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.
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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
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We’ll leave a light on for you,
David
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