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Monday, January 04, 2021

Hilbert’s Hotel: You’re Missing the Best Bit! | Math - Medium

Infinity, enumerating the rationals and elegant solutions to innocuous problems by Ollie Watkinson, Writing on Mathematics and Philosophy, published in published in Cantor’s Paradise.

Photo: Maxime Lebrun on Unsplash
“The infinite! No other question has ever moved so profoundly the spirit of man.” — David Hilbert.

ilbert’s hotel paradox is a beautiful demonstration of the countably infinite. A coach full of the natural numbers, stretching backwards forever, arrives at a hotel with endless rooms and a maths-wizz receptionist. After allocating every guest a room the receptionist returns to her desk only to find another set has arrived! A brief negotiation almost always allows the receptionist to rearrange the current guests, leaving room for the newest. The details of these theoretical negotiations can be both ingenious and infuriating. In building his hotel in 1924 David Hilbert created a brilliant framework for discussing the mathematics and philosophy of infinity.

As is often the case, this paradox is in fact anything but. It can be shown that any coach of numerical guests either can or cannot fit into the hotel, thereby making the set of guests either countably or uncountably infinite.

Many treatments of the Hilbert’s hotel ‘paradox’ have the following structure:...

The italicised at least should have given you a pretty good idea that this solution is flawed. The problem here is that we have mapped some of the rational guests to multiple rooms, the lucky people! Take for example the grid squares corresponding to fractions 1/2 and 2/4 respectively. Being equivalent, both fractions only refer to one guest. We have given this guest both rooms. In fact, given that there is an infinite number of fractions equivalent to every other fraction our receptionist has unwittingly allocated each guest an infinite number of rooms! To satisfy the criteria of a bijection (and the hotel budget) each guest must be allocated one and only one room.

A trivial way of overcoming this hurdle is to simply skip the fractions in the grid that are equivalent to an already-roomed one. Whilst this works, it is not only untidy but makes it difficult to generalise the result to other areas of interest.

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Source: Medium