This is the title of a talk I gave to undergraduates at the CMU Math Club on Sep. 26, 2012.
How high can you count ? In elementary school, we are taught that there are no good answer to that question. For every natural number n, there is a larger number, n + 1. An answer one often hears is "infinity". What would that mean, and would then "infinity + 1" be a larger number ? What about "infinity + infinity" ? In this talk, I will sketch the development of a rigorous theory of "infinitary numbers", called ordinals, that go beyond the natural numbers while keeping many of their properties. I will also try to make sense of the notion of "size" of an infinite set, and show that infinite sets can in a very precise sense have different sizes. This will lead to very interesting questions, such as Cantor's Continuum Hypothesis.