row
(a),...; O, -,,,...;,,,...;...;...
You observe that the row contains all and only the terms
of F; S2 is to be such that, if xS2y, x is on the left of y in
the row, and that any term in the row is an S2 predecessor
of every number on its right.