The infinite has been a perennial source of mathematical and philosophical wonder, in part because of its enormity—anything that large is grand, and provokes awe and contemplation—and in part because of the paradoxes like Galileo’s. Infinity seems impossible to tame intellectually, and to bring within the confines of human understanding. I will argue, however, that Cantor has tamed it. The good news is that Cantor’s mathematics makes infinity clear and consistent but does nothing to reduce the awe-inspiring grandeur of it.