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Grand Hotel Paradox?

DevinKing
Posts: 206
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1/10/2010 5:18:14 PM
Posted: 6 years ago
This is an intersesting paradox:

~From http://en.wikipedia.org...'s_paradox_of_the_Grand_Hotel

Consider a hypothetical hotel with many rooms, all of which are occupied – that is to say every room contains a guest. Suppose a new guest arrives and wishes to be accommodated in the hotel. If the hotel had only finitely many rooms, then it can be clearly seen that the request could not be fulfilled. But the hotel of interest has infinitely many rooms, so if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, you can fit the newcomer into room 1. By repeating this procedure, it is possible to make room for a countably infinite number of new clients: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and in general room N to room 2N, and all the odd-numbered rooms will be free for the new guests.


--Thoughts?
After demonstrating his existence with complete certainty with the proposition "I think, therefore I am", Descartes walks into a bar, sitting next to a gorgeous priest. The priest asks Descartes, "Would you like a drink?" Descartes responds, "I think not," and then proceeds to vanish in a puff of illogic.
popculturepooka
Posts: 7,926
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1/10/2010 5:22:10 PM
Posted: 6 years ago
Yeah, Hilbert's hotel.

William Lane Craig often uses this thought experiment to show that an actual infinite is metaphysically impossible in support of the first premise of the kalam cosmological argument.

It's convincing to me - or at least apparently. Then again I'm biased. (^_^)
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GeoLaureate8
Posts: 12,252
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1/10/2010 5:31:19 PM
Posted: 6 years ago
It's a bogus conceptual argument that doesn't apply to spatial reality. It's only problematic within mathematics.
"We must raise the standard of the Old, free, decentralized, and strictly limited Republic."
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"The worst thing that can happen to a good cause is, not to be skillfully attacked, but to be ineptly defended."
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DevinKing
Posts: 206
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1/10/2010 5:31:26 PM
Posted: 6 years ago
At 1/10/2010 5:22:43 PM, Nags wrote:
What's the paradox?

-- The paradox is that all of the rooms are full yet it is possible to accomodate more people.
After demonstrating his existence with complete certainty with the proposition "I think, therefore I am", Descartes walks into a bar, sitting next to a gorgeous priest. The priest asks Descartes, "Would you like a drink?" Descartes responds, "I think not," and then proceeds to vanish in a puff of illogic.
DevinKing
Posts: 206
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1/10/2010 5:37:09 PM
Posted: 6 years ago
At 1/10/2010 5:33:27 PM, Nags wrote:
That's not a paradox. You can't fill an infinite number of rooms anyway.

-- You can fill an infinite number of rooms with an infinite number of guests.

-- The reason that it is a paradox is because it contains a contradiction which was logicaly (maybe) derived from "true" premises.
After demonstrating his existence with complete certainty with the proposition "I think, therefore I am", Descartes walks into a bar, sitting next to a gorgeous priest. The priest asks Descartes, "Would you like a drink?" Descartes responds, "I think not," and then proceeds to vanish in a puff of illogic.
popculturepooka
Posts: 7,926
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1/10/2010 5:40:46 PM
Posted: 6 years ago
At 1/10/2010 5:31:19 PM, GeoLaureate8 wrote:
It's a bogus conceptual argument that doesn't apply to spatial reality. It's only problematic within mathematics.

How so?
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Xer
Posts: 7,776
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1/10/2010 5:44:11 PM
Posted: 6 years ago
At 1/10/2010 5:37:09 PM, DevinKing wrote:
-- You can fill an infinite number of rooms with an infinite number of guests.

- That's completely theoretical and not rooted in reality at all.
- The rooms wouldn't be infinite if they can be filled. You can't divide infinity by infinity, and you can't fill infinity with infinity. Infinity is not finite, it's not a number.

-- The reason that it is a paradox is because it contains a contradiction which was logicaly (maybe) derived from "true" premises.

I fail to see the contradiction.
GeoLaureate8
Posts: 12,252
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1/10/2010 5:49:03 PM
Posted: 6 years ago
At 1/10/2010 5:40:46 PM, popculturepooka wrote:
At 1/10/2010 5:31:19 PM, GeoLaureate8 wrote:
It's a bogus conceptual argument that doesn't apply to spatial reality. It's only problematic within mathematics.

How so?

