
Matheology § 264 Hilbert's Hotel: checking out.
Posted:
May 11, 2013 3:44 AM


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 reenumerated 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_2n1, ..., q_n+1 and x_n, x_n1, ..., 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.)
Regards, WM

