## Thursday, September 24, 2015

### A New Twist on the Infinity Hotel Problem

9/24/2015

A New Twist on the Infinity Hotel Problem (Hilbert's Hotel):

 The Problem: Hilbert's paradox of the Grand Hotel is a veridical paradox (a valid argument with a seemingly absurd conclusion, as opposed to a falsidical paradox, which is a seemingly valid demonstration of an actual contradiction) about infinite sets meant to illustrate certain counterintuitive properties of infinite sets.  The idea was introduced by David Hilbert in a lecture he gave in 1924 and was popularized through George Gamow's 1947 book One Two Three... Infinity. Consider a hypothetical hotel (Infinity Hotel) with a countably infinite number of rooms, all of which are occupied.  One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms. Suppose a new guest arrives and wishes to be accommodated in the hotel. The Solution: We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from his current room n to room n+1.  After this, room 1 is empty and the new guest can be moved into that room.  By repeating this procedure, it is possible to make room for any finite number of new guests.

 The Problem – RE-THOUGHT: Consider a hypothetical hotel (Infinity Hotel) with a countably infinite number of rooms, all of which are occupied. A hotel with countably infinite number of rooms would have to be an infinitely large building.  The area of the Earth is a finite number of square feet, so even if we use every square foot of the surface of the Earth we could only squeeze a finite number of hotel rooms into the first floor.  So by necessity, the hotel must also be infinitely tall, and have an infinite number of floors. Guests would not be able to complete the move.  Guests could not climb an infinite number of stairs – they would die of old age or exhaustion or old age before they could complete the move.  Even if they used escalators or elevators it would still require an infinite amount of time to travel in them so death due to old age is still a problem. The previous solution in untenable in its assertion that the moves could be completed in one night. Besides the problem of time, there is an issue with population.  The hotel has an infinite number of rooms, but the entire population of the Earth, though large, is still a finite number.  It would be impossible for the hotel to be full and not have a vacancy. Finally, if the hotel were full, then the hotel across the hotel across the street would be empty.  Its vacancy light would be on and the new guests would have gone to that hotel first – and probably would have been offered a significant discount. QED

We’ll leave a light on for you,

David