THE conception of a "limit" is one of which the importance in
mathematics has been found continually greater than had been
thought. The whole of the differential and integral calculus,
indeed practically everything in higher mathematics, depends
upon limits. Formerly, it was supposed that infinitesimals were
involved in the foundations of these subjects, but Weierstrass
showed that this is an error : wherever infinitesimals were thought
to occur, what really occurs is a set of finite quantities having
zero for their lower limit. It used to be thought that "limit"
was an essentially quantitative notion, namely, the notion of a
quantity to which others approached nearer and nearer, so that
among those others there would be some differing by less than any
assigned quantity. But in fact the notion of "limit" is a purely
ordinal notion, not involving quantity at all (except by accident
when the series concerned happens to be quantitative). A given
point on a line may be the limit of a set of points on the line,
without its being necessary to bring in co-ordinates or measure-
ment or anything quantitative. The cardinal number א ₀ is the
limit (in the order of magnitude) of the cardinal numbers 1, 2,
3, . . . n, . . ., although the numerical difference between א ₀
and a finite cardinal is constant and infinite : from a quantitative
point of view, finite numbers get no nearer to א ₀ as they grow
larger. What makes א ₀, the limit of the finite numbers is the
fact that, in the series, it comes immediately after them, which
is an ordinal fact, not a quantitative fact.

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