Recursion과 Zeno's Paradox
We find ourselves in an interesting and bizarre
situation. Taking information from the way that
the Sierpinski Gasket is made (which is
relatively simple), we designed a scheme for
coming up with a long list of numbers to be
added together. No matter how long that list
gets, we can always add more numbers to it,
according to our scheme. And no matter how
many numbers we add together, if we keep to
the scheme, it doesn't seem like we can ever
get the grand total to be more than 1. How can
this be? Will the sum ever actually be equal to
1? It seems like it will get closer and closer. In
fact, it seems that no matter what value you
pick that is very close to 1, you can keep
adding terms to the series so that the sum of
all the terms is greater than that value, but, of
course, still less than 1. But will the sum ever
actually be 1? This question is very similar to
questions posed by the ancient Greek
mathematician Zeno who is credited with setting
forth some famous paradoxes . This kind of
thinking is also very important in the study of
Calculus.
1. Zeno's Paradoxes
(1) The Archer's Arrow
An archer picks up an arrow, places it agains the bowstring,
draws the string back and lets the arrow fly towards the
target. It looks like it will be a bulls-eye, but of course we
won't know for sure until it hits the target.
Before the arrow reaches the target, it must first travel half
of the distance to the target. From there the arrow travels
half of the remaining distance to the target. Quickly, the
arrow travels half of the distance which remains after that,
then half of the distance that is still between it and the
target. In fact, before it can get to the target, the arrow
must always first go half of the distance that remains
between it and the target.
Gee.... Does the arrow ever hit the target?
(2) The Tortoise and the Hare
The Hare was a terrible braggart. It was true that he could
run very fast. In fact, it was doubted that anyone could run
faster. But instead of being quietly proud of what he could
do, he was always challenging the other animals to races
so that he could beat them and make fun of them because
they lost and he won.
One day, he challenged the Tortoise to a race. Of course
she said no, because she walks so very slow. She's faster
than most tortoises---she can actually go half as fast as
the Hare, but that still doesn't give her any hope in a race.
The Hare wanted to race with someone so bad that day,
that he told her he would give her a head start that was
halfway to the finish line. A head start that was halfway
there. Since she could walk half as fast as the Hare could
run, it might just be worth a try. She said yes.
Everyone came out to watch. The race course was set up
from one end of the field to the other. The exact halfway
point was paced off and marked. That is where the Tortoise
started. The Hare started at the "Start" line. At the sound of
the starting gun, they both took off.
The Hare ran as fast as he could and quickly reached the
halfway point where the Tortoise had started. Of course the
Tortoise was no longer there. She had covered half of the
distance to the Finish line by then. The Hare would have to
catch up to her before he could pass her. He took careful
note of where she was, put his head down, and raced to
that point. But when he got there, of course, the Tortoise
was no longer there, because she had traveled half the
distance from there to the Finish line by then. So the Hare
took careful note of where she was, put his head down, and
raced to where the Tortoise was. But he couldn't pass her
when he got there, because by then she was gone. Once
again, she had walked half the distance from where she had
been to the finish line.
They did this all afternoon and on into the night. In fact,
they might be doing it still. Do you think that the race ever
ends? Does the Hare ever catch up to the Tortoise?
문제 1. Archer의 화살 paradox에서의 문제점은
무엇인가 말하시오.
|
문제 2. 토끼는 거북이를 영원히 추월할 수 없다는
주장은 어디에 문제가 있다고 생각하는지 말하시오.
|
2. The Hotel Infinity Paradox
Don't be disappointed or frustrated if this discussion
puzzles you, or if you can't find some simple explanation
that resolves all possible contradictions. From the time of
Zeno to the present, mathematicians and philosophers
have wrestled with the notion of infinity. Always, there has
been one loose end or another which cannot be tied up
neatly with the rest. The story of the Hotel Infifnity reflects
the state of the debate over infinity at the beginning of the
20th century.
Overview of the Hotel Infinity
When turn-of-the-century mathematician Georg Cantor
approached the subject of infinity with mathematical rigor,
he encountered many paradoxes. At the Hotel Infinity, these
paradoxes come to life.
The Paradox of Hotel
Infinity
(Copyright 1991, by Nancy Casey)
So! You want to know about Sam Cantori, his hotel and
those busses. Let me guess! You want to know
everything---all the exactly's and the how-so's. You only
want the truth, you say. The whole truth---complete and
exact. I can tell it to you.