The argument is rooted in numbers, yet people use it to argue against an infinite and eternal Universe. The number infinity is not the same as a spatial infinity.
"We must raise the standard of the Old, free, decentralized, and strictly limited Republic."
-- Murray Rothbard

"The worst thing that can happen to a good cause is, not to be skillfully attacked, but to be ineptly defended."
-- Frederic Bastiat
omelet
Posts: 416
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1/10/2010 5:55:02 PM
Posted: 6 years ago
At 1/10/2010 5:18:14 PM, DevinKing wrote:
This is an intersesting paradox:

~From http://en.wikipedia.org...'s_paradox_of_the_Grand_Hotel

But the hotel of interest has infinitely many rooms, so if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, you can fit the newcomer into room 1.
Yep. This is because infinities are unbounded, and they follow different mathematical rules than finite numbers.
For instance, if you take an infinite value and add 1 to it, the value has not changed at all. You can continue to do this ad infinitum and the value has not changed.

So even though you have added more guests and not gotten rid of any, the number of guests in the hotel has remained exactly the same. This would not have been true in the case of a finite hotel, which is why it is not possible in that case. However, infinity functions in a way so that there is no contradiction.
DevinKing
Posts: 206
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1/10/2010 5:59:08 PM
Posted: 6 years ago
At 1/10/2010 5:44:11 PM, Nags wrote:
At 1/10/2010 5:37:09 PM, DevinKing wrote:
-- You can fill an infinite number of rooms with an infinite number of guests.

- That's completely theoretical and not rooted in reality at all.
- The rooms wouldn't be infinite if they can be filled. You can't divide infinity by infinity, and you can't fill infinity with infinity. Infinity is not finite, it's not a number.


-- I agree that it is true that its not number. Which just brought to my attention that the enire paradox assumes it is a number when it says, "infinite number of rooms".

-- The reason that it is a paradox is because it contains a contradiction which was logicaly (maybe) derived from "true" premises.

I fail to see the contradiction.

-- A contradiction exists if we accept that in this case it is possible to add to an already full hotel. Not to say that I agree with this paradox's conclusion.
After demonstrating his existence with complete certainty with the proposition "I think, therefore I am", Descartes walks into a bar, sitting next to a gorgeous priest. The priest asks Descartes, "Would you like a drink?" Descartes responds, "I think not," and then proceeds to vanish in a puff of illogic.
mongeese
Posts: 5,387
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1/10/2010 6:16:04 PM
Posted: 6 years ago
At 1/10/2010 5:37:09 PM, DevinKing wrote:
At 1/10/2010 5:33:27 PM, Nags wrote:
That's not a paradox. You can't fill an infinite number of rooms anyway.

-- You can fill an infinite number of rooms with an infinite number of guests.

False. If you truly have an infinite number of rooms, and the guests are arriving at a finite rate (as the story implies), there will always be more room. Sure, you could say that an infinite number of rooms minus an infinite number of guests means no rooms, but infinity minus one is still infinity, and minus another one is infinity, and infinity minus one ad infinitum will remain infinity.
DevinKing
Posts: 206
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1/11/2010 2:26:47 PM
Posted: 6 years ago
At 1/10/2010 6:16:04 PM, mongeese wrote:
At 1/10/2010 5:37:09 PM, DevinKing wrote:
At 1/10/2010 5:33:27 PM, Nags wrote:
That's not a paradox. You can't fill an infinite number of rooms anyway.

-- You can fill an infinite number of rooms with an infinite number of guests.

False. If you truly have an infinite number of rooms, and the guests are arriving at a finite rate (as the story implies), there will always be more room. Sure, you could say that an infinite number of rooms minus an infinite number of guests means no rooms, but infinity minus one is still infinity, and minus another one is infinity, and infinity minus one ad infinitum will remain infinity.

--False, The guests which fill those rooms are instantly already there. Also, saying infinity minus <finite number> is irrelevant. If you subtract infinity then nothing exists.
After demonstrating his existence with complete certainty with the proposition "I think, therefore I am", Descartes walks into a bar, sitting next to a gorgeous priest. The priest asks Descartes, "Would you like a drink?" Descartes responds, "I think not," and then proceeds to vanish in a puff of illogic.
mongeese
Posts: 5,387
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1/11/2010 6:04:02 PM
Posted: 6 years ago
At 1/11/2010 2:26:47 PM, DevinKing wrote:
At 1/10/2010 6:16:04 PM, mongeese wrote:
At 1/10/2010 5:37:09 PM, DevinKing wrote:
At 1/10/2010 5:33:27 PM, Nags wrote:
That's not a paradox. You can't fill an infinite number of rooms anyway.

