In this chapter and the next, we consider what earlier thinkers have said about when an infinite series, or other form of infinity, is impossible. There are three main views to consider: (i) that an infinite series can never be completed by traversing the steps in succession, (ii) that a condition cannot obtain if it has an infinite series of preconditions, and (iii) that there can exist only potential infinities, never an actual infinity. These theories are not mutually exclusive; one could endorse two or even all three of these views at once, as I believe thinkers about the infinite have often done. Nevertheless, I shall discuss the three ideas separately, the first two in this chapter and the third (which requires the most extended discussion) in the next chapter.