A warning to you, though: if you only want to hear things
that are easy to believe, why don't you just tell me now and
save me the trouble of wasting my breath. And if you are
the type that likes to say words like ``impossible'', or if you
think you are going to interrupt and try to tell me that things
I saw with these two eyes couldn't have happened, or
things that I made with these two hands couldn't have been
built---well, why don't you just get up right now and go
swap stories with Mr. Kronecker. And when you get there,
don't let him say a word until he tells you how he learned to
talk. Ha!
But if you want to listen, I'm glad to tell you all the stories I
know, and if you help me, I'll try to remember the rest. If we
can get all the remembering done, maybe you could build it
all back up again. Then you could have some of the
fun---because, oh, it was fun we had at the Hotel Infinity.
It was a long time ago that Sam hired me. I think he picked
me because I had already lived and worked a long time and
could do many things. He didn't have to teach me how to
build a hallway, or a room, or how to put in pipes for water,
or wires for electricity. I could build with metal just as easily
as with wood, and had worked on motors and engines since
I was 9, so there was nothing in the putting together of the
building or the busses that I couldn't do.
Being on the job with Sam was different than with any other
boss I've had. I'd get a staircase or a hallway all framed in
the way he said, then he'd check the blueprints he kept
locked in the safe, and have me make tiny changes---a
little twist for the steps, a slant on the ceiling, a bit of lean
on the wall. None of his changes made any sense for
anything except making a good building bad. But I did what
he said. I did what he told me. None of this ``Why do I have
to do it this way, boss?'' or ``I think I know a better way,
boss,'' or ``This doesn't make any sense to me, boss.''
None of that. It was all ``Yes, sir,'' and ``Right away, sir,''
and ``What next, sir?''
Sam Cantori paid me to do, he didn't pay me to think. Once
I started thinking about what it was I had done, it came to
me that I had done some things that were downright
impossible, things I'd stand here today and tell you couldn't
be done, except that I did 'em myself.
There didn't seem to be anything orderly about the way we
worked either. One day it was floors, doorknobs, and fan
belts. Next day he'd put me to working on transmissions,
sinks, and wallpaper. If he tilted a mirror a little crooked, or
had me raise one end of a step, he might then say, ``Now
go do that to all the rest of the busses and in all the rooms
while I study the plans and decide what to do next.''
All the time I worked, the questions I never asked built up in
me somehow, because I remember the morning when I
started to be different. I showed up for work, right on time
like I always do, carrying my tools. And there was Sam in his
office. All the blueprints were gone, all the plans, all the
lists, all the sawdust, the nails, the screws, the paint, the
grease, the sawhorses, the gears. The floor was swept
clean as could be, and what a sight was Sam! His slick and
shiny hair showed the tracks that the comb had left. His
chin was clean-shaven, pink and smooth. The suit he had
on was made of the finest cloth I'd ever seen or touched,
and a cane dangled from his wrist. In one hand was a silk
top hat, in the other a gleaming gold watch. He snapped the
watch case shut, tucked it into his vest pocket and looked
up at me as I came in.
``Sam!'' I cried, ``What's---''
He held up his white-gloved hand for silence, and motioned
me over to his desk. He shoved two papers in my direction.
They were drawings.
``Make these two signs, George. When you've got them
hung up, put your tools on the pile with the other things out
back. I'm going to call Kronecker to haul all that junk away.
Then go to the tailor, the cobbler, and the barber. They're
expecting you. No more of this backbreaking sweat and
sawdust. You're the one who will greet all the guests at the
Hotel Infinity!''
The moment I saw those drawings, something changed in
me, and I turned into a questioning, attention-paying
person. Oh, I didn't speak any questions out loud, but it
was the first time an idea came into my head about not
doing something exactly the same as Sam Cantori told me
to do it. ``Make these two signs and we're done,'' he'd
said. Impossible!
One sign was the big one to go out front: Hotel Infinity, it
said. All rooms One Dollar. That's a strange enough
name, and a price sure to lose money, but that wasn't the
problem. Sam was rich and only doing this for fun anyway,
and he could name his hotel anything he wanted. It was the
other sign, the one that said Vacancy that bothered me.
