The Pinkie Pies Guide to Infinity
“Rarity called it a Pinkie hurricane—the whole town was full of Pinkie Pies, hopping around, shouting: ‘Fun! Fun! Fun!’ Our friends had to hide out in a den in the woods with Fluttershy’s critters. It wasn’t funny at the time, but now that we look back at it—”
Unable to contain her laughter, Twilight paused the story to get over her giggles while Starlight Glimmer stared at her with full attention.
“That’s quite a story.” She was actually thinking: and I thought I was the irresponsible pony using Fiducia Compelus on my friends.
Twilight nodded, still grinning. “It’s the perfect anecdote to illustrate the point that while multitasking can be a useful technique, some tasks can only be done one at a time and trying to be in two places at the same time will drive a pony crazy. I used it when I wrote the introduction to the new edition of The Art of the To-Do List for Random Horse Publications.”
“Well I still think that a sensible use of Similo Duplexis can be beneficial,” said Starlight. “But you’re right of course. What did you do with them all?”
“We sold three hundred and fifty copies at the launch party in Canterlot—”
“No—I mean what happened to all the other Pinkie Pies?”
“Oh—I found a spell to send them back to the mirror pool. The problem was we had to find a way to tell the real Pinkie Pie from all the clones. Fortunately somepony had the bright idea of testing them by making them do something totally not fun. Only the true Pinkie Pie had the determination to endure the challenge. So we got them literally watching paint dry—you should have seen them—sweat dripping down their pink foreheads as they stared at the paint-covered canvas, desperately trying to resist their urge to fidget—it was only minutes until the first one cracked.”
“And then what did you do to them?”
“I cast the spell to send them back—zap!” Twilight bowed her head a sent a pink bolt of light flying out of her horn to illustrate the technique. The optical effect flew across the room at rather less than light speed before disappearing harmlessly before the wall.
“You killed them?”
“No—the spell just sent them back to the pool. You could see their magical spirits flying towards the woods.”
Starlight was troubled by this. “Seems kind of genocidal.”
Twilight stopped smiling. “I never thought of it that way. It can’t have actually killed them could it?” She looked uncertain. “They weren’t the same as the real Pinkie. They had her instincts, but not her concern for her friends. I don’t think they were quite the same as real ponies.”
Like any good student, Starlight’s instinct was to question her teacher.
“But they still had enough sentience that you had difficulty distinguishing them from the real Pinkie. Was it ethical to get rid of them?” She paused for a moment, then realised what she had suggested of her teacher, and tried to backtrack. “I mean—I’m sure it was—since you did it.” She smiled broadly showing all her front teeth. “Who am I to talk to the Princess of Friendship about ethics? Me? The pony who took the cutie marks away from an entire village and screwed up the space-time continuum because of a silly grudge.”
However Twilight had starting thinking.
“That’s a good question. We were so desperate to get rid of them that we didn’t ask what would happen to them. I sort of assumed they would become part of the magical quasi-consciousness of the pool itself.”
“Just forget I mentioned it.”
Twilight wasn’t listening. “Where’s that book that describes the legend of the Mirror Pond?” She jumped up from her chair. “Spike! Spike!”
A few minutes later, after her personal assistant had retrieved an old tome from a secret compartment of the castle library, Twilight was flipping through the pages. Starlight waited patiently while her teacher mumbled “hmm… yes… no… really?.. I’d never have thought…” Eventually she was satisfied with her reading and turned back to face her pupil.
“Apparently the Mirror Pond incorporates an ancient magic to allow it to accommodate all the duplicate ponies it produces.”
“Accommodate them?” Starlight had a vision of a three-dimensional array of Pinkie Pies floating in stasis with their manes drifting in the water.
“It sounds very ingenious.” Twilight smiled with the admiration of a piece of clever magic. “Whoever, or whatever, created the enchantment must have anticipated this problem. It says it can accommodate an unlimited number of duplicates.”
“Unlimited? How can it do that?” Starlight was sceptical. “Surely any assigned magical volume would fill up eventually.”
“I don’t know…” Twilight flipped the pages of the book looking for more information. “Ah-ha. There’s a spell we can cast to reveal where they went.”
“Really?”
“But I don’t understand this bit.” She gave the page in front of her a puzzled stare. “It says it is possible because the union of countably many countable sets is countable. Is that a mistake?” She further squinted at the Old Ponish text. “I don’t know what that means. Why do these old books never give proper references?”
“One way to find out.” Starlight grinned.
“I guess.” Twilight paused as she pondered the risks of experimenting with spells she did not entirely understand. After a moment curiosity got the upper hoof.
“Let’s do it.”
A dual teleportation later the two mares stood at the subterranean pond hidden behind the thickest of brambles and most twisted of vines.
“I last came here with Sunburst,” said Starlight. “Did you know this connects to Maud’s cavern?”
