FYI.

This story is over 5 years old.

Tech

The Hilbert Hotel: An Infinite Thought Experiment

A classic illustration of the mathematical problem of infinity.
Image: YouTube screen-grab

It's probably best not to think too hard too often about infinity. No matter that the concept reappears again and again in the real world, from signal processing to quantum field theory to cosmology, infinity isn't intuitive—a number that is not really a number as well-suited to describing the smallest points of reality and then much, much smaller to the largest. Infinity is uncountable, but also unavoidably real.

Advertisement

The Hilbert Hotel, an illustration first proposed by the mathematician David Hilbert, is a hotel with an infinite number of rooms. For every room number n there exists a room number n + 1. Simple enough, but what if that hotel happened to be booked solid, with every room full with an infinite number of guests. The hotel is full, but is also infinite, which means that we can't very well turn away a new guest. How do we handle that? How do we handle 40 new guests? Or an infinitely full bus of new guests? Or an infinity of infinitely full buses?

In the TED-ed lesson above, Jeff Dekofsky, a high school math teacher by day, offers some solutions, or at least some good ways of thinking about the problem.