Not too strange, just a little bit strange, like those crooked
walls and bending hallways. But this was a different morning.
For some reason I was thinking about things.
Now there's nothing wrong with a Vacancy sign that hangs
under the name of your hotel to let people know there's
room for them. But everybody knows that you have to have
a little sign that says, No to hang up in front of the
Vacancy sign so it says No Vacancy when the hotel gets
full. You get in trouble if you advertise a vacancy, then tell
someone you're all full up and they can't come in. I started
to bring it up with Sam, but he was already on the phone
with the junk man Kronecker negotiating a price for our big
pile of leftover materials and tools. So I made that little sign,
too, and hid it under the pot of the avocado tree. Here it is.
As you can see, we never hung it up.
When I came back from getting myself spruced up like Sam,
Kronecker was leaning against the handle bars of that
propeller-driven solar-powered truck he made out of hot
water tanks and bicycle parts, staring at the big sign in the
parking lot. I knew he was going to be glad to get that junk
pile. He's a funny man, that Kronecker. He won't use
anything he doesn't make himself out of bits and pieces of
everyday things, so all that junk was a pile of treasure for
him. That's why I was surprised when he whirled and shook
a bony finger at me.
``Cantori told me what you two were up to,'' he growled.
``Hotel Infinity, indeed. No such place can be made, yet you
pretend that it can. Such a thing is incorrect, evil, and
wrong. I will see to it that this is stopped. And yes, I will take
your tools, because you two should never be allowed to
claim to build another thing!''
Sure, Kronecker's threat was strange, but Kronecker was
strange. There was no time to argue or worry about what he
said. The first guests were already arriving. It's no surprise
that the place was already beginning to fill up. Where else
can you get a hotel room for a dollar?
Sam stood up from his big leather chair and greeted the
members of the first family in line. ``What is it you like? What
do you like the best?'' he boomed, pumping the hand of the
youngest child.
``Purple!'' came the answer---along with a grin that had
one tooth missing.
``And you?" He reached for the hand of the older child and
gave it a big shake. ``What do you want in your room?''
``Polka dots!''
``Purple with polka dots. Hmmm. Will those be orange polka
dots or crimson ones?'' he asked the parents.
Theh parents turned to each other and shrugged, ``Orange
will be just fine,'' they said.
``Wonderful!'' clapped Sam. ``Room 91 is the one for you.
I'll take your dollar. Then all you need to do is read this very
important paper here. Grownups, please make sure the kids
understand. And when you've signed it, my assistant
George will show you how to find your room."
Now, that paper was a funny thing. Sam always took charge
of that. The paper had all the usual things on it, you know,
clean up your mess, fix what you break, check-out time is
11AM. But there was this one extra thing at the bottom that
Sam always acted like it was so important. It was written in
letters twice as big as the rest, and Sam always made sure
that they saw it. It said, ``AND I PROMISE TO OBEY EVERY
REQUEST THAT IS MADE OVER THE LOUDSPEAKER, NO
MATTER HOW SILLY OR INCONVENIENT IT MIGHT SEEM.''
Well, the silly part was that we didn't ever use the
loudspeaker, even though I knew full well that it went
everywhere in the whole place. I put it there myself, you
know.
Well, my job was to show the people to their rooms. I didn't
have to go all the way there. The place was too huge for
that. Sam would hand me a piece of paper with their room
number on it. I would tell them the number and start them on
their way. When we were just new and people were just
starting to find out about us, I told them numbers like 6, and
28, and 42. In a while, as the hotel got more popular, I was
giving out more interesting numbers like 523, or 954, and
even 1,255. After a long time, I got used to handing out
numbers like, 3,244, and 8,999,223. It was all very organized
the way the hotel was set up, so even though the numbers
got very large, the rooms were easy to find. I liked the big
numbers the best.
One day, when I had just sent a family off to room
4,693,203,355,226 I asked Sam the first question ever. ``How
many rooms do we have in this place, anyway?'' He looked
at me kind of funny. ``Just curious, sir,'' I said quickly.
``You built it, George, you should know,'' he answered.
``Infinitely many.'' I didn't ask any more questions for a long
time. But I did watch, and I did think.
