I made a few good improvements to Al's Infinite Hotel. The conclusion does a pretty good job of showing that the different approaches either end up doubling up guests in a room or leaving one guest in a hall.
The hardest argument is against the idea of a simultaneous shift between rooms. However a rules based system clearly would prevent such a shift. If the rule is that two people cannot occupy the same room, then you can derive the rule that a person cannot shift to the right unless the room to the right is empty. Since the infinite hotel is full by definition, then no-one can shift to the right.
Despite Hilbert's best efforts, n+1 is still greater than n.