Hilbert's paradox of the Grand Hotel - An easier solution?

  • B
  • Thread starter Lars Krogh-Stea
  • Start date
  • Tags
    Paradox
In summary, the paradox is about a hypothetical hotel with infinite rooms and guests, and the question of how new people can arrive when the hotel is already full. Some proposed solutions involve complicated shuffling of guests, but a simpler solution could be to have all the existing guests go outside and then come back in to choose a room from the infinite options. The paradox is often presented as a riddle because of its connection to Cantor's theory of transfinite numbers, which was controversial at the time. Some people may not realize that a bus with one person for every real number would not fit in the hotel.
  • #71
Lars Krogh-Stea said:
I'm not saying that.. I'm saying the digit 1 with infinitely many zeroes before it can be read as a binary number that you can map to room number 1. The digits 11 with infinitely many zeroes before it can be read as a binary number that you can map to room number 3.. And so on. You have already answered yes to the question, when I asked if these combinations would occur in the list. Thats why I said that if you mirror the list, so it reads from right to left, then it's possible to map the passengers to their rooms.

I think most conceptions of an infinite string of 1s and 0s does not have any such thing as infinite 0s, and then a 1. There are actual interesting ways to think about ordering in which you could imagine the set of all strings of infinitely digits and then one extra digit, but the canonical representation there is no "last digit".
 
Mathematics news on Phys.org
  • #72
Office_Shredder said:
I think most conceptions of an infinite string of 1s and 0s does not have any such thing as infinite 0s, and then a 1. There are actual interesting ways to think about ordering in which you could imagine the set of all strings of infinitely digits and then one extra digit, but the canonical representation there is no "last digit".
I agree, but there is a first number. That's why I say "mirror the list".

Example: 100...0 will turn into 0...001. And for this practical application of mapping people to rooms, I see that as sufficient data to get the job done.
 
  • Skeptical
Likes weirdoguy
  • #73
valenumr said:
You're still trying to interpret the stings of ones and zeros as binary numbers. They are not. They are instances of members of a set. There is a mathematical study of boolean logic where such thoughts apply, but it is not relevant here.
If it works, it should be relevant..
 
  • Skeptical
Likes weirdoguy
  • #74
Lars Krogh-Stea said:
Example: 100...0 will turn into 0...001.
Bad example. Where is 100...0 on the number line?
And why not write the latter as just plain 1?

I think we've beaten this horse to death, so I'm closing this thread.
 
  • Like
Likes weirdoguy and PeroK
Back
Top