Countable - Remarkably It Works., Are All Infinite Sets Countable?

Every set that can be counted is countable, but this is no surprise. The interesting case for countable sets comes when we abandon finite sets and consider infinite ones.

An infinite set of numbers, points, or other elements is said to be "countable" (also called denumerable) if its elements can be paired one-to-one with the natural numbers, 1, 2, 3, etc. The term countable is somewhat misleading because, of course, it is humanly impossible actually to count infinitely many things.

The set of even numbers is an example of a countable set, as the pairing in Table 1 shows.

Of course, it is not enough to show the way the first eight numbers are to be paired. One must show that no matter how far one goes along the list of even natural numbers there is a natural number paired with it. In this case this is an easy thing to do. One simply pairs any even number 2n with the natural number n.

What about the set of integers? One might guess that it is uncountable because the set of natural numbers is a proper subset of it. Consider the pairing in Table 2.