One day, Sam said he would be out for a while. He had an
appointment with the head of a tour company that was
probably going to buy the busses for sure. He said that I
was to be in charge while he was gone. I would be the one
who sits at the big desk and calls a big welcome to
everyone who comes in the door: the families, the little
children and their pets, the salespeople, the
convention-goers, the bicycle travelers, the bus loads of
tourists. I couldn't wait.
``Make sure they understand that paper,'' Sam cautioned.
``Don't let anyone in without signing it. And oh, by the
way--there's only one room empty right now. It's 26.'' He
gave a wave and was jauntily off, swinging his cane.
Only one empty room! Thank goodness I had made that
sign. It was going to have to say No Vacancy out there
before this day was over for sure! I kind of wondered how
many people actually were in the hotel just now. That would
be interesting to know. But the office was filling up with folks
like it always did.
At the same time that cars were pulling up in the lot outside,
a few people were dragging in suitcases to leave. A line had
already formed at the right-hand side of the desk where the
sign said, Check In. I had to get that sign up fast. When I
tried to tip the pot of the avocado tree to pull the little No
sign out, it wouldn't budge. The tree had grown too huge
and heavy since I slipped it under there. I was going to need
a lever to get the sign out, and a hammer to hang it up. But
Kronecker had taken away all the tools years ago. It was
useless. What was I to do? I had to think.
As quickly as I began to think, I had an idea. It wouldn't work
for sure, but it would buy me some time. ``Good morning,'' I
said to the first person in the check-in line. He wore a suit
and carried a briefcase. I gave him the paper, made sure he
agreed to all of the rules, including the one about the
loudspeaker, took his dollar, and sent him off to room 26.
Next in line were two young kids wearing backpacks.
``We're traveling by bicycle.'' they said. ``Usually we camp
out, but we heard about this place. The Hotel Infinity. History
making. We could use a shower and a soft bed, so we
decided to stay here tonight. Here's our dollar.''
``Just a minute,'' I said, and swung myself around in Sam's
big leather chair to face the first person in line in front of the
left-hand side of the desk where the sign said Check Out .
The woman handed me the key to Room 4,326.
``Was everything okay,'' I asked her.
``Oh, it was just fine,'' she said. ``I've never slept so well in
a bed shaped like a turtle before.'' She picked up her
suitcase and walked toward the door. The man behind her
started to hand me a key, but I held up my hand, smiled and
said, ``Be with you in just a minute, Sir.'' Quickly, I swung
the chair back around to face the bike trip kids and checked
them into room 4,326. The man who I'd left waiting checked
out of room 4, which is where I checked the next person
into. That's the way it went all morning long. I kept checking
people into whichever room it was that someone just
checked out of. I was lucky. After the last person checked
out, there was exactly one person left to check in. What a
relief!
The lobby was empty. I leaned back into the soft leather of
Sam's chair, propped my feet on the desk and looked at my
reflection in the polished shine of my fancy shoes. I was
proud of myself. Just a little bit of thinking, and I had figured
out what to do. Outside, I could hear splashing in the pool,
the clop-clop of the ponies feet, and the music from the
carousel. People were playing tennis, strolling on the
sidewalks, pitching horseshoes, shooting arrows at the
archery targets, and sipping cool drinks. A perfect day at
the Hotel Infinity. Then, right there before my eyes, a car
pulled up. Oh no! Nobody would be checking out until
tomorrow, and I had forgotten to put up the sign.
Every door of the car creaked slowly open, and a whole
tired-looking family got out, along with two cats, a Saint
Bernard, and a poodle. The girl was carrying a bird cage,
and her brother had a fish bowl. The man pushed the office
door open and tossed a crumpled dollar bill on the desk. He
didn't look friendly.
``We need a room,'' he gruffed. ``A big one. Now. Been
driving since yesterday morning.'' He held his hand out for a
key. ``Which one is it?''
I started to explain that I didn't have a room, but he
interrupted me. He threatened me and scared me. He told
me I was going to go to jail for running a hotel that
advertised a vacancy, and refusing to give him a room. He
said I was discriminating against families, against tired
people, against loud people, against rude people, against
people who had been driving since yesterday morning. He
grabbed the phone and was calling his lawyer, when in burst
Sam.
``George! I've done it!'' he shouted. ``I sold them! I've sold
all the busses to Continue 'Em Tours. All of them, George!
Sold! It's a wonderful day, George!''
