It is demonstrated that a fully occupied hotel with infinitely many rooms the paradox of german power pdf still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. Suppose a new guest arrives and wishes to be accommodated in the hotel.
After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests. 15 or 847, will no longer be occupied. 0 for the people already in the hotel, 1 for the first coach, etc. 4th coach, on the 5th seat. Like the prime powers method, this solution leaves certain rooms empty. This method can also easily be expanded for infinite nights, infinite entrances, etc.
Unlike the prime powers solution, this one fills the hotel completely, and we can extrapolate a guest’s original coach and seat by reversing the interleaving process. First add a leading zero if the room has an odd number of digits. Then de-interleave the number into two numbers: the seat number consists of the odd-numbered digits and the coach number is the even-numbered ones. In this way all the rooms will be filled by one, and only one, guest. The column formed by the set of rightmost rooms will correspond to the triangular numbers. Thus, the process can be repeated for each infinite set.
Doing this one at a time for each coach would require an infinite number of steps, but by using the prior formulas, a guest can determine what his room “will be” once his coach has been reached in the process, and can simply go there immediately. This is a situation involving three “levels” of infinity, and it can be solved by extensions of any of the previous solutions. This room number would have over thirty decimal digits. The interleaving method can be used with three interleaved “strands” instead of two.
Anticipating the possibility of any number of layers of infinite guests, the hotel may wish to assign rooms such that no guest will need to move, no matter how many guests arrive afterward. This ensures that every room could be filled by a hypothetical guest. If no infinite sets of guests arrive, then only rooms that are a power of two will be occupied. Although a room can be found for any finite number of nested infinities of people, the same is not always true for an infinite number of layers, even if a finite number of elements exists at each layer. For example, suppose some people arrive in a set of spaceships which are nested in accordance to the following rules: the smallest ships, each 100 cubic meters in volume, contain ten people.
100 times the volume of each of its ten daughter ships. 1,000,000-cubic-meter ship contains exactly ten 10,000-cubic-meter ships, each of which contains exactly ten 100-cubic-meter ships, each containing ten people. This extends upward infinitely, so that there is no “largest ship”. 375, actually correspond to two passengers, one with an address ending in an infinite string of zeroes, the other ending in an infinite string of nines.
374999 corresponds both to the passenger with address 3-7-5-0-0-0 and to the passenger with address 3-7-4-9-9-9 . If this variant is modified in certain ways, then the set of people is countable again. This time, any given person is a finite number of levels “down” from the top, and thus can be identified with a unique finite address. The set of people is countable again, even if the total number of layers is infinite, because we do not have to consider an “infinitieth layer” in either direction. Initially, this state of affairs might seem to be counter-intuitive. The properties of “infinite collections of things” are quite different from those of “finite collections of things”.