“Rose? Rose!”

Rose heard the sound of snapping fingers, and looked up from her plate to see Alexandria staring back at her, brow furrowed.

“Did you hear a word I just said?” the other girl asked.

“Mrghl.” Rose responded, slumping deeper into her chair.

“Long night?” Borodin asked. Alexandria elbowed him in the side. “What? She’s asleep on her feet, dear sister.”

“Don’t be coarse, Borodin.”

“It was a fair question! Perfectly fair. That lout Dexter probably kept her up all night with his ramblings.” Borodin took a long swig of coffee. He was already on his second cup, again.

“Marghl,” Rose said. It had been a long night, but not quite for the reasons they seemed to think. “Dexter never came back,” she said, looking back down at her plate and pushing the eggs around with her fork.

“Every day in this place seems a little bit brighter,” Borodin said. “Don’t tell me you were up waiting for him?”

Rose wrinkled her nose, not responding. After a moment of visible hesitation, Borodin galantly slid his cup of coffee across the table to her. “Buck up, Rose. Have some liquid spirit. I’m an expert, you could say, in mornings that come too early, and that right there is your solution.”

What had Esper said the word solution meant? To dissolve, or to loosen, or something? Rose reluctantly pulled the cup close to her. Her brain was chugging along jerkily, like an ungreased wheel.

She hadn’t seen Esper since yesterday. The other girl had left the inn early this morning, according to Alexandria, apparently not slowed one whit by the idea of exploring a strange city blind. With Dexter missing, and Robin still holed up in his room, it was down to just the three of them.

“Why the long face?” Borodin asked. Alexandria rolled her eyes.

“Let her wake up a little, Borodin. You’re being a boor.”

“I don’t see why I should. You lot never wait for me to be on my feet in the mornings. It’s terribly funny, isn’t it, when good old Borodin is stumbling around half awake. But does he get to give as good as he gets? Nooooo, that would be boorish.”

Alexandria said something back, but Rose tuned out the bickering, staring down into her black cup of coffee. It was true, now that she thought of it. Borodin was always a mess in the mornings, and this disgusting coffee did seem to help. She tried a sip, mouth puckering as liquid hit her tongue. Who would make a drink that was both sour and bitter?

Borodin laughed at her expression, and she rolled her eyes, pushing the cup back toward Borodin.

It was a cold, foggy morning on the Agora. Rose could barely see ten feet in front of her, but Alexandria had given her clear directions. There were paths carved in the stones, possibly for days like this, and all she had to do was stay on them.

Merzhin’s wife was starting her unit on sets today, and Rose had to be there. Absolutely had to be. Even if it meant going alone.

She’d sort of thought, after the first class, that all six of her friends would stick together. But nobody else had wanted to come.

Dexter and Esper were missing. Robin didn’t seem to want anything to do with them. Borodin had run off to some materials class that was happening at the same time in the Refinery. (Which, to be fair, sounded fascinating.) Alexandria had offered to come, but she’d made the offer so late and so reluctantly that even Rose had gotten the hint.

Rose fidgeted with her cloak. She felt unsettled, for some reason; overflowing with nervous energy. Her eyes darted around as she walked, until finally she could make out a shape emerging from the mists.

Her destination took clearer and clearer form as she approached. It was a small gazebo, sized for maybe a dozen people, but with only three inside of it. All of them were looking at her as she approached. She gulped, her nervous fingers fumbling for her riverstone as she squinted to make them out through the fog.

Was that Esper? No, she decided as she drew closer. Not quite. One of the other students was a woman, maybe in her late twenties. The resemblance to Esper was passing at best – but the woman wore the same all-white clothes and blindfold. That couldn’t possibly be a coincidence.

There was one other student, a tall boy with a hooked nose and close-cropped hair. The third person in the pagoda had to be Merzhin’s wife, by the way she was standing authoritatively at the front. She was young, younger-looking than the woman with the blindfold, although it was hard to tell with people in their twenties.

