Countability (Mathematics)
This is an old revision of this page, as edited by JacobB (talk | contribs) at 22:38, January 6, 2010. It may differ significantly from current revision.
In mathematics, a set is called countable if it can be numbered in such a way that every element will eventually receive a number.
Examples of Countable Sets
Obviously, the numbers 1, 2, 3, etc., are countable, but so are all the integers: 0 we call first, 1, second, -1, third, 2, fourth, -2, fifth, 3, sixth, and so on, going outwards. We're sure to hit every integer this way.
Somewhat surprisingly, the rational numbers are countable as well. See the picture below:
Examples of Uncountable Sets
The real numbers are not countable, nor is any set with positive Lebesgue measure. This is because any countable set of numbers can be completely contained within a set of arbitrarily small measure. A short proof follows: