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No, 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

No, 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 ago
I 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 ago
  Which 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 ago
    Ye 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 ago
      Just 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.
      - rightbyte 5y ago
        > the room numbers are all the natural numbers but they can all be occupied by the even numbers alone Hmm ... that is an interesting take on it. I.e. guests with even customer id:s can have all rooms. Maybe I just have to accept the paradox to be a property of infinite sets by definition.
        danbruc 5y ago
        It's not really a paradox, just some things that work for finite sets no longer work for infinite sets. The sets { A, B, C, D } and { 1, 2, 3, 4 } have the same size because they both have four elements. Another way to see that they have the same size is that you can provide a one to one mapping between them, for example A - 4, B - 3, C - 2, D - 1. For infinite sets the first option no longer works, you can not write down a specific number for the number of elements in a set and then see that two sets have the same size because you wrote down the same number for both of them. But the second option still works, you can provide a one to one mapping between the elements of two sets, for example n - 2n to match all natural numbers with all even numbers. Each natural number n has an associated even number 2n and each even number 2n has an associated natural number n. And this is then just the definition of what it means for two sets to have the same size, there is a one to one mapping between their elements. And this works for finite sets as well as for infinite sets. Everything else are consequences of that. Take the natural numbers and take the natural numbers with the first k of them removed, the two sets still have the same size because you can pair n with n + k even if it is against your intuition that the size of a set does not change when you remove some of its elements.
    - deleted 5y ago
      [deleted]

rightbyte 5y ago

I 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 ago
Which 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 ago
  Ye 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 ago
    Just 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.
    - rightbyte 5y ago
      > the room numbers are all the natural numbers but they can all be occupied by the even numbers alone Hmm ... that is an interesting take on it. I.e. guests with even customer id:s can have all rooms. Maybe I just have to accept the paradox to be a property of infinite sets by definition.
      - danbruc 5y ago
        It's not really a paradox, just some things that work for finite sets no longer work for infinite sets. The sets { A, B, C, D } and { 1, 2, 3, 4 } have the same size because they both have four elements. Another way to see that they have the same size is that you can provide a one to one mapping between them, for example A - 4, B - 3, C - 2, D - 1. For infinite sets the first option no longer works, you can not write down a specific number for the number of elements in a set and then see that two sets have the same size because you wrote down the same number for both of them. But the second option still works, you can provide a one to one mapping between the elements of two sets, for example n - 2n to match all natural numbers with all even numbers. Each natural number n has an associated even number 2n and each even number 2n has an associated natural number n. And this is then just the definition of what it means for two sets to have the same size, there is a one to one mapping between their elements. And this works for finite sets as well as for infinite sets. Everything else are consequences of that. Take the natural numbers and take the natural numbers with the first k of them removed, the two sets still have the same size because you can pair n with n + k even if it is against your intuition that the size of a set does not change when you remove some of its elements.
  - deleted 5y ago
    [deleted]