``Oh, Sam, Sam, no. It's a horrible day,'' I groaned. ``Forget
the busses, we're ruined. Something terrible has
happened.'' Sam's face fell, and I started to explain. ``The
hotel is full, Sam, and there's no place to put this man.
He's mad, he's going to sue. We're going to jail! Oh Sam,
it's so terrible, Sam, I should have told you sooner, we
need to change the sign...''
Sam wasn't even listening. He strolled over to the man and
said, ``Excuse me, Sir, my untrained assistant has made a
mistake. He meant no harm, he just doesn't know how to
find an empty room when the hotel is full. It is true that we
already have an infinite number of guests but this is the
Hotel Infinity. We have an infinite number of rooms, so of
course there is space for you. Please, have a seat, this will
only take a moment." He bustled over to me, fishing into his
vest pocket for a piece of paper which he pressed into my
hand.
``At last, George,'' he whispered, grinning. ``We get to use
the loudspeaker. Go announce this.''
I went over to the table in the corner with the microphone on
it, angry and muttering all the while. ``Untrained assistant!
Haven't I worked for Sam more years than the two of us put
together can remember? Finding an empty room when the
hotel is full. An infinite number of guests. An infinite number
of rooms. An infinite number of too-crazy things is what
we've got here!'' I flipped the switch on the loudspeaker
system, and much to my surprise and relief, it worked after
all these years. I leaned over the microphone and read what
was on the paper.
``Ladies and gentlemen, families and friends. Please recall
that you have agreed to do exactly as we ask when we
make a request on the loudspeaker, even if what we ask
seems silly or inconvenient. This won't be hard and it won't
take long. All we ask is this: would everyone please gather
up your belongings and move to the room whose number is
exactly one greater than the number you are in. Got that?
Everyone move quickly now, yes that's right, to the room
whose number is one greater than the one you are in.''
The people had signed. The people had agreed. So,
although the people were somewhat confused, they began
to move. The trapdoor in the ceiling of room 34 creaked
open, and that family climbed out and moved on to room
35. The businessman in room 578 appeared in the hallway
and asked a boy where room 579 was.
``Oh, my mom and I just left there,'' he said, ``it's back that
way just around the bend in the hallway. We're looking for
room 580 now.''
The third graders on a class trip quickly gathered their
things and moved out rooms 222, 223, and 224 and into to
rooms 223, 224, and 225. The grandparents in room
3,465,892 moved to room 3,465,893. And the lonely old man
who had been in room 1 did not look nearly so glum as he
stood on the ladder, pushed up on the round door over his
head, then climbed up and popped through the floor into to
room 2.
``Well!'' said Sam, as he turned to the man, who still looked
rather angry. ``We are most sorry for the inconvenience,
sir. So sorry. As you can imagine, it has taken many years
for this hotel to get full. This is the first time it has
happened, and my assistant here wasn't aware of the
procedures. Please accept my apologies, and let me return
your dollar so that you can stay in room number 1 for free.
But first, of course, you must sign the paper, here.''
That Sam! He was such a charmer! Oh, those were the
days---before all that fuss with Kronecker heated up.
From then on, more people came than left every day, so
the hotel was always full. We used the loudspeaker to get
everyone to move down whenever we needed an extra
room. It got to be known far and wide that this would
happen, and people began coming to the hotel just to see
how many different rooms they could get to stay in. They
might start out in an upstairs room with dinosaurs on the
wall paper, and then the loudspeaker call would come, and
they would have to move to the room with the checkerboard
bedspreads. If it was a particularly busy day, a lucky family
might, all in one day, get to stay first in the room with the
elephant-shaped bathtub, then in the one with jungle plants
and trapezes. After dinner they might get sent to the room
that has the bathtub slide, and end up going to sleep in the
one with the fold-down beds. People didn't mind moving.
That turned out to be half the fun.
Because I was paying more attention, I was learning to
manage the hotel. Sam could leave me in charge, and I
could handle everything just as smoothly as he could.
Good thing, too, because Sam had to be away more and
more due to all the trouble Kronecker was causing.
I think it was Kronecker's peculiarity about having to know
how things were built up from simple parts that set him off.
And Sam is pretty stubborn, too. He said that if Kronecker
wanted to be a fool and not believe the hotel existed just
because he didn't know how it was made, that was up to
him. Then Kronecker bought up all the newspapers and
wouldn't run Sam's ads because he said they violated
truth-in-advertising laws. That's why they ended up in court.
