 Up to this point, it appears that infinity has been domesticated. This appearance begins to be challenged in section   5, when we canvas some classic paradoxes and puzzles involving infinity. It reappears as both friend and foe in the following sections on some philosophically fecund applications of it. In sections   6 and   7, it is both central to the formulation of probability and decision theory, and the source of more conundrums; we discuss some putative solutions to them. Section   8 presents some problems concerning space and time, as well as some progress that has been made on them—Kant’s antinomies, a Zeno-style paradox concerning measure, developments in non-Euclidean geometries and relativistic cosmology, and in determining whether space is finite or infinite. We conclude in section   9, sanguine overall about our relationship with infinity.