Fun with Infinity

nice
but theres one floor if all the rooms are full and no one can leave then poeple will be perpetually move rooms and die of exuation

Quote from: "the grim squeaker"

nice
but theres one floor if all the rooms are full and no one can leave then poeple will be perpetually move rooms and die of exuation

No individual person will perpetually move; each guest only has to walk the distance from his door to the next door, which is presumably finite.  An infinite number of people move, yes, and it takes infinite time for them all to move (since each one starts a finite time after the previous one), unless you can broadcast the message to all rooms at once.

If you can broadcast, each leaves his room at the same time, and each arrives at his new room at the same time.  Finite-time broadcast to infinite distances violates information theory, but whatever, so do infinite hotels
(but what happens is an infinite number of new guests arrive at the doors and only 200 guests leave?-grim)
Anyway every guest is guaranteed to be able to move into the next room, so nobody dies of exhaustion.(unless there room is the constant-daylight-desrt-no-indoor-plumbing-and-no-way-out-room-hahahahaha-take-that-you-bastard.-grim)

-Erasmus

Okay, you've gotten your new tenant into the hotel.  Unfortunately, a tourbus pulls into the driveway.  The tour guide gets out and tells you he has an infinite number of people on the bus.  They are tired and cranky and need someplace to sleep.  He won't take no for an answer.  How do you fit these people into your hotel?

Okay, I've given it a day or so.  You tell the person in room N to move to room 2*N; now you have infinitely many vacant rooms, but people die from exhaustion on their way to them.  Problem solved!

Thats what i said early, as there is an infinte gap between rooms so unless you had a device that would instantanitusly transport people, if not then there would be no way the get to the outliying rooms.-grim

Quote from: "the grim squeaker"

Thats what i said early, as there is an infinte gap between rooms so unless you had a device that would instantanitusly transport people, if not then there would be no way the get to the outliying rooms.-grim

Er no, there would never be an infinite gap.  There would be arbitrarily large gaps, but each would be finite, and so would take only finite time to get from the starting point to the finishing point.  Not a problem, really.  Just cryogenically freeze the people for whom it will take, say, 50 years or more for them to reach their room.

But again, as Pesto mentioned, that sort of difficulty is not really the crux of the problem.  What we're really doing is demonstrating an interesting feature of infinite sets: you can take a proper subset of them -- that is, you can remove elements -- and the size of the set will not have changed.  You do thing by putting the elements in a one-to-one correspondence.  In the case of the hotel, for example, the one-to-one correspondence is N<-->2N.  The fact that you can do that shows that there's the same number of even integers as there are positive integers, which is kinda counterintuitive.

Another interesting result, along the same lines, is that there are the same number of real numbers between 0 and 1 as there are real numbers in total.  I'll let you try to put these numbers in a one-to-one correspondence with all real numbers.  The next one to try after that is that there are the same number of rational numbers (numbers of the form a/b, where a and b are integers and b is not 0) as there are postive integers.  Next, and hardest, is that there are more real numbers than there are positive integers: you do that by showing that if you put positive integers and real numbers in a one-to-one correspondence, you'll have some real numbers left over, always.  The reason that's cool is it shows that there is more than one infinity, and that some infinities are larger than others.  Mind-blowing.

-Erasmus

Okay, I've given it a day or so.  You tell the person in room N to move to room 2*N; now you have infinitely many vacant rooms, but people die from exhaustion on their way to them.  Problem solved!

Quote from: "Erasmus"

Okay, I've given it a day or so.  You tell the person in room N to move to room 2*N; now you have infinitely many vacant rooms, but people die from exhaustion on their way to them.  Problem solved!

Like I said, we're not worrying about exhaustion, or any of those other pesky inconveniences.

Well, you were able to accomodate the bus, but just as luck would have it, another bus with an infinite number of people pulls into the driveway.  You think, "No problem."  As it turns out, there is a convention in town, and this isn't the only bus.  In fact, there is an infinite number of busses, each with an infinite number of people.  How will you get all these people into your hotel?

No.

Well, you were able to accomodate the bus, but just as luck would have it, another bus with an infinite number of people pulls into the driveway.  You think, "No problem."  As it turns out, there is a convention in town, and this isn't the only bus.  In fact, there is an infinite number of busses, each with an infinite number of people.  How will you get all these people into your hotel?

or you could send them to an alternate universe with an infinite number of empty identical hotels

Quote from: "Pesto"

Well, you were able to accomodate the bus, but just as luck would have it, another bus with an infinite number of people pulls into the driveway.  You think, "No problem."  As it turns out, there is a convention in town, and this isn't the only bus.  In fact, there is an infinite number of busses, each with an infinite number of people.  How will you get all these people into your hotel?

To make things simple, move everybody currently in the hotel hotel into a bus (we already know how to do this).  Now, have the busses all park side-by-side at the hotel's curb, so you effectively have a big grid of bus seats.  Again, just to make things simple, assume that each bus has one long column of seats (people are seated in single-file).  Now you can just assign people to rooms by walking a zigzag path among the busses.  Every time you get to the "top" of the grid, you walk right; every time you get to the left side, you walk down.  As you go, give each rider a room number one higher than the number you gave to the previous rider (put the first rider of the first bus in room 1).

To illustrate:



You just walk along the red path, handing out numbers.  Alternatively, you could have the riders file off the busses by following the red arrow, and fill the rooms in the order in which they arrive in the hotel.

-Erasmus

The pic is broken, but I know what you're talking about.

You're basically using the mapping of the rational numbers onto the natural numbers.

Here's the solution I first heard.  Move all the people in the hotel into the prime numbered rooms.  Assign each bus a prime number.  Give each bus the rooms numbered p^n, for n >= 2.  Then, not only have you put all the people from busses in rooms, but you have an infinite number of empty rooms left over.

What happens if you take someone out a room? Is the hotel full?

Quote from: "Erasmus"

Quote from: "the grim squeaker"

Thats what i said early, as there is an infinte gap between rooms so unless you had a device that would instantanitusly transport people, if not then there would be no way the get to the outliying rooms.-grim

Er no, there would never be an infinite gap.  There would be arbitrarily large gaps, but each would be finite, and so would take only finite time to get from the starting point to the finishing point.  Not a problem, really.  Just cryogenically freeze the people for whom it will take, say, 50 years or more for them to reach their room.

But again, as Pesto mentioned, that sort of difficulty is not really the crux of the problem.  What we're really doing is demonstrating an interesting feature of infinite sets: you can take a proper subset of them -- that is, you can remove elements -- and the size of the set will not have changed.  You do thing by putting the elements in a one-to-one correspondence.  In the case of the hotel, for example, the one-to-one correspondence is N<-->2N.  The fact that you can do that shows that there's the same number of even integers as there are positive integers, which is kinda counterintuitive.

Another interesting result, along the same lines, is that there are the same number of real numbers between 0 and 1 as there are real numbers in total.  I'll let you try to put these numbers in a one-to-one correspondence with all real numbers.  The next one to try after that is that there are the same number of rational numbers (numbers of the form a/b, where a and b are integers and b is not 0) as there are postive integers.  Next, and hardest, is that there are more real numbers than there are positive integers: you do that by showing that if you put positive integers and real numbers in a one-to-one correspondence, you'll have some real numbers left over, always.  The reason that's cool is it shows that there is more than one infinity, and that some infinities are larger than others.  Mind-blowing.

-Erasmus

the probelm with cryogenics is when you unfreeze them the water will expand thus making the person go pop-grim

Besides, math is correct by definition.  Its an analytical truth.

... let me redirect you towards Gödels' works, which logically proved that math isn't actually 'correct by definition'