Suppose we have two finite sets. We have developed enough machinery to tell when one set has at least as many elements than another. But what about infinite sets? For example, we might consider \(\mathbb {N}\) and \(\mathbb {Z}^+\) , and we may ask which one has more elements. Well, we have already developed mathematical concepts that convince us that these two sets have the same number of elements. In this chapter, we investigate the situation for general infinite sets. We will see that there are infinitely many infinite sets, no two of which are equivalent! In a spotlight on The Continuum Hypothesis we learn about the problems Cantor faced and the reception his work received. We’ll see how one of his main questions, whether there is a set of size strictly between the size of \(\mathbb {N}\) and that of \(\mathbb {R}\) , was eventually resolved. In fact, there’s still much research activity in this area today.