At first, 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". The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of transfinite numbers. 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 not smaller than the 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 formula_1.
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