鈥淪ORRY, how do you spell that?鈥� asked the receptionist.
鈥淪, m, i, t, h,鈥� said the portly man in the tweed jacket. He was beginning to
wonder whether spending Christmas away from home was a mistake.
鈥淲e have a booking for a Mr Silverbung鈥攏o, he arrived two days ago and
there are four of them anyway . . . I鈥檓 sorry, Mr Smith, but we have no booking
for you on the computer.鈥�
Advertisement
鈥淏ut I鈥檝e got a voucher,鈥� protested Smith, passing it over. The receptionist
looked at it and sucked in her breath like a motor mechanic about to break the
news that the car needed a new gearbox and it would take at least two months to
order one. 鈥淭hat鈥檚 your problem, you see,鈥� she said sweetly, pointing to one
computer-printed entry.
Smith鈥檚 eyes followed her finger. HILTON HOTEL, TENERIFE, it said. 鈥淪o? This
is Tenerife, isn鈥檛 it?鈥�
鈥淚n a manner of speaking, yes . . . but this isn鈥檛 the Hilton.鈥�
鈥淚 told you that when we turned into the driveway,鈥� said Mrs Smith, a
stick-like brunette wearing a hat that resembled a fruit bat that had dropped
dead in mid-flight. 鈥淭he sign was wrong. But you said the signwriter probably
couldn鈥檛 spell . ..鈥�
鈥淭his is the Hilbert Hotel, not the Hilton,鈥� said the receptionist.
鈥淚 thought it was part of the same chain. I said as much to you just now,
didn鈥檛 I, Hermione . . . 鈥�
鈥淵es, and Washingbert is pretty much where the American president lives, too,
I suppose,鈥� snapped Mrs Smith.
鈥淥h. Then we鈥檇 appreciate it if you could call us a cab to the Hilton,
辫濒别补蝉别.鈥�
The receptionist suppressed a smile. 鈥淚鈥檓 afraid you鈥檒l need an aeroplane,
not a taxi.鈥�
鈥淧补谤诲辞苍?鈥�
鈥淭his isn鈥檛 Tenerife, and there鈥檚 no Hilton within a hundred miles.鈥�
Smith looked puzzled. 鈥淏ut just now you said it was Tenerife.鈥�
鈥淚n a manner of speaking, sir. This is Tenner Reef in the Viagran Islands. So
called because the Duchess of Wapping bought it for 拢10 in 1712. Tenerife
is roughly six thousand miles that way.鈥� She pointed towards a large potted
cactus. 鈥淎 lot of people make the same mistake, sir, if it鈥檚 any comfort.鈥� The
Smiths鈥� visions of a leisurely Christmas break, with roast turkey lovingly
prepared by a top chef and no dishes to do vanished like a snowflake in Death
Valley . . . at noon on Midsummer鈥檚 Day . . . with a blowlamp trained on it . . .
turned on full. 鈥淭he airport鈥檚 closed until the New Year, too,鈥� added the
receptionist. 鈥淪hame, really.鈥�
鈥淥h, well, then鈥擨 guess we鈥檒l just have to pay you for a room and sue
the travel agent when we get home,鈥� said Smith. 鈥淵ou take Armenian Excess?鈥�
鈥淭hat will do nicely. I鈥檒l just check room availability.鈥� Her varnished
fingernails tapped daintily at the keyboard while the Smiths held their breath.
鈥淥h dear.鈥�
鈥淭here鈥檚 a problem?鈥�
鈥淲e鈥檙e completely full, sir. Not even a dog kennel to spare. In fact, the dog
seems to have double-booked his kennel to a pack of huskies.鈥�
Smith gestured at their huge pile of luggage. 鈥淲ell, we鈥檙e not equipped for
camping on the beach. We鈥檝e got to stay somewhere!鈥�
The receptionist nodded in sympathy. 鈥淢mm, . . . I鈥檝e got an idea, sir. But
it will mean a bit of disruption . . . Let me check with the Manager.鈥� She
hurried off. About ten minutes later she returned, smiling broadly. 鈥淵ou鈥檙e in
luck, Room 1 has just become vacant.鈥�
鈥淏rilliant.鈥� A thought occurred to Smith. 鈥淲here are they going, then? You
said the airport鈥檚 closed.鈥�
鈥淚nto Room 2.鈥�
鈥淏ut Room 2 is occupied.鈥�
鈥淣o, they鈥檙e moving into Room 3.鈥�
鈥淎nd where are they going?鈥�
鈥淩oom 4, of course. We鈥檝e relocated the people in Room 4 to Room 5. In fact,
we鈥檝e moved everybody up one room number.鈥�
Smith felt his head spinning. 鈥淏ut鈥ut what about whoever鈥檚 in the last
谤辞辞尘?鈥�
鈥淎h, I see you鈥檙e labouring under a misconception. This is a Hilbert Hotel.
There is no last room. We have infinitely many rooms鈥�1, 2, 3, and so on
forever. Right now, of course, they鈥檙e all full because we have infinitely many
guests. But infinity plus one is still infinity, so we can rearrange the guests
to create an extra room for you.鈥�
Room 1 wasn鈥檛 bad, actually, apart from the pickled onions on the wallpaper.