Twilight wasn’t listening to any small talk. She walked up to the edge of the pool and looked down at the water. Her reflection looked up back at her.
“Now recite the spell with me:
Into a subset selection she stared,
Confused by ideas of infinity squared
And solemnly sweared,
Not to be scared,
By two countable sets of sets to be paired.”
“—what does that mean?” Starlight asked after speaking the words as directed. Before she could think further they plunged onto the water—or maybe the water rose up to meet them—it was hard to tell—and emerged from the surface in an alternative magical world. The light was bright, but a mist prevented them seeing too far away. The cavern and plants had disappeared, but the pool remained. The waterside was littered with deck-chairs, parasols, and a few discarded towels and toys. An oddly-shaped white inflatable unicorn floated in the middle of the water.
Feeling the not-unfamiliar sensation of passing through a mirror to another world, Twilight instinctively looked down between her forelegs and counted her hooves. Satisfied that she was able to walk normally, she took a few steps forward.
“Where are we?” asked Starlight. “Is this the Equestrian Riviera?”
“I—” began Twilight before she noticed a building in front of her. “Is that a hotel?”
In front of them was a tall palatial white building with many windows. The façade extended as far as they could see into the mist to the left and right. The entrance was marked by four gleaming white columns. Above this a sign, painted in large gold letters, read: ‘The Grand Hotel’. To one side of the door, a smaller notice said: ‘No Vacancies’.
Starlight and Twilight walked up to the entrance and, with no other option in sight, went through the revolving door into the hotel.
A familiar face with an oversized grin greeted them from behind the polished marble hotel reception desk.
“Hello my name is Pinkie Pie. Welcome to the Grand Hotel!”
“Pinkie?” said Starlight.
“I think this must be one of the clones,” whispered Twilight.
“Do you want to stay here?” asked the Pinkie clone.
“We… err…” While Twilight and Starlight pondered this question, a door to the side of the desk opened and another Pinkie Pie bounced in.
“Yay! New guests! My name is Pinkie Pie. I am the manager at the Grand Hotel, where new guests are always welcome.”
“But the sign outside said you were full up,” pointed out Starlight.
“We’re always full at the Grand Hotel,” said Receptionist Pinkie Pie.
“And we always have room for more guests,” added Hotel Manager Pinkie Pie.
The two grinning pink faces were met by confused expressions from the unicorn and alicorn. Starlight looked around the room. It was furnished with small ornamental potted trees standing by polished stone pillars, not unlike any luxury hotel lobby. The Pinkie Pies in front of her may have been clones, but their smiles looked authentic, unlike those the followers of her anti-cutie-mark cult had been trained to give to outsiders. Yet here was something very odd, and she had a feeling that she would not relax until she understood what was going on.
“How?” asked Starlight. “You’re always full, and you always have room for more?”
“Stay with us and find out.”
“OK…”
This not-entirely-enthusiastic affirmation was all Receptionist Pinkie Pie needed to launch into the check-in procedure.
“Yay! Fun! You will be staying in room Number 1. Are you going to share the room?—if not—no problem we can get you another. Here are your towels. Please remember if you go for a swim, your room number may have changed when you get back. Breakfast is served from 6.30. We recommend you get down early as it does get a little bit crowded. Now Pinkie Pie will show you to your room.”
Hotel Manager Pinkie Pie ushered the new guests along a corridor. Receptionist Pinkie Pie bounced along behind them, not wanting to miss out on the fun.
“We have an unlimited selection of leisure activities and in-room treatments,” said the manager. “There’s Yoga and Pilates on Monday. We have a creative writing class—who have written some amazing sonnets and plays in their group exercises. And if you’re feeling lonely you can always just go down to the pool and find a new friend by staring into your reflection and solemnly swearing not be scared at the prospect of being doubly mared.”
This final statement alarmed Twilight. “You can’t do that! It can cause a lot of trouble—”
“Yes we can,” said both Pinkie Pies. “It doubles the fun. We do it all the time.”
“But what happens to all the ponies?”
“We can always find room for new friends,” said the receptionist.
“Is that sustainable?” asked Starlight.
“It is at the Grand Hotel,” said the manager proudly. “We have all the rooms we need. We have no trouble with recruitment. We can always get more staff when needed.”
They stopped at the first room, which was labelled ‘1’. She banged a hoof on the door. This was opened by another Pinkie Pie.
“Time to move.”
“We get to move rooms! Fun!”
Pinkie Pie 1 hopped out of her room, banged a hoof on room number 2 and shouted “We’re moving rooms!”
Pinkie Pie 2 stuck a grinning head out of the door to her room and shouted, “Fun!” before bouncing over to room number 3 to repeat the exercise. Pinkie Pie 3 moved along to room 4. This continued ad infinitum with cries of ‘Fun!’ echoing down the corridor.
