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.