A countable set is a collection of items that can be matched one-to-one with the set of natural numbers 0, 1, 2, .... This means that the elements of the set can be listed in a sequence, even if the set is infinite. Examples of countable sets include the set of integers and the set of rational numbers. In contrast, uncountable sets cannot be matched with natural numbers. A famous example of an uncountable set is the set of real numbers, which includes all the numbers on the number line. Countable sets are important in mathematics, particularly in set theory and analysis.