Countable set is a set in which each element can establish a one-to-one correspondence with each element of natural number set n. If each element of countable set is marked with its corresponding natural number symbol, then the elements of countable set can be arranged into an infinite sequence a 1, a2, a3, …an in the order of natural numbers.
For example, all positive even sets are countable sets, and all positive odd sets are countable sets. They can establish the following one-to-one correspondence with natural number set.
A definition of countable set set is "a set that can correspond to a subset of a set of natural numbers". In this sense, uncountable sets are called uncountable sets. This term was coined by Cantor. The elements of a countable set are "countable" as the name implies: although the counting may never end, each specific element in the set will correspond to a natural number. ?
The term "countable set" can also mean a set that can correspond to the natural number set itself one by one. The difference between the two definitions lies in whether a finite set is regarded as a countable set. In order to avoid ambiguity, the countable set in the former sense is sometimes called countable at most, and the countable set in the latter sense is also called infinite countable set.