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Grand Hotel Paradox?
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1/10/2010 5:18:14 PM Posted: 7 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 oddnumbered 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. 
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1/10/2010 5:22:10 PM Posted: 7 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. (^_^) At 10/3/2016 11:49:13 PM, thett3 wrote: BLACK LIVES MATTER! 
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1/10/2010 5:22:43 PM Posted: 7 years ago What's the paradox?

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1/10/2010 5:24:44 PM Posted: 7 years ago At 1/10/2010 5:22:43 PM, Nags wrote: http://en.wikipedia.org... At 10/3/2016 11:49:13 PM, thett3 wrote: BLACK LIVES MATTER! 
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1/10/2010 5:31:19 PM Posted: 7 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."  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 
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1/10/2010 5:31:26 PM Posted: 7 years ago At 1/10/2010 5:22:43 PM, Nags wrote:  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. 
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1/10/2010 5:33:27 PM Posted: 7 years ago That's not a paradox. You can't fill an infinite number of rooms anyway.

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1/10/2010 5:37:09 PM Posted: 7 years ago At 1/10/2010 5:33:27 PM, Nags wrote:  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. 
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1/10/2010 5:40:46 PM Posted: 7 years ago At 1/10/2010 5:31:19 PM, GeoLaureate8 wrote: How so? At 10/3/2016 11:49:13 PM, thett3 wrote: BLACK LIVES MATTER! 
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1/10/2010 5:44:11 PM Posted: 7 years ago At 1/10/2010 5:37:09 PM, DevinKing wrote:  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. 
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1/10/2010 5:49:03 PM Posted: 7 years ago At 1/10/2010 5:40:46 PM, popculturepooka wrote:At 1/10/2010 5:31:19 PM, GeoLaureate8 wrote: 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 
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1/10/2010 5:55:02 PM Posted: 7 years ago At 1/10/2010 5:18:14 PM, DevinKing wrote: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. 
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1/10/2010 5:59:08 PM Posted: 7 years ago At 1/10/2010 5:44:11 PM, Nags wrote:At 1/10/2010 5:37:09 PM, DevinKing wrote:  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.  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. 
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1/10/2010 6:16:04 PM Posted: 7 years ago At 1/10/2010 5:37:09 PM, DevinKing wrote: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.At 1/10/2010 5:33:27 PM, Nags wrote: 
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1/10/2010 7:48:52 PM Posted: 7 years ago It's a silly absurdity, asking for what is infinity +1, +1, +1...

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1/10/2010 8:20:49 PM Posted: 7 years ago

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1/11/2010 2:26:47 PM Posted: 7 years ago At 1/10/2010 6:16:04 PM, mongeese wrote:At 1/10/2010 5:37:09 PM, DevinKing wrote: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.At 1/10/2010 5:33:27 PM, Nags wrote: 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. 
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1/11/2010 6:04:02 PM Posted: 7 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: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.At 1/10/2010 5:33:27 PM, Nags wrote: 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. 
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1/11/2010 6:25:53 PM Posted: 7 years ago At 1/11/2010 6:04:02 PM, mongeese wrote: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.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: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.At 1/10/2010 5:33:27 PM, Nags wrote: 
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1/11/2010 6:28:30 PM Posted: 7 years ago At 1/10/2010 6:16:04 PM, mongeese wrote: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. 
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1/11/2010 6:29:35 PM Posted: 7 years ago If any number times infinity is infinity and any number times zero is zero, what is zero times infinity?
in the blink of an eye you finally see the light 
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1/11/2010 6:45:57 PM Posted: 7 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. 
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1/11/2010 6:56:56 PM Posted: 7 years ago At 1/11/2010 6:29:35 PM, wjmelements wrote:Undefined. Also, all REAL numbers multiplied by zero are zero. Infinity is not a real number. 
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1/12/2010 5:13:39 AM Posted: 7 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. 