Although this problem is called "paradox", in fact it is not contradictory, just contrary to our intuition. When there are infinite rooms, "every room is full" and "can't accommodate new guests" are not equivalent.

The properties of infinite sets are different from those of finite sets. For hotels with limited rooms, the number of odd rooms is obviously always less than the total number of rooms. However, in Hilbert's imaginary hotel, the number of odd rooms is the same as the total number of rooms. Mathematically, it can be expressed that the potential of a set containing all rooms is the same as that of a subset containing all odd rooms. In fact, all infinite sets have this property, and all infinite sets have the same potential as some of their subsets. For countable set, its potential is recorded as (Alev zero).

In addition, we can also say that for any countable infinite set, there is bijection from this set to the set of natural numbers, even if this set (such as the set of rational numbers) itself contains the set of natural numbers.