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

278 MATHEMATICAL PHILOSOPHY
zero and all positive rational numbers are Si limits of V;
that zero and nothing else is an S2 limit of V; and that
y has no Sa limit whatever. Two additional facts are
worth noting here: one of them is that, under D2 zero
and every positive real number is a limit of F; the other
is that, under D3, zero and every positive real number will
be an S limit of 7, if S's field be the class of positive reals
and zero, and if S arrange the terms of F in the order of
increasing magnitude.
I shall leave it to you to practise to your heart's content in applying D3 to such various variables and sequences
as you can readily find or devise.
Presently, I shall ask you to consider a fourth conception of limit. Before doing so, I wish to call your attention to a curious nice little dispute that now and then
arises respecting the notion of limit as defined by D3
or by a virtual equivalent of D3. The dispute arises out
of confusion due partly to the row method of indicating
sequences and partly to the custom of speaking figuratively of a variable as if it actually changed, varied,
increased, decreased, and so on, instead of merely representing " any one " of the terms of some specified class.
I can best present the matter by means of an example
or two. Consider the three sequences indicated by the
three number rows.
(rl):,...;
(r2): T,,, t,...; I
(r3):., t,,..; I
You note that the three sequences are distinct and that
their fields are distinct. If the ranges of the variables
FV, V2, F3 contain respectively the same terms as the
fields except I, you easily see that, under D3, I is a limit