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

302 MATHEMATICAL PHILOSOPHY
the equivalence in question actually exists, you have now
merely to observe that we can associate the first fraction of the row with the integore i, the second with 2, and
so on, thus using each fractior4 and each integer once
and but once. Such an astonishihg result makes one
wonder whether every infinite- clssirs denumerable. The
answer is, No. It is well known-;t.hat the class of real
numbers,-even the class of thEèirrational numbers, even
the class of points in a microscopically short line-segment,
-is non-denumerable. Such classes are infinite but they
are infinite of higher order. it is known that infinities
rise above infinities in a summitless hierarchy. At present,
the denumerable grade and that of the real numbers are
the most important. In time to come such may not be the
case; no one knows enough to say.
Here we must make a distinction. A class is a multitude-not a magnitude such as length, for example, or
weight or area or volume or distance or the like; an infinite class is thus an infinite multitude; it has its root in
the question-how many? A class is not a variable in
the ordinary sense of this term,-it is a fixed thing,-a
constant,-a datum given once for all in the world of
thought-in logic the members of a class do not succeed
each other in time-they coexist; and so you see that an
infinite class is a static infinity. Long before this conception of infinity established itself in mathematics there was,
as there is now, another conception of infinity,-of a
sort of dynamic infinity,-namely, the conception of a
changing magnitude or function capable of growing to
exceed any given amount denotable by any integer however large. The idea may be conveyed as follows: let
n denote any given positive integer, no matter how