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Hilbert's paradox of the Grand Hotel

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Re: Hilbert's paradox of the Grand Hotel

#11

Post by lgandersen »

URSpider94 wrote: post #5

A related derivation of the "uncountably infinite" is Cantor diagonalization: https://en.m.wikipedia.org/wiki/Cantor%27s_diagonal_argument Put in more concrete terms (hard to say when we are talking about infinities): there are an infinite number of integers. For each integer, there are an infinite number of real numbers (decimals) between n and n+1. For each of those doubly-infinite real numbers, there are an infini…

Just to clarify a little bit: For each integer n there is also an infinite amount of rationals (fractions) between n and n+1 and the rationals is also a countable set like the integers (just as 'infinite', so to speak). Quite counter-intuitive, I think.

I usually interpret the first steps in Hilbert's hotel as the first steps in the ordinal hierarchy (https://en.wikipedia.org/wiki/Ordinal_number).

Re: Hilbert's paradox of the Grand Hotel

#12

Post by rootedbox »

Maybe this is my ignorance of the understanding of countably infinite. But if a hotel has infinite amount of rooms, and all rooms are full. Then why would anyone ever show up to take another room.. To me it would seem that all persons are in the hotel already.

Re: Hilbert's paradox of the Grand Hotel

#13

Post by Retra »

rootedbox wrote: post #12

Maybe this is my ignorance of the understanding of countably infinite. But if a hotel has infinite amount of rooms, and all rooms are full. Then why would anyone ever show up to take another room.. To me it would seem that all persons are in the hotel already.

You might have an uncountably infinite number of people to draw from. Or really, just 'more people' in another form.

Let's say your hotel is filled with even numbers. There are plenty of numbers not in the hotel that you could add.

Re: Hilbert's paradox of the Grand Hotel

#14

Post by NotAPerson »

rootedbox wrote: post #12

Maybe this is my ignorance of the understanding of countably infinite. But if a hotel has infinite amount of rooms, and all rooms are full. Then why would anyone ever show up to take another room.. To me it would seem that all persons are in the hotel already.

Imagine you have one person for each (positive) integer, given a unique integer ID at birth, and a hotel with countably infinite rooms, each with a unique room number.

The hotel could have someone in everyone room if every person with an even number as their ID was staying there. That is, for every room number, twice the room number is a unique even number, so there's a 1-to-1 correspondence between the number of rooms and the even integers.

You could then have someone with an odd number ID show up looking for a room, and would have to rearrange from a stay-in-half-your-ID lineup.

It's a (arguably defining) property of infinite sets that they contain a strict subset (at least one guy not in the subset) with the same "size" as the whole set.

So the evens and the integers are an example of this, with both having the same "size", even though the evens are contained in the integers.

Re: Hilbert's paradox of the Grand Hotel

#15

Post by drdeca »

rootedbox wrote: post #12

Maybe this is my ignorance of the understanding of countably infinite. But if a hotel has infinite amount of rooms, and all rooms are full. Then why would anyone ever show up to take another room.. To me it would seem that all persons are in the hotel already.

Suppose people are labeled with positive integers, and rooms are labeled with positive integers.

Suppose first the people labeled with even integers show up, and they each go in the room with half their number. Then each room will have a person.

Then the person with number one shows up, and none of the rooms are empty.

If there are א people that exist, not all collections of א people have all people.

I think I don't have an intuitive understanding of how your intuition works.

Re: Hilbert's paradox of the Grand Hotel

#16

Post by unclenoriega »

It seems to me that the only thing that makes this a 'paradox' or is counterintuitive is the idea that a hotel with infinitely many rooms can be full. Is there some math concept that allows this or is it just semantics? It sounds like some sort of "quantum" effect where there's only a room there if you look for it.

Re: Hilbert's paradox of the Grand Hotel

#18

Post by rootedbox »

NotAPerson wrote: post #14
rootedbox wrote: post #12

Maybe this is my ignorance of the understanding of countably infinite. But if a hotel has infinite amount of rooms, and all rooms are full. Then why would anyone ever show up to take another room.. To me it would seem that all persons are in the hotel already.

Imagine you have one person for each (positive) integer, given a unique integer ID at birth, and a hotel with countably infinite rooms, each with a unique room number. The hotel could have someone in everyone room if every person with an even number as their ID was staying there. That is, for every room number, twice the room number is a unique even number, so there's a 1-to-1 correspondence between the number of roo…

But the paradox says...

infinite number of rooms are all occupied by a person.. then a person shows up. There is one to one ratio here... of all rooms are occupied by a person.. infinite rooms.. infinite people.. if the infinity of people are all ready in the infinity of rooms..

Then who is showing up? Everyone is already in the rooms.. So no need to worry about moving anyone.

If we change it to be numbers it still doesn't work.. If we have an infinite amount of slots.. and in each slot is a number.. and all slots are full.. how can we make room for another number..

If you say that oh well there were only even numbers in the slots.. well then what you told me isn't true.. all slots aren't full with numbers.. only even numbers..

I understand what the paradox is trying to explain about sets.. but to me it just falls apart as a metaphor.

Re: Hilbert's paradox of the Grand Hotel

#20

Post by thedufer »

rootedbox wrote: post #18
NotAPerson wrote: post #14

Earlier quoted context omitted.

Imagine you have one person for each (positive) integer, given a unique integer ID at birth, and a hotel with countably infinite rooms, each with a unique room number. The hotel could have someone in everyone room if every person with an even number as their ID was staying there. That is, for every room number, twice the room number is a unique even number, so there's a 1-to-1 correspondence between the number of roo…

But the paradox says... infinite number of rooms are all occupied by a person.. then a person shows up. There is one to one ratio here... of all rooms are occupied by a person.. infinite rooms.. infinite people.. if the infinity of people are all ready in the infinity of rooms.. Then who is showing up? Everyone is already in the rooms.. So no need to worry about moving anyone. If we change it to be numbers it still d…

You're making very strong assumptions about the number of people there are. How did you decide that's a countable infinity? No such assumption is made in the problem statement. If there are an uncountably infinite number of people then they wouldn't all fit in the countably infinite number of rooms, so of course there are people that can show up.
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