(No, not in the design. On the wallpaper.) The Smiths had just unpacked when the
phone rang. It was the receptionist.
鈥淢r Smith, I鈥檓 sorry to disturb you, but would you mind moving to Room
21?鈥�
鈥淲hy, doesn鈥檛 that have pickled onions?鈥�
鈥淚鈥檓 sorry? Er, anyway, a coachload of people has arrived unexpectedly and we
need 20 more rooms. I鈥檓 asking everybody to move up 20.鈥�
They could scarcely refuse, having just benefited from a similar
rearrangement. It was a nuisance repacking, though. When they got to Room 21, Mr
Smith began to unpack, but his wife stopped him. 鈥淎lfred, I am having one of my
premonitions鈥攍et鈥檚 wait half an hour.鈥�
After 29 minutes, the phone rang. 鈥淚鈥檓 very sorry, sir, but it turns out that
the coach is a Cantor Coach which contains infinitely many passengers. We need
to create an infinite number of spare rooms.鈥�
鈥淚 don鈥檛 see how you can manage that,鈥� said Smith. 鈥淪urely infinity plus
infinity must be infinitely bigger than infinity!鈥�
鈥淣ot at all, sir. We want you to move to Room 42. Everybody goes to the room
with twice the current number, leaving all the odd-numbered rooms empty. The new
people can go into those.鈥�
Scarcely had the Smiths lowered their luggage to the floor of Room 42,
however, when there was a knock on the door. It was the bellboy. 鈥淚鈥檝e been told
to help you move to Room 1723.鈥�
鈥淲hat the devil . . . ?鈥�
鈥淒idn鈥檛 the receptionist phone, sir? No, she鈥檚 a bit tied up鈥攊nfinitely
many infinitely large coaches have arrived. Now, it鈥檚 all a bit complicated, but
apparently we can fit them all in if the people in Room 1 stay put, then one
group from coach 1 goes into Room 2, then the people from Room 2 go into Room 3.
Then another group from coach 1 goes into the next room followed by two groups
from coach 2, and the people from Room 3 go into Room 7. Then another group from
coach 1 goes into the next room followed by another group from coach 2 followed
by three groups from coach 3, and the people from Room 4 go into Room
13鈥攕orry, I said it was complicated, but the upshot is that you need to be
moved to Room 1723, and that鈥檚 why I鈥檓 here.鈥� The Smiths wearily picked up their
bags.
Night fell, and the influx of unexpected visitors seemed to have stopped.
They went down to dinner. The table was infinitely long鈥攇uests sat in
pairs facing each other at seats whose numbers matched their rooms.
The vegetables were served.
鈥淏russels 蝉辫谤辞耻迟蝉!鈥� came an anguished cry from seat 1. 鈥淚 hate Brussels
蝉辫谤辞耻迟蝉!鈥�
鈥淭hat鈥檚 OK, pass them to me,鈥� said seat 2. 鈥淥h, drat, I鈥檝e just remembered, I
don鈥檛 like them either!鈥�
鈥淣o worries,鈥� said an Australian at seat 3. 鈥淐an鈥檛 stand the ruddy things
meself, mate, but if you give `em to me I鈥檒l pass `em on. Yep, throw yours in
too. Get a move on, though, mate鈥攚e `aven鈥檛 got all night.鈥� Within half a
minute he had passed the unwanted vegetables to seat 4, together with his own
sprouts. A quarter of a minute later they were at seat 5. An eighth of a minute
later at seat 6, and so on. A rapidly growing pile of Brussels sprouts whizzed
past the Smiths a split second before a minute had passed, accumulating theirs
as it did so, and that same split second later, the Brussels sprouts had
vanished from the dining room.
鈥淲here did they go?鈥� asked a puzzled Smith.
鈥淔ell off the top end.鈥�
鈥淏ut there isn鈥檛 a top end. It just goes on forever.鈥�
鈥淚 know. But nobody鈥檚 got any sprouts. Ask anyone, if you don鈥檛 believe
尘别.鈥�
鈥淐an鈥檛 be bothered,鈥� said Smith. 鈥淏ut this sausage is excellent. I wish I had
补苍辞迟丑别谤.鈥�
鈥淎llow me,鈥� said seat 1724, passing over a sausage.
鈥淏ut now you haven鈥檛 got any sausages yourself.鈥�
鈥淣o matter. I鈥檒l get two from seats 1725 and 1726. Then 1725 can get two from
1727 and 1728, and so on. If we hurry up, we can all have two sausages.鈥� He
paused. 鈥淥r would you prefer three? Nineteen? A trillion?鈥�
鈥淣o, no, thanks all the same,鈥� said Smith. But he ended up with six puddings,
even though only one was served to each guest. Everybody ended up with six
puddings, except the person at seat 14 333 652, who was greedy and had
infinitely many.
The next morning, one of the guests, a Mr Russell, decided they ought to form
clubs to help pass the time. A bridge club, a chess club, a club for underwater
mountaineers, a club for people who didn鈥檛 like clubs. In fact, every possible
club, finite or infinite in membership, should be represented鈥攕ubject only
to an obscure hotel rule that if a guest belonged to a club, then everybody else
in their room was an honorary member. So the clubs were made up of roomfuls of
people, not individuals.
