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

394 MATHEMATICAL PHILOSOPHY
many such sequences,-sequences (of rationals), that isthat approach perfectly definite somewhats as limits
which (limits), however, are not rational numbers. The
total ensemble of definites thus defined or definable is the
system of irrational numbers. And these, taken together
with the rational numbers from which they are thus
derived or derivable, are commonly said to constitute the
system of real numbers. This use ofthe terms " rational,"
" irrational " and "real," though dictionally somewhat
unfortunate, is historically justified. I need not say that
as employed in mathematics, these terms have completely
lost whatever metaphysical connotation they may once
have had: rational does not signify reasonable; nor
irrational, unreasonable; nor is a "real" number any
more real metaphysically than is any other sort of number.
Well,-to return from this cautionary digression,-it is
the real numbers that constitute the Grand Continuum.
Perhaps it were better to say that the system of real
numbers is the basal instance of the Grand Continuum
for other continua essentially like it are derived from it
as the model. You are aware that it is common to give
the name continuum to any segment of the Grand Continuum, where, by segment, I mean any two real numbers
together with all the numbers that lie between them in
value. Thus the numbers zero and I, with the real numbers between them, constitute a continuum of second
order. The most convenient and vivid example or
representation of such a continuum is the ensemble of
points constituting a straight line-segment as ordinarily
conceived-as conceived, that is, in such a way that by
taking an arbitrary point of the line for origin and employing an arbitrary unit of length or distance, a one-to-onê
correspondence can be set up between the points of the