 An insomniac, counting imaginary sheep to try to get to sleep, will never run out of natural numbers to do the job: 1, 2, 3, …. There is no bound on the set of natural numbers. This is our first infinite set. It is perhaps a natural thought that there is just one infinity for counting infinite sets, which we might symbolise ‘\(\infty\)’. The thought may seem even more natural when we define an infinite set as one that has the same size (in a sense to be made precise shortly) as a proper subset of it. In fact, the thought could hardly be more mistaken: as we will soon see, according to mathematical orthodoxy—namely contemporary set theory and the attendant notion of cardinality of a set—there are infinitely many infinities. This prompts a series of questions: Is there a smallest one? Yes, as we will see. Is there a largest one? No, as we will see. What can be said about the spacing between the infinities and how far infinities extend? Well, we will see. We can also ask parallel questions about the infinitely small.