You can see that there are infinitely many natural numbers 1, 2, 3, ...$1,2,3,\ldots $, and infinitely many real numbers, such as 0$0$, 1$1$, pi$\pi$, etc. But are these two infinities the same?
Well, suppose you have two sets of objects, e.g. people and horses, and you want to know if the number of objects in one set is the same as in the other. The simplest way is to find a way of corresponding the objects one-to-one. For instance, if you see a parade of people riding horses, you will know that there are as many people as there are horses, because there is such a one-to-one correspondence.
We say that an set with infinitely many things is "countablecountable," if we can find a one-to-one correspondence between the things in this set and the natural numbers.
E.g., the integers are countable: 1 <-> 0$1\leftrightarrow 0$, 2 <-> -1$2\leftrightarrow -1$, 3 <-> 1$3\leftrightarrow 1$, 4 <-> -2$4\leftrightarrow -2$, 5 <-> 2$5\leftrightarrow 2$, etc,. gives such a correspondence.
However, the set of real numbers is NOT countable! This was proven for the first time by Georg Cantor. Here is a proof using the so-called diagonal argument.
