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<title>Harry Potter and the Diagon(alization) Alley - Comfortably Numbered</title>
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<a href="/"><span class="left-word">Comfortably</span> <span class="right-word">Numbered</span></a>
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<article id="postcontent" class="centered">
<section>
<h1>Harry Potter and the Diagon(alization) Alley</h1>
<center><em><p>My attempt at <em>Harry Potter</em>/<em>Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen</em> crossover fanfiction.</p>
</em></center>
<h4>Thursday, February 26, 2015 · 6 min read</h4>
<p>It is a well-known fact among wizarding circles that the Gringotts Wizarding
Bank has an infinite number of vaults, numbered from 1 onwards. Equally
well-known is the fact that in the name of what goblins call efficiency and
everyone else calls parsimony, each of the vaults is currently occupied.</p>
<p>As a result, there has formed a situation which Muggle economists describe as
“scarcity”. Gringotts vaults, like heirloom-quality furniture, are prized in
families. They only change hands at readings of wills of dead great-aunts.</p>
<p>With that in mind, it was, of course, reasonable for Mr. Hill Bertok to scoff
at the old man who demanded of him a vault one chilly February morning.</p>
<p>Mr. Hill Bertok was a run-of-the-mill businessgoblin and teller at the
Gringotts Wizarding Bank. The old man was Professor Albus Dumbledore,
headmaster of the Hogwarts School of Witchcraft and Wizardry.</p>
<p>Mr. Bertok did not, of course, notice this at first. Had he realized who the
old man was, he would perhaps have shown a little more respect before scoffing.
The scoff, however, had been scoffed already, and a scoff once scoffed cannot
be recalled.</p>
<p>Having conceded this, Mr. Bertok attempted mildly apologetic gesture with his
eyebrows and presently returned to his paperwork. When he looked up a few
minutes later, the old man was still there.</p>
<p>“Dumbledore,” he said, “as a goblin I hold you in much higher regard than most
wizards. But I still think you’re crackers.”</p>
<p>“Very well, Mr. Bertok, but many millions of lives depend on the security of
the object I have with me. Perhaps you can make an exception?”</p>
<p>“Exception! What object could possibly require so much security?”</p>
<p>“It’s a rock, Mr. Bertok. A very important rock.”</p>
<p>“…you <em>are</em> crazy, Dumbledore. I’ll tell you what: if you can find an empty
vault, you can have it.”</p>
<p>“Mr. Bertok, I believe I have this problem sorted out already. If you’ll lead
me down, perhaps I can demonstrate.”</p>
<hr>
<p>“Crazy, completely and utterly bonkers,” thought Mr. Hill Bertok as he led the
old man through the labyrinthine passages of Gringotts.</p>
<p>“713 sounds like a nice round number,” said Dumbledore, “Let’s stop here. Do I
have your permission to use this vault? It’s a harmless procedure, really.”</p>
<p>“You are out of your mind.”</p>
<p>“Need I remind you that I am the Chief Warlock of the Wizengamot?”</p>
<p>“You need not,” seethed a very reluctant goblin, adding (under his breath)
“Albus Dumbledore pulling rank—what is the world coming to?”</p>
<p>“Very well then. <em>Presto-incremento!</em>“ For a sliver of a second, Vault 713 of
the Gringotts Bank glowed a pale electric blue. Then, with a gentle rumble, the
door swung open, revealing absolutely nothing whatsoever.</p>
<hr>
<p>“<em>You asinine old man!</em> That vault contained more galleons than your demented
old brain can count!”</p>
<p>“Fear not, Mr. Bertok, you will find all of your galleons. They have simply been
transported to Vault 714.”</p>
<p>“Impossible! Vault 714 contains statues of gold; you cannot fit a mountain of
galleons in there!”</p>
<p>“Patience. The statues of Vault 714 have been moved to Vault 715. And—before
you get started again—the contents of Vault 715 have been moved to Vault 716.”</p>
<p>“Excuse me?”</p>
<p>“Each vault’s contents have been moved into the next vault. Ad infinitum. That
left Vault 713 open for me.”</p>
<p>“Wait, but… how can… <em>our clients!</em> How do you propose to notify them of
the change?”</p>
<p>“I had the foresight to dispatch some Ministry owls ahead of my visit.”</p>
<p>“You cannot send owls to an infinite number of clients, Dumbledore.”</p>
<p>“It may interest you to know, Mr. Bertok, that the Ministry happens to have a
fleet consisting of an infinite number of owls. They’ve been quite useful: just
last week we used them to—ahem—<em>persuade</em> a Muggle family to send their
nephew to Hogwarts.”</p>
<p>And with those words, Professor Albus Dumbledore dropped a small grubby bag
inside the vault and closed it.</p>
<hr>
<p>The news spread faster than the plague. The next morning, the tellers of
Gringotts Bank were greeted at the front desk by Stan Stunpike. Unfortunately,
they were also greeted by an infinite number of passengers on his infinitely
long bus. These were restless passengers who wanted their magically-created
Gringotts vaults and would not leave without them.</p>
<p>Owing to the lack of space and general stuffiness that was developing in the
Gringotts lobby—and the inevitable threat being posed to his employment—Mr.
