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Hilbert’s Infinite Hotel

The following is a guest post by Science World Future Science Leader (FSL), Léal Makaroff, about a recent FSL session where the students stepped into the strange world of mathematics.

Among the various problems we discussed during the February 5 session of FSL Discover, the Hilbert's Infinite Hotel, by mathematician David Hilbert, stood out as one of the most fascinating. This imaginative illustration of the properties of infinite sets describes a hotel with an endless number of rooms, labelled 1, 2, 3, and so on, some of which may be occupied and some vacant. While availability at Hilbert’s Infinite Hotel may vary, it often accommodates an infinite number of guests, creating problems that left us FSLers marvelling at the strange mathematics.

To begin, we considered a situation in which every room was full. This meant that behind every room number there was a guest, leaving no room for newcomers. However, due to the mindboggling properties of infinity, a way could be found to accommodate an extra person. Because there were an infinite number of rooms, all that had to be done was to shift each guest up one room. As the guest in room #1 moved to the second room, the second to the third, and so on, the first room became empty.

This solution blew our minds. How could one simply add a number to endless infinity? How could an empty room just appear like that? In the land of infinity, such an apparent contradiction is possible. Despite its defiance of our normal intuition, Hilbert’s Infinite Hotel served as a good example of how regular mathematical operations break down in the move to the mathematics of infinity.

How it works:

To accommodate newcomer A, in mathematical terms, we would move guest n to room n+1. Written as infinite sets, this means:

Before, we had a set of guests: 1, 2, 3, 4, 5, 6, 7...
After, we have a set of guests: A, 1, 2, 3, 4, 5, 6...

Successfully accommodating the newcomer!

Keep up to date with what our FSL students are doing, by checking out their student blogs.