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

MORE ABOUT LIMITS 281
a non-quantitative, purely ordinal notion,-and that D1
and D2 do not. I mean that D3 contemplates the variable's range as being the field (or a part of the field) of a
series, or sequence, and that D1 and D2 do not. As
ontologists you may no doubt contend that the terms of
any given class and hence the terms of any given variable's
range are, quite independently of our intention or will,
arranged once for all and eternally in every variety of
sequence of which they are capable. I am not disputing
the justice of that contention; conceding it to be just,
granting the eternal existence of all the sequences possible
for a given range, I am merely signalizing the fact that D1
and D2 disregard them each and all, and that D3 does not;
D3 regards the variable's range as an ordered class of
terms; D1 and D2, disregarding order, regard the variable's range as an orderless collection. We may say,
then, that D1 and D2 are quantitative definitions and that
D3 is mixed-both quantitative and serial.
It is natural to ask whether the term " limit " sometimes denotes a purely serial conception. The answer is
affirmative. The following definition presents such a
definition of the term.
D4: Let F be a variable whose range R is included in
the field F of a sequence (series) S; if an F term t be such
that, given any S predecessor t' of t among the R terms,
there is an R term between t' and t, or such that, given
any S successor t' of t among the R terms, there is an
R term between t and t', then t is an S limit of F.
Let us at once cite some simple examples. Consider
the sequence.
Si: I, 2, 3, 4,... (ad infinitum), -, 3.
Let predecessor-successor mean left-right; let the terms