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

INFINITY Sos
large; in the fraction n, let x be a variable representing
x
a real number; let us treat x as an infinitesimal-a variable having zero for its limit; as x grows smaller and
smaller, the function n grows larger and larger; if we
x
prescribe any finite amount-an amount, that is, such that
we can denote it or a larger one by a positive integer,
then, as x decreases towards zero, n will come to exceed
x
the prescribed amount, however large; we express this
obvious fact by saying that, for x approaching zero, n
x
approaches infinity or becomes infinite or is an infinite
variable or function; by such speech mathematicians do
not mean that there is a definite quantity called infinity
(oo ) and that the ratio becomes equal to it when x takes
the value zero; for when x takes this value, the indicated
division becomes meaningless and the ratio ceases to
exist; what the speech means-and it means nothing else,
-is, as said, that for x decreasing as indicated, the ratio
becomes larger than any prescribed finite amount. Such
is the conception of an infinite function or variable,a dynamic infinity, as we may call it to distinguish it from
the other,-the static infinity. The dynamic type has its
root in the question-how much? It is obvious that the
two conceptions, though radically distinct, are intimately
related. I shall leave it to you to compare them deeply
if you will.
The "new infinity," as it is sometimes called, means
the static infinity. It was introduced into mathematics
something more than a half-century ago by Bernhard
Bolzano, Richard Dedekind and Georg Cantor. Long