Example
The set of odd whole numbers less than 1 000 000 is a countable set.
Educational Note
Students can easily confuse “finite set” and “countable set” because we associate a countable set with a set where we can count the elements. The confusion comes from the concept of “knowing how to count”. When we count objects, we associate each of them with a number: 1, 2, 3, 4, 5, etc., which means that between the first object and the next, we know we are not forgetting one. However, knowing how to count numbers does not mean that there is a finite number of them. With some patience (!) we can count an infinite amount of numbers.
Therefore, the sets of whole numbers and integers are countable; we call them infinite countable sets. The set of rational numbers is also countable, just like the set of decimal numbers, but this one is less obvious! On the other hand, the set of real numbers is not countable; it’s an infinite uncountable set. If we wanted to count them, starting from 0, we would not know how to find the next real number (meaning the one that comes right after it) without forgetting another number between 0 and this number, hence the impossibility of counting them.