Lots of people from the hotel came to see, but of course
they didn't all fit in the courtroom.
Sam stood up in front of the judge and declared, ``At this
moment there are infinitely many people inside the building
that Kronecker claims everyone should ignore." He pounded
his fist on the witness stand. ``He has never set foot in the
door! If he did come inside, see the people come and go,
try and count them if he likes, and watch our operation, he
would see how it works. It is indeed infinite. Why, I'm sure
we could find room for any number of new guests right
now.''
``Preposterous!'' shouted Kronecker ``He tells me to come
visit the infinite as though it is something waiting to be
discovered or explored, skipping over entirely the fact that
ordinary people can not figure out a way for it to get there.
If he cannot tell me how that place is constructed, it is
neither real nor useful and people should stay away! And
furthermore---''
``Listen to this ridiculous man!'' burst in Sam. ``I am sure
he cannot tell me how to construct a stomach, yet this has
never prevented him from staying away from his lunch!''
There was a lot of shouting back and forth like that, and the
judge kept having to pound his gavel and make them calm
down. In the end, though, he pretty much sided with Sam
when he said, ``He claims that there are infinitely many
people inside, and that there is still room for more. I agree,
Mr. Kronecker, that this is quite puzzling, but just because
something is puzzling, we can't be telling people it is
dangerous and they should stay away. There are ways, Mr.
Kronecker, that you could prove that what Mr. Cantori
claims about his hotel is not true. If there aren't infinitely
many people there, as you claim, just count them and tell
us how many there are really. And it also seems to me that
if someone is turned away because the hotel is full, you
would have proved your point about Mr. Cantori's
advertising being false. If you can present the court with this
kind of information, we will reopen this case, but for now I
rule that you must allow Mr. Cantori to advertise. Case
Dismissed!"
Sam laughed and looked at Kronecker, ``Come on over and
count. Or come on over and wait for us to turn a guest
away. You will be at either task for infinitely many days!'' He
paused, and scratched his head. His voice got quieter. ``Or
maybe a little longer. I'm not sure. I'll have to think about
that.''
Kronecker shook his bony finger and threatened revenge,
but things settled down. A good thing, too, because I don't
know what I'd have done if I'd been there alone the day that
one of Sam's busses pulled up---fully loaded and needing
rooms for everyone.
I told you I made those busses just like I made the hotel,
back when all I did was follow orders, before I started asking
questions. Did I tell you that those busses were made just
like the hotel? That the busses held infinitely many people?
Sam saw it pull up before I did. He rushed right out and
began talking to the driver. I knew it would be time for my
sign for sure. I hoped the driver wouldn't be too mad---or
the people on that bus either. Where else is a bus loaded
with infinitely many people ever going to be able to let all the
passengers off so they can take a shower and sleep in a
nice warm bed? Oh, I hoped Kronecker wouldn't get wind of
this.
Well, wasn't I surprised when Sam Cantori shook the
driver's hand and started waving the people off the bus. As
he walked inside he was fishing into his vest pocket with his
eyes all bright. He handed me a note, and flashed me a
grin. ``Loudspeaker time, George. By Zeno! I'm glad you
made those busses. I don't see how we'd have ever gotten
to this any other way!''
The lobby was filling up with people, and more and more of
them were pouring out of the bus. No time to study the
note---I just went to the loudspeaker and read it off.
``Ladies and gentlemen, families and friends. Recall that
you agreed that you would follow our requests on the
loudspeaker, no matter how silly or inconvenient they seem.
Here is what we must ask you to do. Please look on the
outside of your door and copy your room number down on
a piece of paper. Everybody got that? Now, everyone,
please multiply that number by 2. Yes, double the number,
and write down your answer. Check your work please, make
sure that it is correct. That's your new room number, folks.
If you would please, gather up your things as quickly as you
can, and move into the room that has your new number.''
And they did. The person in room 1 moved to room 2. The
family in room 2 moved to room 4, the kids in room 3
moved to room six, and so on in all the rooms. The room
4,886 was now occupied by the people that had been in
room 2,443. The driver was motioning people off the bus,
there were people everywhere, infinitely many people! Where
would we put them all?
