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Hilbert's grand hotel as infinite number of pairs 
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#19
Jan2014, 12:51 PM

P: 13

You'll tell me to move the guests one room back, in order to fix the infinity, but I'll say NO, I've already mapped them and I have one extra room. Who is right then, you or I? 


#20
Jan2014, 12:57 PM

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#21
Jan2014, 12:57 PM

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#22
Jan2014, 01:58 PM

P: 13

OK, I understand and thank you all
Now my question is: Why Hilbert didn't give the simplest answer; unpair the set, add as much as you want and pair it again? 


#23
Jan2014, 03:54 PM

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Let's start with all rooms occupied. Person1 > Room 1 Person2 > Room 2 ... PersonN > Room N etc. There is a onetoone pairing between occupants and rooms. Another guest arrives. Where should he go? The hotel manager asks each person who already has a room to move to the next higher numbered room. The pairing is now like so: New person > Room 1 Person1 > Room 2 Person2 > Room 3 ... PersonN > Room N+1 etc. This is also a onetoone pairing between occupants and rooms. For each person I can say what room he or she is in, and for each room, I can say who is in the room. Now for the sake of simplicity in the next step, let's renumber the people: Person1, Person2, etc. At this time a bus with an infinite number of people arrives. What to do? As already mentioned, the manager asks each room occupant to move to the room whose number is two times his/her current room number. This frees up rooms 1, 3, 5, ..., all the oddnumbered rooms. The people from the bus file in and each person is assigned one of the oddnumbered rooms. The pairing is now (BP  person from bus, P = person already present): BP1 > Room 1 P1 > Room 2 BP2 > Room 3 P2 > Room 4 ... BP_N > Room 2N1 P_N > Room 2N etc. This too is a onetoone pairing. For each person I can say what room he or she is in, and for each room, I can say who is in the room. 


#24
Jan2014, 05:10 PM

P: 13

Very good explanation. Now, I understand that the result of infinity minus any number is still infinity, but I still cannot wrap my mind around subtracting from paired sets. Although infinity as number has infinite value, when paired it's bound to the other side as equal value (size). Let's take Galileo's argument that S = {1,4,9,16,25,...} is the same size as N = {1,2,3,4,5,...} because there is a onetoone correspondence: 1 ⇔ 1, 2 ⇔ 4, 3 ⇔ 9, 4 ⇔ 16, 5 ⇔ 25, ... If we subtract 1,2,3,4,5 from N it will still be infinite on its own, but it will be with 5 less than S (smaller size), which will brake the pairing. I wonder how to deal with that 


#25
Jan2014, 06:10 PM

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It doesn't matter if we break a pairing when we remove or add some elements to one of the sets. All that we need to do is to establish a new pairing that is also onetoone. Here's a pairing of the set {6, 7, 8, 9, ...} with {1, 4, 9, 16, ...}. 6 ⇔ 1 7 ⇔ 4 8 ⇔ 9 9 ⇔ 16 .... n ⇔ (n  5)^{2} ... 


#26
Jan2114, 01:57 AM

P: 13

Zeno's paradox is a result from mixing time with distance, not taking in account the speed. Hilbert's paradox is a result from mixing pairs with pair parts. Left and right as parts of one pair are bound together and you should not be able to create new pair by adding only left or only right part to it. An infinite paired set is infinite in number but its parts are limited to each other (each left shoe has as a pair a right shoe, and there are no singles available, otherwise the set wouldn't be called paired set) Regardless whether the number is limited or infinite, the pairs in the row are created and cannot be increased by adding only to one side of the pair. One would argue that we don't increase infinity, because its value cannot be increased or decreased by adding or subtracting, but by not taking in account the properties of the paired infinite set we have a moment between the pairings when one of the sets is with number greater than the other. Again, "with number greater than the other" is not correct use for infinity, so we rather call it unpaired number (one of the sides contains unpaired number of shoes). That unpaired number implies limit to both sides when we look at them as pair parts. That's why I say that a pairing should occur only once if we don't want to create a paradox. In Hilbert's case we have complete infinite number of pairs (no singles available) and any adding to one of the sides will break that completeness. Hence the paradox. 


#27
Jan2114, 03:05 AM

P: 13

A pair has a property which should not be confused with the properties of its parts. My opinion is that we should think of pair's property as of the result from 1 liter of water paired with 1 kg of cement. So, if we pair 1 liter of water with 1 kg of cement we'll get +/ 2 kg of concrete. Now, think of the rooms as of buckets of water, and the guests as of 1 kg of cement. You can add as much as you want to each infinite set before the pairing, but nothing you can do after you put the cement in to the water. Second pairing is impossible. 


#28
Jan2114, 04:52 AM

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P: 839

You can argue this is a veridical paradox, in that you have to reject your intuition of finite sets when dealing infinite sets. That's the entire point of Hilbert when he created this: to teach his students intuition of countable infinity. But it is certainly not a falsidical paradox. Edit: * this probably is a bit too harsh. The problem is that what you are claiming is whether something is an allowed operation or not. The problem is that the definition of "infinite set" is what allows us to do this, so your argument distills down to rejecting the axiom of infinity. And that is a philosophy question, not a mathematics one. 


#29
Jan2114, 07:30 AM

P: 13

Don't get frustrated if you cannot answer some of my questions and logical points. Most probably that is because my points are so lame that you cannot make sense of them But still, if the authorities were never questioned, the science wouldn't go that far 


#30
Jan2114, 07:35 AM

P: 13

I didn't question the axiom of infinity, but the way we deal with it. I think that the "concrete" example clarified my point. 


#31
Jan2114, 09:48 AM

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#32
Jan2114, 09:59 AM

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The question has been asked and answered (ad infinitum, so to speak). I am closing this thread.



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