-- You can fill an infinite number of rooms with an infinite number of guests.

False. If you truly have an infinite number of rooms, and the guests are arriving at a finite rate (as the story implies), there will always be more room. Sure, you could say that an infinite number of rooms minus an infinite number of guests means no rooms, but infinity minus one is still infinity, and minus another one is infinity, and infinity minus one ad infinitum will remain infinity.

--False, The guests which fill those rooms are instantly already there. Also, saying infinity minus <finite number> is irrelevant. If you subtract infinity then nothing exists.

Doesn't the paradox assume that you have to move the guests to accomodate the new guests? Essentially, you constantly double the number of guests. Infinity will NEVER be reached, and therefore, the rooms will never run out.
omelet
Posts: 416
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1/11/2010 6:25:53 PM
Posted: 6 years ago
At 1/11/2010 6:04:02 PM, mongeese wrote:
At 1/11/2010 2:26:47 PM, DevinKing wrote:
At 1/10/2010 6:16:04 PM, mongeese wrote:
At 1/10/2010 5:37:09 PM, DevinKing wrote:
At 1/10/2010 5:33:27 PM, Nags wrote:
That's not a paradox. You can't fill an infinite number of rooms anyway.

-- You can fill an infinite number of rooms with an infinite number of guests.

False. If you truly have an infinite number of rooms, and the guests are arriving at a finite rate (as the story implies), there will always be more room. Sure, you could say that an infinite number of rooms minus an infinite number of guests means no rooms, but infinity minus one is still infinity, and minus another one is infinity, and infinity minus one ad infinitum will remain infinity.

--False, The guests which fill those rooms are instantly already there. Also, saying infinity minus <finite number> is irrelevant. If you subtract infinity then nothing exists.

Doesn't the paradox assume that you have to move the guests to accomodate the new guests? Essentially, you constantly double the number of guests. Infinity will NEVER be reached, and therefore, the rooms will never run out.
If you're only adding guests at a finite rate, then you're right that the hotel would not have infinite guests after any finite amount of time unless the hotel started with guests. I'm pretty sure the hotel starts with infinite guests in this example, though.
omelet
Posts: 416
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1/11/2010 6:28:30 PM
Posted: 6 years ago
At 1/10/2010 6:16:04 PM, mongeese wrote:
Infinity minus one ad infinitum will remain infinity.
If you're only referring to a limit as X approaches infinity, then you're correct.
If you're talking about an actual infinity minus an actual infinity, the result is undefined.
wjmelements
Posts: 8,206
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1/11/2010 6:29:35 PM
Posted: 6 years ago
If any number times infinity is infinity and any number times zero is zero, what is zero times infinity?
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mongoose
Posts: 3,500
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1/11/2010 6:45:57 PM
Posted: 6 years ago
The problem is the premise that you've filled a hotel with infinite rooms. If this is true, then you can't add more people.
It is odd when one's capacity for compassion is measured not in what he is willing to do by his own time, effort, and property, but what he will force others to do with their own property instead.
omelet
Posts: 416
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1/11/2010 6:56:56 PM
Posted: 6 years ago
At 1/11/2010 6:29:35 PM, wjmelements wrote:
If any number times infinity is infinity and any number times zero is zero, what is zero times infinity?
Undefined.
Also, all REAL numbers multiplied by zero are zero. Infinity is not a real number.
Floid
Posts: 751
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1/12/2010 5:13:39 AM
Posted: 6 years ago
Consider a hypothetical hotel with many rooms, all of which are occupied

Then by definition the hotel must have a finite number of rooms.

But the hotel of interest has infinitely many rooms

If the hotel had "infinitely" many rooms, then there are an infinite number of empty rooms at all times, no matter how many guest you have (even if that number was infinite!). Why?

Well vacancies = number of rooms - number of occupants

Obviously infinity - any real number = infinity... so that case is trivial.

But what if you had infinite occupants? Does infinity - infinity = 0? No.

Lets assume:

infinity - infinity = 0

But now lets add one to each side: 1 + infinity - infinity = 0 + 1. Well we know infinity + 1 = infinity so what we now have is:

infinity - infinity = 1

Oops! It turns out infinity - infinty is undefined because infinity is just a conceptual limit and not an actually defined term.

So the "paradox" is really just a misuse of mathematical terms to create a bogus premise.