Any set which can be put in a one-to-one correspondence with the natural numbers (or
integers) so that a prescription can be given for identifying
its members one at a time is called a countably infinite (or denumerably infinite)
set. Once one countable set 
is given, any other set which can be put into a one-to-one
correspondence with 
is also countable. Countably infinite sets have cardinal
number aleph-0.
Examples of countable sets include the integers, algebraic numbers, and rational numbers. Georg Cantor showed that the number of real numbers is rigorously larger than a countably infinite set, and the postulate that this number, the so-called "continuum," is equal to aleph-1 is called the continuum hypothesis. Examples of nondenumerable sets include the real, complex, irrational, and transcendental numbers.
