Can we think of Hilbert's Grand Hotel problem as of infinite number of pairs; room/guest pairs? If we pair infinite number of left shoes with infinite number of right shoes, we will have infinite number of shoe pairs. In such pairing there will not be left any unpaired shoes, because that would brake the infinity in both the sets. That would mean that there is no way to add one left shoe and get new pair of shoes in the shoe pair infinity. Same way, it would not be possible to add even one only guest in Hilbert's Grand Hotel. Should we not treat Hilbert's hotel as two paired infinite sets, where adding to one of the pair sides will brake the infinity in both sides?