Rose approached slowly, eyes darting between the three strangers. The master turned to her and smiled. Her features were pale and delicate, which might be part of why she looked so young.

“Ms. Rose, I presume?” she asked.

“Um,” Rose replied, a bit of a squeak in her throat.

“I thought so. My husband mentioned you might be coming. My name is Gwendolyn Breton, Master of Logic.”

The woman was wearing long robes that fit just a bit too tightly to seem formal. She held one end and dipped her legs in a curtsy. Rose blinked back at her, not sure what to say, since the woman already knew her name.

Gwendolyn breezed past her silence gracefully, turning back to the other two. “Let’s begin, then. Rose, this is Eiline Bellatrix, and Duncan Roth. I expect you three will be the only students this cycle. Interest in my courses seems to be waning.”

Rose turned to look at the blindfolded girl. So she was related. The blindness must run in the family, somehow. She should ask her whether she knew that Esper was here. Or would that be rude?

The girl turned her head around, giving a polite wave to nobody in particular. “A pleasure,” she said.

While Rose was debating how to respond, Gwendolyn pushed on. “Eiline, Duncan, this is Rose of Summervale. Now that introductions are out of the way, let’s begin.”

Rose pushed thoughts of Esper’s sister – cousin, maybe? – out of her mind, and turned to face the front. Gwendolyn walked a few feet away to stand by the side of an enormous sheet of flat stone held in a wooden frame. Rose was pretty sure she’d figured out how these things worked by now. The enormous flat stone was very hard. Gwendolyn would use a soft stone of a different color to make markings on it, then wipe the dust off when she was done. Ingenious.

Rose retrieved her notebook, pulling her cloak tight against the damp morning cold. The thick mists around their pagoda made it seem like the four of them were the only people in the world. She flexed her fingers, working blood past the knuckle and into the cold tips.

Beside her, Duncan took Eiline’s hand. As Gwendolyn began to write on the flat rock, she saw him drawing patterns in her palm. That was clever. Maybe she could do that for Esper, in Merzin’s class?

“This class will cover the basics of sets,” Gwendolyn said, back turned to them as she bagan to write strange symbols on the board. “A set is simple: it’s just an unordered collection of objects. We will add more formality to this definition later. Now, you might ask, why do we care so much about unordered collections of objects?”

She turned back to them. “For one very simple reason: sets can be used to construct almost any logical objects we desire. Even very fundamental logical objects, such as numbers, can be constructed from sets. Once you understand how these constructions work, you will be able to extend them, or even create your own logical constructs for new problems.”

Rose wrinkled her forehead. What did that even mean?

“Let me give you an example. We will construct the natural numbers, and then extend them. Does everyone know the natural numbers? Rose?”

Rose swallowed as Gwendolyn’s eyes swept across the three of them to stare directly into hers. “Um,” she said quietly. “It’s another word for the whole numbers, isn’t it? Like 1 or 2, but not 1/2?”

Gwendolyn smiled. “That’s right, dear. Now, let’s define 0 first.”

Gwendolyn turned back to the board, and Rose sat a little higher, putting her shoulders back. “0 is simple: it’s the empty set. The set containing nothing. Easy enough. We will write this {}. Now, let’s define 1. 1 is the set containing 0, which we will write interchangeably as {0} or {{}}. Sets can contain other sets, and this is how all constructions will go.”

“Now, if we want to define 2, there are a handful of options. The simplest is to write 2 as {{{}}}, or {1}. This construction is easy but lends little insight. Instead, we will write 2 as {{{}}, {}}, or {1, 0}. You most likely see the pattern already: every natural number is the set of all natural numbers smaller than it.”

Rose dutifully wrote down what Gwendolyn was saying in her notebook. It seemed stupid, though. She already knew how numbers worked. What was the point of writing them with squigly lines?

