Steven: It seems to me, on reflection, that the Hilbert Hotel argument does not actually rely on an induction and so the moves, as you say, could happen simultaneously. I think this is a real difference between this and the "Covering the Rationals" example. That helps! Thanks again. Ashley

Steven: Surely claiming that the extra guest can be accommodated is the same as claiming that there is a time (state) when everyone has moved. If no such time (state) exists, the claim that the extra guest can be accommodated cannot be said to be true. Probably all you can actually claim is that you cannot establish for sure that the hotel is full (a different and weaker claim!) Put it this way, I would try and find a different hotel :-) Thanks for your comments!

Steven: I am quite happy that you cannot use this kind of inductive reasoning to make a conclusion about a transfinite state. This is what I was trying to express in the third of my possible resolutions. However, it seems to me that the usual Hilbert's Hotel reasoning about a new guest (The hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1) also depends on arguing about a transfinite end state, as otherwise there are always 2 guests in some room. So this must be invalid too.

Thanks Guillaume. The reasoning I used that the number of gaps is less than the number of intervals placed is essentially mathematical induction. (If it is true after placing n intervals, and placing an interval cannot increase the number of gaps by more than 1, then it is true after placing n+1 intervals.) Does this mean that mathematical induction sometimes fails if applied to an infinite sequence of cases, because "weird things happen"? The "Hilbert's Hotel" example uses a very similar kind of inductive reasoning but is generally held not to be fallacious. Why not?

Thank you all for your help. From your answers, I think the following are all true: - The number of rationals in [0,1] is countable - The number of gaps is less than the number of rationals (as the number of intervals placed is the same as the number of rationals, and placing an interval never creates more than one gap) - The number of gaps is uncountable. Have I got this right? Many thanks Ashley