Sam pulled me aside and whispered, ``It's OK, George, I told
them they could take any odd-numbered room that they
wanted. There are infinitely many odd numbers, you know,
so it should come out just right, don't you think?''
Well, it did come out just right. And people kept coming. For
years and years they came to see and visit and change
rooms at the Hotel Infinity. Sometimes it was a carload of
people that would come, sometimes just one person on
roller skates or on a unicycle, sometimes a regular bus, and
sometimes one of Sam's busses for infinitely many people.
It always worked. Once five of the Continue 'Em Tour
busses came at once. Another time there were 10. And one
time over a hundred and fifty of those bus loads pulled up
at once. No problem! We could always fit them in. Until the
day when---well, you've probably all heard plenty about
that.
Some say that Kronecker had a hand in the disaster. Maybe
he did, but you want the truth, so I can only tell you what I
know. He was the one who rented all of the Continue 'Em
Tour Busses that day, filled them up and sent them over. I
know that because the Continue 'Em Tour director called
and told us. She knew all about the hotel, of course, but
even she was a little concerned about what could happen if
all of the busses showed up at once. I told her not to worry,
and passed the message along to Sam just to let him
know, and to my surprise, he looked alarmed!
He hurried to the safe and unlocked it for the first time in
oh-so-many years. The crinkled papers he pulled out were
covered with mathematical-looking writing.
``When I discovered this trick, I was amazed that such a
thing could be possible,'' he said. ``Now, how does it go
again?''
I could hear rumbling in the parking lot. The busses were
arriving. He began to shuffle faster and faster through the
brittle pages, and as he did, they crumbled into hundreds,
thousands, maybe infinitely many flakes of old yellowed
paper.
``It's lost!'' he cried out. ``The secret is lost! If I cannot
remember, Kronecker will have beaten us,'' and he buried
his head in his hands.
``Sam,'' I said, ``It cannot be so bad. We have
accommodated many busses before. We will just do the
usual trick. The people like the loudspeaker. They don't
mind moving. So what if it is 200 or 500 or even a thousand
busses. How many of those busses did we build anyway?''
He looked up at me in a way he never had before. He spoke
to me, not like I was his assistant, his builder, or his
employee. Sam spoke to me like the old friend he had
become.
He said my name quietly. ``George, you don't know, do
you?''
I shook my head. I had stopped trying to count things so
long ago. Of course I didn't know.
``George, I have maybe done a terrible thing keeping it all so
secret, secret even from you. I left it up to you to notice,
George, but I didn't really think you would. You didn't notice,
did you, that every time we finished a room in the hotel we
finished a bus.''
I stared at him. ``A bus for every room?'' No, I hadn't
noticed.
``Yes,'' he said. ``As many busses as there are rooms.''
``Then that means...''
``Yes,'' he said. ``Infinitely many busses. And they are
beginning to line up outside now. I cannot remember what to
do to fit all those people in. We are finished. Let Kronecker
say he has beaten us.'' He stood up and hobbled over to
the avocado tree. ``I need your help,'' he said, slipping the
brass tip of his cane under the lip of the pot. We leaned
hard together and the heavy tree tipped a bit. With a deft
kick of his once-shiny shoe, he sent that little sign I had
made skittering out across the floor.
``I'd always been amused by your little secret,'' he smiled
sadly. ``Go ahead. Put it up. Let the Hotel Infinity say No
Vacancy. Kronecker can take a picture for all his
newspapers.''
Next to that potted tree, he stood, not straight and tall in
the manner of the Sam I will always remember, but hunched
over, his fine clothes hanging loosely from the bony hump
between his shoulders. He turned and went through door
number 8. The draught from the hallway that leads to all the
rooms whose numbers begin with `8' caught the blueprints
and they floated like giant feathers to the floor. The
fragments of crinkled paper whirled and fluttered like snow.
The last I saw before the door closed was a slow-moving
figure that didn't carry his cane over his wrist in that old
jaunty way. He used it to support himself as he hobbled
along.
``Sam is old,'' I said to myself. The door clicked shut
behind him, and the nail that fastened the wooden `8' to the
door must have come loose, because the number tipped
over on its side. ``This hotel is old, too,'' I said. ``So am I.
But I'm not too old to think.''
