When is full not full - part 2

This is a continuation of the post of 3rd November.  I think my problem is that I'm being too logical.  We have two statements - (1) all rooms are occupied, and (2) I can create an unoccupied room, and clearly these two statements are mutually exclusive - they cannot both be true.  Either all rooms are occupied, or they're not. So the paradox doesn't make sense on a logical level but, because we are dealing with infinity, it does on a mathematical level, and Hilbert (who was a real person) was a mathematician.  There will always be a "next room" for each occupant to move into, but that room will be occupied, so that occupant will have to move into the next room, which will be occupied, and so that occupant will have to move into the next room, which will be occupied, and so on ad infinitum.  In fact it's worse than that - Hilbert claimed he could not only accommodate one extra guest, he could accommodate an infinite number of extra guests.  How? Well he would ask each guest to move to the room whose number was double that of the room they were in - so 1 would go to 2, 2 would go to 4, 3 would go to 6, 4 would go to 8 and so on. This would free up all the odd-numbered rooms, and as there are an infinite number of odd-numbered rooms, he could accommodate any number of extra guests.  And I don't know about you, but my head hurts!