Conference rusure::math

723.0. "Cantor and the Greedy Gnome (or the Marble Parable)" by RDVAX::PERRONE () Mon Jun 29 1987 12:01

    Note number 722 reminded me of a very interesting but quite elaborate
    story about infinity that I once heard.  The abridged version of this
    story follows.
    
    
    A very greedy gnome (Yes, a gnome!  Remember its only a story.)
    had a countably infinite number of marbles which he carried around
    with him in his pocket (which expains why he limped! anyway...)
    Cantor, the great set theorist and marble collector, was known to
    possess the largest and most beautiful marble in the world and of
    course the greedy gnome wanted it.  So Cantor and the gnome entered
    into a contract which went something like this:  Cantor, who was
    also an avid box collector, had a countably infinite number of boxes
    which he had numbered one to infinity.  The agreement was that if
    the gnome could put each of his marbles into each of Cantor's boxes
    in three different ways then the gnome would get Cantor's marble
    otherwise Cantor would get the gnome's marbles.  The rub was that
    the gnome had to put his marbles in the boxes the way Cantor directed.
    At the beginning of each of the three phases of the agreement the
    gnome was given the option of calling off the bet.  The gnome said
    to himself, "Hey!  There is a one to one correspondence between
    the elements of any two countably infinite sets.  I can't loose!"
    
    Cantor and the gnome got together the next morning to fulfill their
    parts of the contract.  The first phase as Cantor outlined it was
    for the gnome to put his first marble in the first box, second into
    the second and so on until all of his marbles were in boxes.  Now
    this may seem like an impossible task for most humans but the gnome,
    as his name implies, was a gnome; and we all know that gnomes are
    very fast.  In fact, when doing repetitious tasks gnomes speed up
    by a factor of two with each successive task (and that's why there
    are so many gnomes).  It took the gnome one minute to put the first
    marble into the first box (he was being careful) and thirty seconds 
    to but the second marble in the second box.  He finished the first
    phase of the agreement in two minutes.  The gnome then took his
    marbles out of the boxes and awaited the next phase.  Cantor next
    told the gnome to put his first marble in the second box and the
    second marble in the first box, the third into the fourth and the
    fourth into the third and so on.  The gnome completed the second
    stage in two minutes and then removed all of his marbles and awaited
    the final stage.  Cantor now told the gnome to put his first marble
    into the first box; and then to put the first marble into the second
    box and the second marble into the first box; and then to put the
    first into the third, the second into the second and the third into 
    the first; thus the n-th marble would always initially be placed
    into the first box and would bump all of the other marbles down
    one box.  The gnome did this in four minutes (even gnomes get tired!)
    
    However when the gnome went to get his marbles the first box was 
    empty and so was the second.  And so were all of the others!
    Cantor gathered up his boxes and walked away.
    
    What happened?   
    	[The answer will be entered as a reponse to this note.]
    
    

Before I even finish reading the text of .0, I'm willing to bet
that the gnome loses all of his marbles :->.

    re .0
    




    
    
    

























    Assume the gnomes k-th marble was in some box. 
    It couldn't be the first box because k was moved to the second box
    after start + 1/2**k minutes.
    It couldn't be in box 2 because k was moved to the third box after
    start + 1/2**(k+1) minutes
    It couldn't be in box 3 after start + 1/2**(k+2) minutes
    ............
    It couldn't be in box n after start + 1/2**(k+n-1) minutes
    therefore it couldn't be anywhere. Gnome looses!
    

    Re: 723.3
    
    Touche'!  
    
    The moral of this story is that *order* is important when dealing
    with infinite sets.
    
    If I add 1 and -1 together an infinite number of time the sum depends
    on the order in which I do it (even if I use the same number of
    ones)
    
    1-1+1-1+1-1+1-1+... = oscillates
    1+1-1+1+1-1+1+1-... = diverges to infinity
    1-1-1+1-1-1+1-1-... = diverges to negative infinity.
    
    Different results even though I added the same number of positive
    and negative ones each time (albeit infinite!)

    The gnome may have lost, but Cantor didn't win the marbles either!

    re .0
    
    here is my version of the Gnome. The parable is called be Gnome
    before midnite.
    
    There are a countable infinity of balls marked 1,2,3,......
    
    At one minute to midnite the Gnome puts ball #1 in a bucket
    At 1/2 minute to midnite the Gname takes out # puts #2,#3 in bucket
    At 1/4 minute to midnite the Gnome takes out #2,#3, puts in #4,5,6,7
    .....
    At 1/2**k minute to midnite the Gnome takes out #2**k-1 thru #2**k
    - 1 and puts in #2**k thrue #2**k+1   - 1.
    
    etc
    
    How many balls in the bucket at midnite.
    
    Answer 0 even thought the number of balls in the bucket increase
    exponentially every 1/2**k minute.

    re 723.5
    
    Ok, ok.  I should have known that someone would have noticed that
    Cantor didn't get any marbles  :-)
    
    There are at least two ways of handling this problem (1) we claim
    that Cantor didn't really want the marbles in the first place and
    was just trying to teach the greedy gnome a lesson or  (2) we can
    use the unabridged version's explanation, to wit:  Cantor tells
    the gnome to use every other marble in the third phase of the 
    contract - thus in the end, the gnome could never fill the boxes
    nor would he ever run out of marbles.
    

>    use the unabridged version's explanation, to wit:  Cantor tells
>    the gnome to use every other marble in the third phase of the 
>    contract ...
    
    At this point the gnome is likely to cry "foul".  The gnome is no
    fool, and would never have agreed to the game if such a rule were
    allowed.  Then pick up his marbles and go home.