What is the math concept that allows for a hotel with infinitely many rooms in the first place? One way of (somewhat) formalizing this is to say that you have

What is the math concept that allows for a hotel with infinitely many rooms in the first place? One way of (somewhat) formalizing this is to say that you have a set containing infinitely many rooms, and another set containing infinitely many tenants. Intuitively, it does not seem unreasonable that you can give every room its own tenant. Once you do this, then the hotel is "full" in the sense it is impossible to find an empty room. Of course, we can also think about giving each tenant her own room, which also seems possible. If we can do both of these things, we say that the set of rooms is the same size (or cardinality) as the set of tenants. It is worth noting, that not all infinite sets are the same size [0]. For the purpose of Hilbert's paradox, I believe it is far to assume that we are dealing with countably infinite sets [1]. The mathematical result in Hilbert's paradox is that if you add elements to a countably infinite set, the result is still countable infitite. [0] https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument [1] That is to say, sets which are as large as the set of natural numbers.