Can the infinite be tamed? Many people say that the human mind cannot comprehend the infinite. And from Zeno to Bertrand Russell, mathematicians and philosophers who have tried have been plunged into paradox. This course moves from philosophical perplexity about such paradoxes to mathematical theories that define different notions of infinity, compare infinite sets, and discern an infinite progression of distinct transfinite numbers. Whether these theories are coherent - and more than a fantasy - remains a matter of controversy. As a mathematical inquiry, this course inculcates techniques of abstraction, definition, proof and calculation. It also invites reading, discussion, and writing on some of the most fascinating and persistent of philosophical problems.