“Now, there are a number of interesting directions one could go from here. First, we can define an order: a number x is larger than another number y if x contains y. We can define an increment operation. To turn a number x into x+1, we simply construct a set with all the elements of x, plus x itself. I suppose I should mention two operations we will use frequently: union and intersection. The union of two sets is all the elements that occur in either of them, the intersection of two sets is all the elements that occur in both of them. So x+1 is just the union of x and {x}. Does that make sense? Any questions so far?”

The wrinkles in Rose’s forhead deepened. It did make sense, in a twisty, useless sort of way. What in the world was the point?

“Rose?” Gwendolyn said, not turning around from the board. “Do you have a question?”

Rose hesitated, tapping her finger on the spine of her notebook. “How do you add two numbers?” she asked. “With these sets, I mean?”

Gwendolyn laughed. “You can add two natural numbers easily enough. First you have to define x-1. x-1 is the largest element of x, which is well-defined because we already defined an order. Now, x+y is just (x+1)+(y-1). You can perform this transformation repeatedly until the number on the right is 0, at which point you’re done.”

That was asinine. It didn’t answer the question at all, just gave a process for answering the question that took longer the bigger the numbers were. Although, now that she though about it, normal addition took longer the bigger the numbers were too. Just not so unreasonably long.

“Now,” Gwendolyn said. “We have a lot to cover today, so I’m going to move quickly. So far we’ve defined the natural numbers. We’re going to extend the natural numbers to include various types of infinities, just to give you a taste of how powerful our construction is. We’ve defined 0, and we’ve defined an increment operation. Next, consider the set of all numbers which can be constructed by performing the increment operation a finite number of times. We’ll call it w.”

Rose was pretty damn sure that didn’t make even a tiny bit of sense. She reached through her cloak for her glass, gripping it tightly.

“We can, naturally enough, perform the increment operation on w as well. We can thus get w+1, w+2, and so on. If we consider the union of w and all numbers which can be reached by a finite number of increment operations applied to w, we get 2w. You probably see the trick now. Similar constructions will get you to w^2, w^w, and so on. Sets of this form are known as the ordinals, and are important for a technique known as transfinite induction…”

Rose looked down at her notebook. She’d stopped writing, at some point. She went back and copied down everything Gwendolyn had said, squeezing her glass as tightly as she could to try and wring some sense from it. A cold wind rose up through the mist. It stole more heat from her one exposed hand, and she shivered. The words were clear, nothing seemed wrong. But she was forced to confront the fact, as she copied down word after word, that…well…

She didn’t get it.

Rose pushed through the front door of the Tawny Portus, warm air blowing past her face and out the door behind her. Alexandria was sitting at what had become her usual table. She had papers laid out in front of her. Esper sat beside her, eating a late breakfast with slow and delicate care, but the others were nowhere to be seen.

She lowered the hood of her cloak, brushing hair from her face as she walked over to join them. Her hand came away damp. The weather on this mountain really was something. Too many days of this and her notebook would be ruined.

“Robin?” Esper asked, when she was within earshot of the table. “Or perhaps Rose?”

“Rose,” Alexandria said, not looking up from her papers.

“How could you tell?” Rose dropped her bag over the back of a chair.

“Your steps, my dear. Light and dejected, which usually means Robin.”

Rose sighed, sitting down in the chair and propping her chin up on her hands. An uncomfortable silence hung in the air for a moment, broken only by the cracking of the fire and the sounds of Esper’s fork.

“How was your lesson?” Alexandria asked.

Rose chewed on her bottom lip. “It was good. Really good.”

“You seem a little off today.”

“I’m just tired.” Rose propped her chin up on her hands, staring at the fire. She felt her fingers searching through her cloak, looking for some sort of treasure, but she wasn’t sure what she needed.

Alexandria made a noncommital hmming noise. “Did you ask Master Breton – er, the Mrs. Master Breton, when she’s teaching the lesson on inductive arguments?”

“What?” Rose asked, looking up.

“Did you ask the Mrs. Mas–”

“Oh. Oh, no, I forgot.” She’d gathered her things and left, right after class. That really wasn’t like her, to forget one of the three things Merzhin had told her she needed. She’d been so excited last night.

