Joel David Hamkins, Professor of Logic, Oxford University This lecture is based on chapter 3 of my book, Lectures on the Philosophy of Mathematics, published with MIT Press, https://mitpress.mit.edu/books/lectures-philosophy-mathematics. Lecture 3. Infinity We shall follow the allegory of Hilbert’s hotel and the paradox of Galileo to the equinumerosity relation and the notion of countability. Cantor’s diagonal arguments, meanwhile, reveal uncountability and a vast hierarchy of different orders of infinity; some arguments give rise to the distinction between constructive and nonconstructive proof. Zeno’s paradox highlights classical ideas on potential versus actual infinity. Furthermore, we shall count into the transfinite ordinals. See lecture course information, including the schedule of topics, at http://jdh.hamkins.org/lectures-on-the-philosophy-of-mathematics-oxford-mt20.