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

e74 MATHEMATICAL PHILOSOPHY
D3, zero (or null) for limit is called an infinitesimal-a
term of great importance in most branches of mathematics. We will return to it if time permits.
Let us choose another difference-kind and see what
happens when V, F and S have the same meanings as
above. Observe that the denominators of the F terms are
positive integers, for 0 may be written 0/n where n is a
positive integer. We may compare the F terms with sole
reference to the values of their denominators-with
reference, that is, to the difference-kind k' in respect of
which we compare the magnitudes of positive integers
as such. Of k' the only amounts d are: null, I, 2, 3, 4,...;
hence the smallest k'-neighborhood d of O is that for which
d= I; as no R term differs from 0 (0/n) by an amount
of difference-kind k' more than null and less than I, it is
seen that no R term t' differing by more than null of the
difference-kind k' from 0 is in the neighborhood of 0 for
which d ==; therefore, 0 is not an S k'-limit of V. And
so is justified the mention of k in D3.
We have just seen that, though a t be an S k-limit of F,
it may not be an S k'-limit of F if k and k' be not the
same. We may now show that, though a t be an S k-limit
of 7, it may not be an S' k-limit of F, if S and S' be
different sequences. Consider the numbers in the row
~, 4... (ad infinitum), -, O.
Let S' be a sequence having the class of these numbers
for its field and let S' be such that, if a is an S' predecessor
of b, then a and b are in the row, a on the left of b, and
that any number in the row is an S' predecessor of all the
numbers on its right and an S' successor of all the numbers
on its left. S', as you see, is now completely determined