Alexandria finally looked up from her papers, raising a single elegant eyebrow. “That really isn’t like you.”

Rose sighed.

“Did something happen? Did you run into Dexter?”

Rose sighed again, more dramatically. “No.”

Her fingers were still grasping through her cloak. With a dash of conscious effort, she directed them toward her stone. It was always a safe choice, when she felt a little off-kilter, even if it didn’t quite feel right at the moment.

“Where is our friend and companion from Melan?” Esper asked, her voice soft. “I had a question I wished to ask of him.”

“We haven’t seen him,” Alexandria said. “Not since yesterday.”

Esper made a hmmming noise. “So it goes, I suppose, so it goes.”

“What do you think he’s doing?” Rose asked.

“Who can say? We were together for such a brief time, down below.” Esper raised a hand in front of her face, stretching the fingers and closing them one by one. “In times of danger, men want only one thing. In times of peace and plenty, they want ten thousand.”

“Philosophy?” Alexandria asked.

Esper nodded. “From the mainland. When we were down below–”

“Yes, yes, we get it,” Alexandria said.

“As you say.”

Something tickled at the back of Rose’s brain. She released her stone, and brushed her fingertip against her glass instead.

In times of danger, men want only one thing. In times of peace and plenty, they want ten thousand.

They all wanted the same thing, at least in theory. To enter the tower proper. No, even that wasn’t true; Robin was considering staying in the town.

But down in the catacombs, there had been urgency, danger, and so there had been unity. Now, they were safe and warm, with years and years to manage their entrance. She brushed her glass again, the idea crystalizing.

She was lonely. It didn’t make any sense, she barely knew these people. They were strangers, and she spoke with at least one every day. She shouldn’t care if she spent time with them, and if she did care, she should be satisfied with what she had.

But something about the six of them spending every moment together, sleeping on hard stone and waking up to more puzzles – well, she missed it. She missed feeling like they were on a team.

“What do you want?” she asked Esper, the question coming out of her mouth so suddenly it surprised even her.

Esper turned her head in Rose’s vague direction. “Hm?”

“What do you want? There’s peace, there’s plenty; what do you want?”

Esper shrugged, her white robes rising and falling. “Don’t worry about me, dear Rose. I wish only to observe.”

Rose rolled her eyes. That was nonsense. Maybe she could puzzle it out, though? Dexter would know, or at least he would have a cynical take, which would be a good enough start.

It always came back to Dexter. What did he want? He’d as good as told her, she supposed, with his puzzle on the night of the storm. He had some sort of project, some obsession. It involved paper too thin to sell on the surface. It needed candles for some reason. Based on his question to Merzhin, it likely violated one of the edicts.

“I can tell you what I want,” Alexandria said crisply. She gather together the papers in front of her, tapping them into a perfect stack. “I want a plan. We know what the masters want from us to enter the tower. Now we just need to figure out how to get there.”

Rose smiled. Alexandria, at least, wore her desires openly. Borodin did too, she supposed, in his own sort of way.

“By the way,” Rose said, a thought from earlier coming back to her. “I think I met one of your sisters, Esper. Eiline was her name?”

Esper stiffened, just for a moment. Rose had never seen the other girl anything but calm and languid. “Oh?” Esper said. “Tell her hello from me, next time you see her.”

“I will,” Rose said slowly. “You don’t want to meet her? I’m sure you have a lot to talk about. She’s in my course on sets, every week.”

“She might be able to give us some help,” Alexandria added. “How close were you two?”

“Perhaps I will go with you, some time,” Esper said, her voice oddly stiff. She stood, stretching her long arms over her head. “If you’ll excuse me, though, I had an early morning. I think I might take a brief rest.”

The town around the tower – did it even have its own name? – used an odd address system. It was organized radially, with circular streets wrapping around the tower at its center, and wider avenues cutting through the streets like spokes on a cart wheel.

