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Topic: Matheology § 264 Hilbert's Hotel: checking out.
Replies: 16   Last Post: May 13, 2013 3:29 AM

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Posts: 18,076
Registered: 1/29/05
Matheology § 264 Hilbert's Hotel: checking out.
Posted: May 11, 2013 3:44 AM
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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

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.)

Regards, WM

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