Kurt is talking about finite numbers, even though they are large. You all need to know that there are much larger numbers.
Aleph-null (
) is by definition the cardinality ("size") of the set of all natural numbers, and (assuming, as usual, the axiom of choice) is the smallest of all infinite cardinalities. A set has cardinality
if and only if it is countably infinite, which is the case if and only if it can be put into a direct bijection, or "one-to-one correspondence", with the natural numbers. Such sets include the set of all integers. There are an infinite number of integers, but between any two successive integers there is no other integer.
The cardinality of the set of reals numbers is
. Between any two real numbers are an infinite number of real numbers. It follows from Zermelo-Fraenkel,
ZFC, set theory that the celebrated continuim hypothesis,
CH, is equivalent to the identity
CH is independent of ZFC: it can be neither proven nor disproven within the context of that axiom system. That it is consistent with ZFC was demonstrated by
Kurt G?del in 1940; that it is independent of ZFC was demonstrated by
Paul Cohen in 1963.
There are bigger numbers than aleph aero and aleph one -- there's aleph two, etc., etc.
I taught a course on this earlier this year.