You’d think a street and avenue pair would give you a corner. But instead it addressed the parallelagram-shaped block outward and clockwise from that corner. Near the tower, you could just check all the numbers on the block. By the time you reached the edge of town, near the manufactory, these blocks were so large they had alleys of their own subdividing them, and it turned into a chaotic mess.

When Rose finally found the address Merzhin had given her, the sun was already low in the sky. Even at the edges of town, buildings stretched two, sometimes three stories into the sky, casting the streets and alleys below them in shadow. The address led to a small shop.

There was a wooden sign hanging in front of the door, with a picture of green liquid in a triangular glass. Rose peered in the window, trying not to look suspicious. It was lined with shelves. The shelves were filled with bottles, jars, and paper-wrapped rectangles, all of them oddly colored and carefully arranged. There was an old man behind the counter, sitting in a high chair and reading a book. He had a long white beard that drooped so low it was laying on the pages. He had to push it out of the way every time he turned a page, but he didn’t seem to mind.

Rose checked the address for the third time. Hadn’t Merzhin said she was looking for another student? This couldn’t be the right place. She glanced at the long shadows on the ground.

Rose gripped her iron and pushed through the front door. A bell tinkled, somewhere in the back, and the old man looked up.

“Ho!” he said, his voice surprisingly strong for his aging frame. “Can I help you, little miss?”

“I’m looking for a student with a book,” Rose said. Did that sound stupid? That sounded stupid. She really should’ve asked Merzhin for a name.

“What?” the old man said, cupping a hand to his ear. “Come a little closer.”

Rose approached the counter, trying to think of something less stupid to say. She took a deep breath, and almost coughed. Something sharp and acrid was in the air.

“I’m looking for another student, to borrow a book. Master Breton sent me.” she said, once she was closer to the old man. He bobbed his head, as if it were the most natural request in the world.

“Thank goodness,” he said. “I thought you were a customer. Lady Lo’loth is just down the stairs over there. He waved vaguely at a door on the side of the store, just in front of the counter, and went back to his book.

“Thank you,” Rose said. She turned toward the door, but hesitated. Her curiosity got the better of her. “Why would you be unhappy to have a customer?”

“Hm?” The old man looked up again.

“You, um, you said ‘thank goodness’. Why wouldn’t you want a customer? If it isn’t rude to ask?”

The old man smiled, and his eyes crinkled at the edges. “You’ve misunderstood me, young miss. Do you even know what I sell here?” Rose looked around again, at the strangely colored liquids and powders in odd containers. She shook her head.

“I sell essences. Do you know what those are?”

Rose did, in a vague sense. Her mother had told her that everything was made of something else, even everyday objects like stone and wood. One of the tower’s mysteries was that they could break objects down to their component parts, and put those parts back together in secret ways, producing entirely new substances.

She looked around again, with more interest. “Tell me more,” she said.

The old man laughed. “And that, young miss, is why I was happy to learn you weren’t a customer. Exactly that impulse. Essences are terribly dangerous. If you don’t know precisely what you want to buy, and precisely what you’re going to do with it, I won’t sell you any.”

Rose was a little disappointed, but she certainly couldn’t argue with that. “And how would one learn precisely what to do?” she asked.

“Read books. Ask questions. Study under Master Woarck, if you can stomach the work. Apprentice under another student. Knowledge is like this town; many avenues that all lead to the same place.”

Rose nodded. “Thank you. You said to go down the stairs through that door?”

The old man seemed a little surprised, for some reason, but he nodded and turned his eyes back to his book. Rose allowed herself one more longing glance at the shelves of magical essences, then locked that longing up in a box and stuffed it away. She’d learn the tower’s secrets, and she’d come back.

She slipped through the door on the side, the acrid smell diminishing as she did so. Past the door was a cramped stairwell heading up and down. Up made sense; there were three stories to the building. But she hadn’t realized it had a basement, too.

Fingers still wrapped around her iron, she started down the stairs, hand trailing lightly on the rail.

