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> moving every guest from their current room n to room n+1 The whole setup of the paradox is bad. There a infinit amount of rooms, all are full, but somehow at
by rightbyte 5y ago
> moving every guest from their current room n to room n+1
The whole setup of the paradox is bad. There a infinit amount of rooms, all are full, but somehow atleast one room is not. The guests and rooms operate under different rules.
- MereInterest 5y agoThat's sort of the entire point of the paradox. Every room is full at the start. Not "somehow at least one is not", but every single room is full. Then, by having guests move to different rooms, you can rearrange such that there is an empty room. The guests are markers to indicate a bijective function between room numbers. Each guest represents a (before, after) tuple.
- rightbyte 5y agoA point of a paradox can also be to invalidate the assumptions made. I mean, mathematicians are getting away with way too much nonsense by using diffucult words. The hotel paradox is essentially equivalent to: I can add a natural number to the set of all natural numbers which is not in the set allready.
- morelisp 5y agoNo, the paradox is that there are the same number of even natural numbers and natural numbers, yet the former is a strict subset of the latter.
- rightbyte 5y agoYe sorry I was refering to the hotel paradox. Edited.
- MereInterest 5y agoI think I would call that a contradiction rather than a paradox. A conclusion that is a contradiction means that one of your starting assumptions is incorrect. A conclusion that is a paradox means that your intuition about the problem was incorrect.
- danbruc 5y agoNo, you are trying to use your intuition for finite sets on infinite sets. Subsets of infinite sets can have the same size as the entire set. There are as many odd natural numbers as there are natural numbers even if your intuition tries to tell you that this can not be true because the natural numbers additionally contain all the even natural numbers.
- rightbyte 5y agoI am buying that. In the hotel case, it would be like claiming I can add a natural number to the set of all natural numbers which is not in the set allready. All guest are allready at the hotel ...
- lupire 5y agoWhich number are you adding? In the problem, you are adding 1 to the set, but it's already there, so you remove it by replacing it by adding 2, and n by n+1 for all n
- rightbyte 5y agoYe well my point is there is no number to add that is not allrady in the set and that the hotel clerk can't change that by rearranging guests. I.e. there are no guests that are not allready checked in (each guest need a room with a natural number on the door). Edit: I.e. the error in the paradox is assuming that there can be any more guests to check in when the hotel is full.
- danbruc 5y agoJust imagine all the even numbers are currently in the hotel, zero is in room zero, two is in room one, four is in room two...2n is in room n. Now all the odd numbers arrive at the hotel, maybe one after another, maybe all at the same time, but you can apply one of the procedure to fit them in. You must not confuse the numbers occupying each room with the numbers of the rooms, the room numbers are all the natural numbers but they can all be occupied by the even numbers alone. Because, as I said, there are as many odd numbers as there are even numbers as there are natural numbers.