 At this point, one may be tempted to shout three cheers—or perhaps infinitely many of them—for infinity. Indeed, one may get the impression that we can’t live without it. At the same time, there are various apparent problems with infinity, and it starts to look less congenial. As they pile up, one might get the impression that we can’t live with it. Infinity, as we shall see, gives rise to numerous paradoxes that have preoccupied philosophers for millennia. Any praise of infinity must be tempered with circumspection and caution.