The infinite subway paradox

The first in a series of essays on the infinite subway paradox

Every morning in the island city of Infinitopolis, a train begins at the downtown terminal, station 0, and then proceeds uptown to station 1, station 2, station 3, and so forth—there is a station for every natural number—until finally arriving at the uptown terminus, station ω.

Different size trains are used on different days, with some able to accommodate only a fixed finite number of passengers in total, while others are finite but extensible, having a stretching capacity to accommodate any desired finite number of passengers, although not infinitely many at once. Fortunately, when these expandable trains run there are special helpers at every station wearing clean white linen gloves, tasked when necessary with gently pushing the boarding passengers onto the train, taking advantage of the stretchy extensible nature of the trains. Meanwhile, on other days there are larger trains able to accommodate an actually infinite collection of passengers at once, although some can hold only a countable infinity of passengers, while others hold uncountable multitudes.

Please enjoy this series of essays on the infinite subway paradox. In this first essay, I shall introduce the main paradox, in elementary form, with passengers getting on and off the train at all finite stations—how many passengers will arrive at station ω? How many can arrive there? We shall exhibit various paradoxical outcomes. Subsequently, we shall explore a fuller range of paradoxical behavior, aiming ultimately to provide a necessary and sufficient condition on the finite disembarkment and embarkment counts that allow for a given arrival count at ω, with suitable passenger itineraries. After this, we shall extend the subway line further into the transfinite, realizing more sophisticated versions of the paradox, extending at first to higher countable ordinals but then ultimately to uncountable ordinals, where a surprising phenomenon emerges.

Let’s get started!

A catastrophic explosion at infinity…or a ghost train?

On a particular day recently, there was a peculiar pattern of passengers embarking and disembarking the train. Let me describe it.