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I was watching this video How An Infinite Hotel Ran Out Of Room, by Veritasium.

The video says that it is not possible to fit names made of infinite strings of $\{A,B\}$.

We know we can fit infinite people in the hotel, we also know that there solution that works and solution that don't work. For example if you have infinite buses with infinite people if we gave a room one bus at a time we would be full after the first bus, on the other hand if we make a matrix and translate the diagonal to a single line (as shown on the video) we can fit everyone.

Following this if we ignore the name and treat the set as a stack then you would fit all of them because it's then a infinite stack and we can use their hotel room number to identify them, would this not be a possible solution?

Ran G.
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Dev_anon101
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2 Answers2

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An infinite string is uniquely identified by a function $s:\mathbb{N}\rightarrow \{A,B\}$, such that $s(n)$ is the $n$'th character in the string.

The number of such functions is known to be: $|\{A,B\}|^{|\mathbb{N}|}=2^{\aleph_0}$.

Now, its also known that $2^{\aleph_0}>\aleph_0$, and hence there is no way to assign strings to hotel rooms, and more formally:

there is no one-to-one function between the set of all strings, (denoted $\{A,B\}^\omega$) and $\mathbb{N}$.

John L.
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nir shahar
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There is, in fact, a solution. Convert the name to a binary number such that A=1 and B=0.

Then, either subtract each person’s name from infinite 1111111 or just remove all 0s before the first 1. Either way, the set becomes all binary numbers from 0 -> infinity.

At that point, just add 1 and convert to decimal. Then you have all decimal numbers 1 -> infinity.

You can use the trick in Veritasium’s video to move all current occupants to double their number, then take all ABBA’s friends and take their name-based number • 2 - 1 to make a set of odd numbers 1 -> infinity and now they all have rooms based on their original names.

For my video about this: https://youtu.be/o1JXUKhUzh4

Wishbone8121
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