

Cantor visits Hilbert's Hotel - RiderOfGiraffes

This was something I wrote a while ago, and I'm putting it here in response to the item about infinitely many monkeys and its connection to infinity.<p>Suppose I have a hotel with lots and lots of room. Every room has a number on the door, and every positive whole number appears as the room number of some room. Call them R1, R2, R3, <i>etc.</i><p>I also have a very large number of delegates for
a conference. Each delegate is assigned a room,
and there are no rooms left unoccupied. Each
delegate keeps his or her room key with them at
all times. We'll call them D1, D2, D3 in the
obvious way.<p>There are many, many sub-committees that have to
convene. In fact, every possible sub-committee.
Someone decides that each sub-committee should get
its own room, so someone else spends a sleepless
night drafting an assignment of each committee to
a different room.<p>We now show that there is a committee that has been left out.<p>Consider the committee C that is made up as follows.
For each delegate Di, if they are on the committee that
convenes in their room, leave them out of C. If they
are not in the committee that convenes in their room,
put them in C. C is the committee of those delegates
displaced when their room is used.<p>Ask - where does committee C meet?<p>If C convenes in Rn, then let's consider Dn, and ask
if they are on C.<p>If Dn is on C, then they are displaced when C meets.
But that means they are not on the committee meeting
in Rn, which is C. That's a contradiction.<p>So Dn must not be on C. Hence Dn is not displaced when
C meets, so they must be on the committee in Rn, which
is C. So they are on C. Another contradiction.<p>So C can't meet anywhere.<p>So we have shown that any assignment of committees
to rooms must omit at least one committee.<p>If we now think about the collection of all possible
committees, we can see that every time we try to
assign them to rooms, at least one must remain
unassigned. We can never pair off committees to
rooms.<p>As with cups and saucers, if whenever you pair them
up you always, always have a saucer left over, there
must be more saucers than cups.<p>So in some sense, there must be more committees than
hotel rooms.<p>And this is true, even though there are infinitely
more hotel rooms.<p>Like in Vegas ...
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tokenadult
I'll put up some references here to Hilbert's Hotel

<http://www.bbc.co.uk/dna/h2g2/A4080467>

<http://www.bbc.co.uk/dna/h2g2/A593552>

<http://faculty.cua.edu/glenn/187f09/hilbert_hotel.pdf>

to help people follow your argument. I see you are getting into the power set
when you write, "There are many, many sub-committees that have to convene. In
fact, every possible sub-committee."