Bertok decided to do something about it. After the first 50 new clients,
though, he began coming to the realization that making all of these accounts
would take an infinite amount of time.</p>
<p>He didn’t have an infinite amount of patience, but he knew someone who did. He
sent an owl to Dumbledore.</p>
<hr>
<p>“Hush, calm down, everybody.” The lobby was beginning to smell like the inside
of a used coffin, so Dumbledore decided, for the first time in his life, to
quit the whole pedagogical spiel and solve the problem.</p>
<p>“<em>Presto-doublinato!</em>“ he cried, and the lobby began to rumble as an infinite
number of doors began to open.</p>
<hr>
<p>“<em>Now</em> what have you done?”</p>
<p>“Vault 1’s contents have been moved to Vault 2. Vault 2’s contents have been
moved to Vault 4. Vault 3’s contents have been moved to Vault 6. In general,
each vault has been moved to the vault with double the number.”</p>
<p>“How does that help?”</p>
<p>“All the odd numbered vaults are empty. Since each of your angry bus passengers
has a seat number, you assign them a vault based on those. The person in the
first seat is assigned the first odd number (1), the person in the second seat
is assigned the second odd number (3), and the person in the hundredth seat is
assigned the hundredth odd number (199).”</p>
<p>Mr. Hill Bertok briefly considered hiring Dumbledore in the HR department.</p>
<hr>
<p>Wizards, it has been found, have friends. As a result, after the miracles of
the second day, news spread like proverbial wildfire and the parking lot behind
Gringotts was quickly filled with an infinite number of infinitely long buses.</p>
<p>Mr. Bertok knew what to do this time, and before you could yell “cardinality”,
Dumbledore had apparated to the lobby, holding a big bucket of black paint.</p>
<p>Without a word, he walked out to the parking lot and began painting large numbers
on the buses’ windows.</p>
<p>“Hey, Mister, what do you think you’re doing?” cried an understandably
distressed bus-driver. He was quickly and efficiently turned into a frog by
Gringotts’ security team, and Dumbledore continued with his painting,
undeterred.</p>
<p>After a few tense minutes, the parking lot resembled this (pardon my
badly-illustrated row of buses):</p>
<pre><code>+----+----+----+----+----
< 01 | 02 | 04 | 07 | ...
+----+----+----+----+----
< 03 | 05 | 08 | ...
+----+----+----+----
< 06 | 09 | ...
+----+----+----
< 10 | ...
