I’ve always hated this supposed paradox, because how is it possible for a hotel with infinite rooms to be full? Even with infinite guests, there will always be room for more. Because, you know, there are infinite rooms.
It’s not always discussed as such, but Hilbert’s Hotel is a mathematically well defined topic and can be proved rigorously. An infinite set of rooms can be the set x1, x2,…xinf, and people can be y1, y2…yinf. You can pair every entry in these two sets. x1&y1, x2&y2,…xinf&yinf. You can’t number a room without having a person in the room, and you can’t find a person who doesn’t have a room.
The hotel should really just fill its even numbers rooms first, then if another person turned up after you already have an infinity of guests, you just house this start of a new Infinity in room 1 etc
What if they filled in multiples of 3? They get vacant space for infinitr people twice. Well just fill people as multiple of infinity, so that infinite times infinite people can fill it in
I’ve always hated this supposed paradox, because how is it possible for a hotel with infinite rooms to be full? Even with infinite guests, there will always be room for more. Because, you know, there are infinite rooms.
It has to do with countably infinite sets.
The analysis on Wikipedia does a better job of explaining the concept: https://en.m.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel#Analysis
The whole point is that it’s something we can prove mathematically that is highly unintuitive.
It’s mostly just a way of communicating the bizarre nature of infinite series and other problems related to infinity. Just fun thought experiments.
It’s not always discussed as such, but Hilbert’s Hotel is a mathematically well defined topic and can be proved rigorously. An infinite set of rooms can be the set x1, x2,…xinf, and people can be y1, y2…yinf. You can pair every entry in these two sets. x1&y1, x2&y2,…xinf&yinf. You can’t number a room without having a person in the room, and you can’t find a person who doesn’t have a room.
If you have n rooms and n guests there are no empty rooms. Let n go to infinity and there should still be no empty rooms.
The trick of the Hilbert hotel is that if you add a guest to a hotel with countably infinite guests the number of guests does not change.
If there is no vacancy then the hotel has infinite full rooms and no empty ones.
The hotel should really just fill its even numbers rooms first, then if another person turned up after you already have an infinity of guests, you just house this start of a new Infinity in room 1 etc
What if they filled in multiples of 3? They get vacant space for infinitr people twice. Well just fill people as multiple of infinity, so that infinite times infinite people can fill it in
Let’s leave some space for all the types of infinities and only fill every hundredth room
There are infinite types of infinity. Crazy right?
Well every infiniteth room is better