Hilbert's Paradox of The Grand Hotel - Analysis

Analysis

These cases constitute a paradox not in the sense that they entail a logical contradiction, but in the sense that they demonstrate a counter-intuitive result that is provably true: the statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms (an analogous situation is presented in Cantor's diagonal proof).

Some find this state of affairs profoundly counterintuitive. The properties of infinite "collections of things" are quite different from those of finite "collections of things". The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of Transfinite Numbers. Thus, while in an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is no smaller than total "number" of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets (sets with the same cardinality as the natural numbers, this cardinality is (aleph-null).

Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers—those numbers which can be written as a quotient of integers—contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: There is a bijection from the naturals to the rationals.