+----+----
|... |
</code></pre><p>“As you can see,” he began (the outdoor parking lot was airy enough for his
pedagogical side to shine), “I’ve numbered each bus window diagonally. If I
keep this up, each window will get a number.</p>
<p>“Since I’ve already shown you how to deal with that with the
use-all-odd-numbers trick, I think I’ll take my leave now. It’s almost
lunchtime.”</p>
<p>He disappeared with a pop, leaving one frog and an infinite number of very
confused wizards.</p>
<hr>
<p>Like all good things, this one had to come to an end. The end came when an
inhabitant of a Hogwarts painting overheard a conversation about Gringotts’ new
policy. And so, the next morning, the portrait of Gringott the Goblin was
inhabited by an infinite number of painting-people. In front of them was a
dashing, athletic-looking young gentleman. A caption floated above his head. It
read “C. Antor. Tennis player.”</p>
<p>“Greetings, Mr. Bertok. I’m Charles Antor, representing the paintings.”</p>
<p>“Welcome to Gringotts, Mr. Antor.”</p>
<p>“Well, we came to ask: can you work your magic and give us vaults, too?”</p>
<p>“What does a <em>painting</em> need a vault for?”</p>
<p>“If you’re going to denigrate us, we’ll take our business elsewhere.”</p>
<p>Mr. Bertok hesitated. “Alright, we will need you all to line up so that we can
number you.”</p>
<p>“I’m afraid we can’t do that.”</p>
<p>“What?”</p>
<p>“We aren’t numbered with counting numbers like 1, 2, 3… We’re numbered with
real numbers. Decimals and fractions. Each of us identifies himself or herself
with a number between zero and one.”</p>
<p>“So? Why can’t you all stand in order?”</p>
<p>“Well, suppose we did order ourselves in a line, and suppose you assigned us
all vaults.”</p>
<p>“Alright, then what?”</p>
<p>“What is the number of the painting at the front of the queue?”</p>
<p>“<em>What?</em> How can that possibly matter?”</p>
<p>“Please bear with me. Just invent a number between zero and one. It doesn’t
matter which one.”</p>
<p>“If you say so. The painting at the front of the queue is painting 0.234567.”</p>
<p>“Well, the first digit of my number is a 1, not a 2 like his. So can we agree
that I’m not at the front of the queue?”</p>
<p>“I suppose so.”</p>
<p>“Good. Now, who’s second?”</p>
<p>“Should I make up another number?”</p>
<p>“Sure.”</p>
<p>“0.1111111.”</p>
<p>“Well, my number’s <em>second</em> digit is a 9, not a 1. So I’m not second in line
either.</p>
<p>“I don’t see where you’re going with this.”</p>
<p>“Mr. Bertok, no matter how you number us paintings, there will always be
someone whose first digit is different from the first person’s first digit. And
whose second digit is different from the second person’s second digit. And
whose third digit is different form the third person’s third digit. And so on,
and so forth.</p>
<p>In other words, he cannot have <em>any</em> position in your queue, and so he can’t
be in the queue.”</p>
<p>Mr. Bertok scratched his head. “But if there’s no way to number you folks,
there’s no way to assign you vaults.”</p>
<p>“Exactly. In a weird way, there are ‘more’ of us than there are of wizards in
Stan’s bus. Even though both numbers are ‘infinity’.”</p>
<p>“Gosh, who know paintings could be so complicated? You aren’t even real.”</p>
<p>“On the contrary, Mr. Bertok, we are as real as can be.”</p>
<hr>
<p><strong>Epilogue.</strong> Mr. Hill Bertok, now inspired, went on to study infinities. After
an interesting encounter with two old wizards named Banach and Tarski, he
discovered a means of mining an infinite amount of gold. Surprisingly enough,
he ended up living happily forever after.</p>
<p>Charles Antor went on to become a tennis star among the paintings.
Unfortunately, he met his match when a certain B. Russell proved to the referee
that none of his sets could exist.</p>
<p>The events of this story thoroughly confused Stan Stunpike, who decided to take
an early retirement from bus-driving and instead perhaps go herd a finite
number of goats in Mongolia. Fortunately, a pair of psychiatrists, Calkin and
Wilf, convinced him that he can fit all the rational people in the world onto
his bus. Though he tried hard, he never really got the hang of it, and one day
his bus was found—destroyed—with an infinitely large tree growing out of the
windshield.</p>
<hr>
<blockquote>
<p>This post was inspired by something my computer science teacher said. I
forgot what exactly it was that he said. Most ideas were shamelessly stolen
from a chapter in Ian Stewart’s delightful book, <em>Professor Stewart’s Cabinet
of Mathematical Curiosities</em>.</p>
</blockquote>
<hr>
</section>
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