> Hilbert's Hotel, last chapter: checking out. > > Please leave the room on the day of your departure at the latest till > 11:00 a.m. > > Hilbert's hotel is not luxurious but expensive and notorious for > frequent change of rooms. Therefore many guests prefer Math's Motel. > Before checking out a guest must have occupied room number 1 (because > the narrow halls are often blocked). No problem, guests are accustomed > to that habit. > > The guest of room number 1 checks out at half past 10 and all other > guests change their rooms such that all rooms remain occupied. The > second guest checks out at quarter to 11. And all guest switch rooms > such that no room is empty. And so on. At 11 a.m. all guests have left > Hilbert's Hotel. Every room is occupied. > > A fine result of set theory. It can be improved however to have a real > mathematical application, enumerating the sets of rational and of > irrational algebraic numbers. > > First enumerate the first two rationals q_2 = 1/2 and q_1 = 1/3. Then > take off label 1 from 1/3 and enumerate the first irrational x_1 and > attach label 2 to the first rational 1/2. 1/3 will get remunerated and > re-enumerated in the next round by label label 3, when 1/2 will leave > its 2 but gain label 4 instead. So 1/2 and 1/3 will become q_4 and > q_3. > > Continue until you will have enumerated the first n rationals and the > first n irrationals > > q_2n, q_2n-1, ..., q_n+1 and x_n, x_n-1, ..., x_1 > > and if you got it by now, then go on until you will have enumerated > all of them. Then you have proved in ZFC that there are no rational > numbers. (If you like you can also prove that there are no algebraic > irrational numbers. But that's not a contradiction, of course.)
The only thing you have proved is your stupidity, as usual. There is some procedure occurring in stages indexed by natural numbers: stage 1, stage 2, stage 3 and so on; here, in stage i guest j who initially was in room j switches into room j-i resp. leaves the hotel if j-i<1. These stages then by themselves do not entail anything about a purported state of affairs reached after running through all these stages. Getting to such a state of affairs requires taking a limit, which in turn requires first of all that such a limit is defined and exists. But no such limit procedure has been defined for Hilbert's hotel. To be sure, even the easy task of accomodating one additional guest if all rooms are occupied already requires a limit procedure (at least if it is seen as occurring in stages, where in stage i the guest from room i switches into room i+1) if the hotel wants a certification (as popular today) that it is working according to ZFC. But that is an unproblematic limit procedure (because in this case the guest in room j does not change any more after stage j, and so this can be taken for the limit after running through all stages) and it is ignored in the popular fairy tale. In your nonsense story things are totally different (because now the guest in each room changes in each stage and there is nothing reached in a finite stage which can be taken for the limit after all finite stages) and your conclusions about a final outcome are completely unwarranted.
But all this is only for readers which might fall prey to your nonsense; you have been told all this so often in the course of almost 10 years now without showing the least sign of understanding that meanwhile you could be regarded as a certified idiot.