# continuum hypothesis

In 1874 Georg Cantor discovered that there is
more than one level of infinity. The lowest
level is called **countable infinity**; higher levels are known
as **uncountable infinities**. The natural
numbers are an example of a countably infinite set and the real
numbers are an example of an uncountably infinite set. The continuum
hypothesis (CH), put forward by Cantor in 1877, says that the number of
real numbers is the next level of infinity above countable infinity. It
is called the continuum hypothesis because the real numbers are used to
represent a linear continuum. Let *c* be the cardinality of (i.e.,
number of points in) a continuum, aleph-null,
be the cardinality of any countably infinite set, and aleph-one be the next
level of infinity above aleph-null. CH is equivalent to saying that there
is no cardinal number between aleph-null
and *c*, and that *c* = aleph-one. CH has been, and continues
to be, one of the most hotly pursued problems in mathematics.