There was honking and shouting in the parking lot. The
drivers who were awaiting their instructions for unloading the
busses were getting impatient, and more busses were
pulling up. I had disobeyed Sam when I made that little sign,
and I was ready to disobey him again. First I made the usual
announcement that emptied out all the odd-numbered
rooms. That gave me some time to think.
``Infinitely many odd numbers,'' I said to myself. ``Infinitely
many rooms empty, I know that much for sure. How can
there be more than infinitely many people on those
busses?''
Then, I started out slowly toward that parking lot, and with
every step I thought harder and deeper. When I got to the
first bus, I had an idea that was worth a try.
I told the first couple of drivers to pass the word down that
all the people would file off the busses and fill up the
odd-numbered rooms. Now all I had to do was get all the
folks to move off the busses in an orderly way so no one
would get skipped, and no one would have infinitely many
people in line in front of them. Then they could fill the odd
numbered rooms in order.
``OK,'' I said. ``Now, we'll take the first person off the first
bus and the first person off the second bus.'' A lady with a
briefcase. A man and his dog. No problem. ``Now, we get
the next two people off the first bus, one person from the
second bus, one person from the third bus, and one
person from the fourth bus.''
You know, people who come to the Hotel Infinity are all so
wonderful. Other folks might have minded the wait, but
these people were cheerful, like it was a game. People who
came to the Hotel Infinity counted on doing something
unusual.
After the first person got off the fourth bus, we worked our
way backwards, taking one person off the third bus, the
second bus, and the first bus. Then one more off the first
bus, and we worked our way back up, taking one person
from each bus that was already in the game, and adding a
person from the fifth bus and a person from the sixth bus
this time. Back and forth we went, always letting two people
off the first bus, and starting to empty two more busses
each round.
It looked like it would work. More and more people were
moving. At each round, more busses came into the game
and the people began to file out. The odd numbered rooms
in the hotel were slowly filling up. Word spread down from
driver to driver about how the busses were being unloaded.
You could hear whispering in the busses as the people
passed the word to one another about how it was all
working.
At least I think it was all working. I think it would have
worked. After they didn't need my help on the busses
anymore I went inside and sat in the big leather chair. The
rooms in the seven hundreds were about half full when
people started rushing out saying they smelled smoke. I
didn't wait to sound the alarm. A big wooden building that
old, and all those nice people---I couldn't put them in
danger. I got on the loudspeaker and told everyone to get
back on the busses--including the people in the
even-numbered rooms. The guests were all out and the
busses were pulling away for safety when the big blaze
started to roar.
Maybe the building was just too old and it was hopeless.
Maybe the fire fighters could have saved it if it had been
made in a different way. You know there were all those
twists and slants I told you about. Ever so slight, but they
mattered. They only thought they could head down Hallway
Eight to get around the other side of Hallway Seven where
the fire seemed to have started. Of course it seems like it
ought to go together that way, but nothing fit together
inside that building the way you'd think it would fit. I guess
that's the reason why it all fit.
Anyway, it's good they went down Hallway Eight instead of
Hallway Two like they should have. Because they found
Sam, hunched and hobbling, rounding the curve towards
the eight millions. Oh, he gave him a time, he did, holding
them at bay with his cane and hollering at them to get out,
that there was no vacancy here, and that the Hotel Infinity
couldn't be on fire because there was no Hotel Infinity
anymore. They took him to the hospital figuring he was in
shock about the fire.
Well before long it was true. There was no Hotel Infinity
anymore, just hot ashes, coals, and thin plumes of smoke.
I've been thinking since then that if we could have saved
just one hallway we'd still be in business. But we didn't. All
that's left is the sign that stood up outside, and this little
sign here that I never put down after Sam handed it to me.
I'm going down to the hospital to try and see Sam. I go
every day about this time. So far, though, he refuses to see
anyone. If he'll talk to me today I'm going to ask him if
maybe it would be OK if I put the little sign up now.
Hotel Infinity, No Vacancy. People will see it and wonder
if that's possible, wonder how such a thing could be true
문제 3. 위의 Hotel Paradox의 문제점은 무엇인가
알아보고, 이와 유사한 Paradox를 각자 생각하여
보시오.
|
Some books to read about Infinity
- Dunham: Journey through Genius
- Rotman: The Ghost in Turing's Machine
- Barrow: Pi in the Sky
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