Everyone moves at the same time. Is there any guest that can't move? No. So what's the problem? You seem to imply that 'infinite' means 'everything'. That's not what it means. For example, there are infinite even integers, right? But that does not mean that all integers are even. So, we can have infinite guests in the hotel, and still guests outside. No, there are infinite guests in the hotel. If the guests aren't infinite, then they must be finite. So how many are there? Correct. But the whole point of Hilbert's Hotel is that we can manipulate that association. Look at this direct association: (R2,G1) + (R3,G2) + (R4,G3) .... All previous guests are accounted for, but R1 is now a spare room.