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The Infinite Hotel Et Al


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lol Pip

and silly dst getting it right.


Two more;

[b][u]1.[/u][/b] Does [b]∞ = [/b][b]∞[/b] ?

[u][b]2.[/b][/u] Imagine a sequence;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.....ad infinitum
and another, using just the even numbers;
2, 4, 6, 8, 10.....ad infinitum

which is longer?

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Infinity does not equal infinity, unfortunately.

And the first set would be twice as long. Now that is a brain twister! One infinity twice as large as the other?

Take 2:

I suppose that their LENGTH is equal, because both of the lengths are infinity. But still, one of those infinity is not necesarily equal to the others infinity, but they are both infinity.

This revolves around the fact that infinity is not a number but a concept.

Awi

Edited by awiiya
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I would say no. In my opinion, infinity starts with nothing and ends with nothing but pure infinity. This is a very philosophical yet honest question to pondered about, to me, infinity does not deal with numbers, because if you would consider it as numbers, you won't have an honest and concrete answer. Let's think of it as an object: a Circle (represents the Hotel): we do not know as to where is its starting point nor the ending, but all we know is that it is an unknown and continuous line of endless mark, a Curve (represents a room): it is a part of the circle, that if when you cut a Circle by a number of times into very tiny pieces that makes an infinite number of curves, it will still remain a part of the infinity (do not think of an endless spiral because of the infinite rooms, it is a Circle and will remain as a Circle), and when you take one out, it will still remain an infinity, why? Because infinity is not numbers, it is more than numbers, it is a form of quantity where quantity transcends the boundaries of the known digits, it's really hard to explain though but I hope you get the point. By adding another curve to the circle, is just not possible, it will ruin the order of the infinity therefore bringing chaos to the shape and the Circle itself will not be a Circle anymore since you have added 1 piece of a curve, a lot of changes will be seen on the Circle and thus, unbalancing the infinity. However, by adapting a curve to another curve, that might be possible, just like if you lay a circle of cards, and put another card on top of the other, it might, but that card will not be counted as an addition, but more likely an imitation of that single card. I know this is a very complicated stuff, but I tried my best to explain my opinion.

So, my conclusion is, you cannot find a vacant room on that Hotel that has infinite rooms, but there are a lot of different kinds of people in each of those rooms, and you can find an infinite quantity of people that are kind enough to share his/her room with you.

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There are two widely accepted views of Infinity
1. Infinite Infinity

This is where infinity is infinite so

∞ + 1 = ∞

2. Finite Infinity

∞ + 1 != ∞
(Note: != means does not equal)

Finite Infinity is used in Further Pure mathematics, in Number Sets and Banach Spaces
Wheras we would normally describe space as infinite because there is no conceivable end.

So if we then apply both ideas to the problem.

Dst would be correct if you were going to be using a value for infinity that was infinite. So if you just moved each person one room up you would have one room free.

But if you were to use Finite Infinity then there would always be an infinite rooms full, Therefore there wouldn't be a room spare.

However Both these forms of Infinity are only valid in certain circumstances, Just like light is sometimes seen as a particle (a photon) and sometimes seen as a wave because both these ways of viewing light is correct, but in certain circumstances its easier to view light in one way or the other.

NOTE: if you are more intrested why these both fit and the proof i would definitely try and google Professor Timothy Gowers
While attending one of my yearly lectures in London he was speaking about infinity and its use in his PHD thesis.

Edited by Chewett
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The hotel uses the regular infinity Chewett. It's a well known 'problem'.

As for finite finitism, I haven't encountered that one yet. The first five full length articles or video's google showed, all had either server problems or were inaccessible from Belgium? :P
In finitism it is quite the opposite too: you have a finite number N with the property N+1 = N. Sometimes N + 1 = - N or some combination of both can be used too.

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[quote name='Shadowseeker' date='20 February 2010 - 03:04 PM' timestamp='1266678242' post='54829']
Then what is N in that case? Curious, since common maths won't give a solution to that.
[/quote]



OOOOOwwwww I know !!!!

N + 1 = - N (add N to both sides)
2N + 1 = 0 (subtract 1 from both sides)
2N = -1 (divide by 2)

N = -1/2



Hmmmm that doesn't look like infinity ...

Maybe you were using the real numbers (-1,0)
in which case its half way there.

Cutler

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[quote name='Shadowseeker' date='20 February 2010 - 04:04 PM' timestamp='1266678242' post='54829']
Then what is N in that case? Curious, since common maths won't give a solution to that.
[/quote]
It is unknown. Finitism is a bit controversial after all. Probably there are estimates involving the age of the universe, light speed, Avogadro constant
and the Planck length.

There are practical applications, most known use a modulo mathematic: a + N = a. If I remember it right, if N is a prime number, you can create a consistent geometry with it (with n² points, each point being part of the same amount p lines and each line containing the same amount r points (p,r are functions of n)).

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Unfortunately, DST's answer seems to have a slight problem: it would require cooperation of infinite guests (and as stated, it would take infinite time to complete). What if everyone except for a finite few refused to move?

A simpler way to solve the problem would be to tell one stupid guest to move to a room situated at the end of the hall. Since there is no end, he'd never get there. So a room is free.


I think this is proper to quote here:

"Only two things are infinite, the universe and human stupidity, and I'm not sure about the former." - Albert Einstein

Edited by apophys
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^ Well, you could just tell all new arrivals other than VIPs like Grido to move to the end of the hall...
Then there is no end to the number of people you can take in. Of course, eventually they'll get tired of walking and will sleep in the hallway... but this infringes on the practicalities of an infinite hotel, so I won't speak further. ^_^

Edited by apophys
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[quote name='Prince Marvolo' date='21 February 2010 - 12:32 PM' timestamp='1266751971' post='54883']
What if an infinite number of persons arrive? XD
[/quote]
You move all the current guests to a new room with number equal to twice their old room number. Then an infinite amount of new arrivals can be put into odd numbered rooms.

Usually it is stated as a hotel with infinite number of rooms where an infinite coach arrives, bringing infinite guests. The fun is when infinite many infinite coaches arrive at once and you still get to squeeze them into the same fully booked hotel.

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