Galileo's Paradox of Infinity

How can a part be the same size as the whole? Joel David Hamkins walks through Galileo's famous paradox, which reveals a deep tension between two seemingly obvious principles: that sets matched one-to-one must be the same size, and that the whole must be greater than any proper part. Beginning with a simple dinner party analogy for one-to-one correspondence, Hamkins traces the history of this puzzle through Aristotle's wheel paradox, Bolzano's geometric observations about arcs and line segments, and Galileo's original 1638 dialogue about natural numbers and perfect squares. Along the way, he shows how finite line segments can be matched point-for-point with infinite lines, and how circles of different sizes contain equally many points. The lecture concludes with the modern resolution: the Cantor-Hume principle and the Cantor-Schröder-Bernstein theorem, which together provide a coherent framework for comparing the sizes of infinite sets.