Hotel Manager Pinkie Pie removed the key from the door of room 1 with her mouth and passed this to Starlight, then she hopped around the room straightening a few ornaments, before jumping the bed to test the springiness.
“We hope you enjoy your stay. Is there anything else you need?”
“Err… no thanks,” replied Starlight. “You know, you just got all of your guests to move rooms...”
“Yes!”
“...You put yourself—or at least your guests—to an infinite amount of trouble to create a new room.”
Hotel Manager Pinkie Pie beamed proudly while Receptionist Pinkie Pie vigorously nodded in agreement.
“We spare no expense for our guests at the Grand Hotel.”
“We can always find room for another guest.”
“Or another two guests!”
“Or any number of guests!”
“We can accommodate an infinite number of new guests!”
Twilight looked sceptical at this claim.
“How can you find an infinite number of rooms?” She was writing notes on a pad of paper in an attempt to clarify the logic. “If you have N new guests, then you can move all the guests from their current room number M to room number N+M. That will free up enough new rooms, but if N is an infinite number, you can’t do it.”
Manager Pinkie Pie broadened her smile even further.
“The Grand Hotel is not a finite establishment. We don’t have limits. Watch.”
Before Twilight or Starlight could respond, Pinkie Pie walked up to a fire alarm button on the wall and smashed the glass with a hoof. Doors flew open all along the infinite corridor.
“Evacuate the hotel!”
“Is it a monster attack?”
“Or a drill?”
“It’s just for fun!”
“Fun!”
“Fun! Fun! Fun!”
“Did somepony say fun? Where’s the fun?”
Pinkie Pies emerged from their rooms and trotted, skipped, and jumped down the corridor. Twilight and Starlight were swept along by the herd rushing out of the hotel.
“What’s going on?” Starlight stammered.
“We’re going to show you that you how we can accommodate an infinite number of guests, by moving the infinite number of guests in this one into another fully-booked infinite hotel! Fun!”
“Is that possible?” cried Starlight.
Twilight flew up in the air to be free of the Pinkie stampede. She looked at the notes on her pad. “Maybe it is… Mathematically…”
By the pool side Hotel Manager Pinkie Pie stared into the water and recited a new spell:
“Into her own reflection she looked
With no fear of being doubly booked
And faithfully promised not to be shocked
At a unphysical logical construct!”
After she had spoken these words, an infinite number of Pinkie Pies jumped into the pool taking Starlight and Twilight with them. They were magically reflected into another dimension of the pool (without getting wet) and arrived at another water-side hotel, apparently identical to the one they had left. The infinite Pinkie herd bounced up to the hotel entrance and filed through the revolving door one by one.
Twilight and Starlight bypassed the line by teleporting into the hotel lobby, where Hotel Manager Pinkie Pie (whether she was the same one they had spoken to before was not clear) was directing an infinite number of new guests in groups.
“—you will be in room 1, you in room 3, you take room 5, you are in 7...”
“Where do the guest’s currently in those rooms go?” asked Starlight.
“Easy!” said Pinkie Pie. “Everypony in the hotel moves to the room with the number twice of their current room. Every other room becomes vacant and we can fit in all the new guests!”
“I see,” said Starlight, watching every Pinkie Pie happily sign their name in the guest book and bounce off down the corridor to find their new room.
“Fun! Fun! Fun!”
Twilight nodded. “I get how it works. You create an infinite number of empty rooms by moving the pony in room N to room 2N. . All the odd-numbered rooms are then empty.”
“This is starting to make my head hurt,” said Starlight. “Are there really an infinite number of Pinkie Pies here?”
“I guess so,” said Twilight. “The clones know how to use the magic of the pool to duplicate ad infinitum, but the pool is able to contain such an infinity. It really is a very clever piece of magic.”
Ever the questioning student, Starlight was sceptical. “But can we prove that? There could just be a very large finite number of Pinkies Pies here?”
“Do you want to stay and count them?”
“Err… maybe not.”
“I think it’s time to go home.”
A short trans-dimensional teleportation later and Starlight and Twilight were back in the castle library. They fell down onto the soft cushions. Starlight put her head back and closed her eyes with a sigh of relief. Twilight took out her notebook and pen and started scribbling notes.
“This has been really fascinating. Now I understand what the book was referring to. The pool is clearly designed with a robust mathemagical logic so it can accommodate an infinite number of clones. You can add an infinite set, or even an infinite number of infinite sets, to it, as long as it is a countable number of individual ponies.”
“That’s all mathematically correct,” said Starlight. “It’s just not physically possible.”
“It doesn’t have to be. This is not a physical reality, but a magical illusion created by the pool.”
Starlight sighed. “I think that’s enough adventures in mathematics for one day. What shall we do now?”
Twilight smiled. “We could go and seen the real Pinkie Pie and get some cupcakes at Sugarcube Corner.”
“Sounds like fun.”
Author's Note
For more details, see: A Pinkie Parable of Hilbert’s Hotel
