Many of the most interesting problems involving the infinite concern the structure of space and time. Some of these problems involve large infinities: Could there have been a countably infinite number of past days? Could there be a countably infinite number of future days? Could there be no upper bound to the length of straight lines through space? Other problems involve both large and small infinities: Can we make sense of the suggestion that space is composed of points? Can we make sense of the suggestion that there is no lower limit to the size of spatial volumes and spatial intervals? Can we make sense of the suggestion that time is composed of instants? Can we make sense of the suggestion that there is no lower limit to the size of temporal intervals?

Questions such as these have bothered philosophers since the dawn of philosophy. We shall begin our discussion with some puzzles that are due to Zeno of Elea, the pupil and principal defender of Parmenides of Elea. (Note that Zeno of Elea should not be confused with Zeno of Citium, the founder of Stoic philosophy. Zeno of Elea was born around 490 b.c. Parmenides was born around 515 b.c.) While the puzzles that have come down to us from Zeno seem to admit of satisfactory answers, they raise – or, at any rate, suggest – quite deep questions about geometry and topology that are not at all easy to settle.