I begin by exploring the old adage that ‘the Greeks abhorred the infinite’, and show why and how they did so. I focus in particular on Pythagoras and his followers, who divided the world into two fundamental cosmic principles: what they called Peras (the Limited, or the Finite) and Apeiron (the Unlimited, or the Infinite). I also consider Zeno of Elea, who, after wrestling with the notion of infinity and discovering various associated paradoxes involving motion, came to the conclusion that motion itself was impossible. But the primary focus of this lecture is Aristotle, who sought a reconciliation between the idea that things can go on for ever and the Greeks’ abhorrence of any such idea. Aristotle famously distinguished between the potential infinite and the actual infinite, and argued that the former, and the former alone, was the acceptable face of infinity. Aristotle’s views held sway for thousands of years. In particular, they had a very significant impact on medieval thought. This was largely through the intermediary St Thomas Aquinas, who attempted to reconcile Aristotle’s teachings on the infinite with the doctrines of the Catholic church. He enjoyed some success in this endeavour, and, once the church had embraced Aristotle’s teachings as the new orthodoxy, philosophers stepped out of line at their peril. I shall signal in particular Galilei Galileo, who did just that. He dared to add some new paradoxes of his own to discussion of the infinite: these foreshadowed later thinking on the infinite (as I shall discuss in Lecture Three) and showed that Aristotle’s views were not as straightforward as they had come to seem.