This is the title of a talk given at the Harvard mathtable on Mar. 27, 2018.
It is a little known fact that there is a subway line connecting Logan airport to the Hilbert hotel. This line has infinitely-many, and in fact uncountably-many, stops. At each station, exactly one passenger gets off (if the train is not empty), and a countable infinity of passengers get on board. How many passengers will eventually get to the Hilbert hotel?
In this talk, we will make the question precise and find the surprising answer to this puzzle! This will be done by sketching the development of a rigorous theory of "infinite numbers", called ordinals, that go beyond the natural numbers while keeping many of their properties. Time permitting, we will investigate connections with more classical problems, such as generating Borel sets of reals and studying the automorphism tower of a group.