鈥淭he clubs must have meeting rooms,鈥� said Russell.
鈥淎驳谤别别诲.鈥�
鈥淏ut all rooms are occupied. Maybe we should move . . . 鈥�
鈥淣o!鈥� yelled the Smiths. 鈥淏ut we don鈥檛 need our rooms during the daytime, not
if we鈥檙e all going to be at clubs anyway. Why don鈥檛 we just agree that each club
meets in one of the rooms?鈥�
鈥淓xcellent idea,鈥� said Russell. 鈥淲e鈥檒l assign each club to a room, and to
avoid wasting space, every room will host one and only one club.鈥� And he went
away. Two hours later he returned. 鈥淗mm, hit a bit of a snag,鈥� he explained.
鈥淒on鈥檛 think it can be done.鈥�
鈥淲hy not?鈥�
鈥淲ell, it was all going swimmingly until I got to the Absentees鈥� Club.鈥�
鈥淲hich is?鈥�
鈥淭he club formed by all the people who do not belong to the club that meets
in their room.鈥�
鈥沦辞?鈥�
鈥淲ell, it has to meet in someone鈥檚 room鈥擣red鈥檚, say. So does Fred
belong to the Absentees鈥� Club, or not?鈥�
鈥淯p to him, isn鈥檛 it?鈥�
鈥淎pparently not.鈥�
Smith thought it through. 鈥淲ell, if Fred does belong to the Absentees鈥� Club,
then he doesn鈥檛 belong to the club that meets in his room. Fair
别苍辞耻驳丑鈥攐辞辫蝉.鈥�
鈥淧谤别肠颈蝉别濒测.鈥�
鈥淭he Absentees鈥� Club meets in his room. So if he belongs to it, then he
doesn鈥檛. OK, then he doesn鈥檛 belong to the Absentees鈥� Club鈥攐h. That means
that he does belong to the club that meets in his room. Which is . . . 鈥�
鈥淭he Absentees鈥� Club.鈥�
鈥淪o if he belongs to it he doesn鈥檛, and if he doesn鈥檛, he does.鈥�
鈥淏it of a problem, isn鈥檛 it?鈥� said Russell.
鈥淓xactly. Which leads me to conclude . . . 鈥�
鈥淭hat it鈥檚 a silly club and shouldn鈥檛 be allowed?鈥�
鈥淧erhaps. A deep philosophical point. The trouble is, if we go down that
route, which clubs do we allow? We might have to disband the bridge club, for
all I know, and that way lies disaster, I can tell you. No, I conclude that the
infinity of all clubs formed from infinitely many people is bigger than the
infinity of people.鈥�
Smith looked worried. 鈥淚鈥檓 not sure you can have one infinity bigger than
another,鈥� he muttered.
鈥淟et me tell you about the even bigger infinity of associations of clubs,鈥�
said Russell darkly.
The rest of the holiday passed festively, and the Smiths relaxed and put on
10 kilos apiece. During the day, the Synchronised Swearing Club met in their
room, but they were out at the Goat-scrubbing Club one day, the
People-Who-Didn鈥檛-Arrive-in-a-Coach Club the next. They had a great time.
Finally, it was time to check out. After 1722 parties of guests had been
dealt with, the Smiths finally found themselves at the head of the queue.
鈥淭hat will be VI$ 264 317.47,鈥� said the receptionist. 鈥淩oughly
拢6000 sterling. Our standard charge, the same for every room.鈥� Smith
swallowed bravely. It was all right, the travel agent would have to pay up
eventually. He handed over his credit card. The check-out girl gave it the kind
of look normally reserved for strange mounds of fungus in dark corners. 鈥淪orry,
sir, but your card expired at midnight on New Year鈥檚 Eve.鈥�
Smith sighed. 鈥淚鈥檓 sorry, but that鈥檚 the only way I have to pay.鈥� The
check-out girl glared at him. He looked round frantically, like a rabbit in the
headlights of a juggernaut laden with lead blocks . . . with a very fat driver .
. . who nursed an irrational dislike for rabbits. But Mrs Smith was having an
animated discussion with the people from Room 1724.
鈥淟et鈥檚 go, Alfred,鈥� she said, taking him by the arm.
鈥淏ut鈥攖he bill, Hermione. Who鈥檚 paying the bill?鈥�
鈥淩oom 1724 is.鈥�
鈥淏ut that鈥檒l cost them a fortune.鈥�
鈥淣o, because their bill will be paid by Room 1725. Whose bill will be paid by
1726, whose bill will be paid by 1727, and so on.鈥�
鈥淵ou mean, all bills are paid but we get the holiday free?鈥�
Mrs Smith settled the dead fruit bat firmly on her head and started working
out how everybody could go by taxi to the airport and end up with twice as much
cash as they鈥檇 had to begin with, while giving every taxi driver a hefty tip.
She gave her husband a smile. 鈥淎lfred, after everything else that has happened,
are you surprised?鈥