This argument proves there are different sizes of infinity. Actually, there are infinitely many sizes of infinities bigger than that of $\mathbb{N}$. The argument is as follows. If you can prove you cannot construct a list of all the real numbers, it means the set of real numbers has to be a bigger infinity than the natural numbers. Let's try to make a list of all real numbers between $0$ and $1$. We'll assume, for the sake of contradiction, that we can. Our list might start like this: 1. $0,141592$ 2. $0,500000$ 3. $0,333333$ 4. $0,718281$ 5. etc. Cantor realized that even if this list goes on forever, you can always construct a new number that is guaranteed not to be on this list, using the following method: 1. Take the first digit of the first number. Add $1$ to it. 2. Take the second digit of the second number. Add $1$ to it. 3. Take the third digit of the third number. Add $1$ to it. 4. Take the fourth digit of the fourth number. Add $1$ to it. 5. etc. Is this new number on the list? It cannot be the first number, because it differs in the first decimal place. It cannot be the second either because it differs in the second decimal place, and so on. You can *always* construct a number that is not on the list. Therefore, we cannot create a complete list of all the real numbers. Therefore, the set of real numbers is an infinity bigger than that of the natural numbers. This is the diagonal argument (because you pick numbers in a diagonal pattern). Sets that can hold more elements than other infinite sets are called *uncountable sets*, the size of which is *uncountable infinity*. On the other hand, $\mathbb{N}$ is an infinite countable set, the size of which is countable infinity. The Cantor Set is famous example of an uncountable set.