The door to the shop’s basement was heavy, made of some wood Rose didn’t recognize. She hesitated for a moment, then knocked decisively three times.

There was no answer from inside.

She stood, shifting her weight from one foot to the other. Was it rude to knock again? How long were you supposed to wait, exactly?

After an uncomfortably long time, she raised her fist to rap a little louder on the door. Before she made contact, there was a slick noise of metal on wood, and the door pulled open.

Rose’s was brought face to face with a black-skinned girl, exactly the same height as her. The girl’s eyes were intense, even angry, and she was wearing what looked to be a single cleverly-wrapped length of purple cloth in the shape of a dress.

None of that mattered, though. Her ears were what caught Rose’s attention. They were pointed, the skin toward the top-back folding over itself to form a crease. Rose’s heart sped up as her brain began to make connections. Elves weren’t supposed to exist outside the tower, were they? Was she a half-elf?

“Yes?” the girl asked. Her voice was melodic, like the diction Esper sometimes used, but more sibilant. Rose realized she’d been staring at the other girl with her hand poised in the air for an uncomfortable length of time.

She flushed, clasping her hands behind her back. “Um,” she said. “Hello–”

“Faster,” the girl said.

Rose could do faster. “Master Breton sent me to borrow a book on binary arithmetic and operations. Could you–”

“No. I’m using it. Good day.”

The door slammed shut in Rose’s face, close enough she felt the wind brush her hair back. She bit her lip. That wasn’t quite how she’d hoped the conversation would go.

Still. A real elf. Maybe. Rose had a brief internal struggle over what to do. Politeness formed an alliance with nervousness and social aversion to argue for leaving, and all three were promptly crushed by the iron first of curiosity. Necessity cheered awkwardly from the back, grateful things had broken the way they did.

She raised her hand and knocked on the door again. There was no answer. She knocked louder, banging until her hand hurt.

The door flew open.

“Yes?” the girl said. Her voice and face were angry, which juxtaposed oddly with the musical tone of her speech.

“Are you using it right now?”

“What?”

“Are you using the book right now? Could I come in and read until you need it?”

The girl stared at her, baffled.

“Also,” Rose continued breathlessly, “are you a real elf? What’s it like in the tower? Are you born with ears like that or is it surgical?”

The girl stood there for a moment with her mouth open. When she finally spoke, she practically snarled at Rose. “You are the rudest, most insufferable bumpkin to have ever darkened my doorstep. I will not let you read my book, and if you continue to bang on my door I will have you tossed head-first off the mountain. Good day.”

The door slammed shut again.

“You look – mmph – happy,” Robin said. He had to pause mid-sentence to shove a forkful of food in his mouth. He was out of his room for once, sitting next to Esper and Alexandria in the inn’s common room, and tucking away potatoes as fast as he could. Eating was the only time Rose saw him, really, and only because he didn’t want to get his books greasy.

Rose sighed, a little more melodramatically than she might have liked, and sat down next to him. “Mrgh.” she said.

Alexandria glanced up at her. It seemed like she was always here, whenever she wasn’t at class, holding court at their little table. It was starting to feel like the only constant in Rose’s life. Rose gave her a small shrug, and Alexandria made a sympathetic face, going back to whatever useless paperwork she was reading.

Rose sighed again.

“What’s the problem?” Robin asked, popping more potatoes in his mouth.

“You’re talkative today,” Alexandria said.

“It’s funny. My whole life, I always thought – Robin, if people would just leave you alone, let you have a moment to yourself to read in peace, maybe life wouldn’t be an endless miserable slog. And you know what? I was right!”

Robin took a long drink of hot water, making an ugly face as he put it down. “I’ll never get used to this stuff. Anyway, the last few days have been the best time of my life. Solitude is amazing. So, yes, I’m on a little bit of a high, and I’m maybe even feeling a tiny bit friendly. What’s the problem?”

Huh. Rose filed that little tidbit away for later. It connected, to thoughts she’d had the other day, about what different people wanted. Was Robin really so simple?

He was waiting for an answer. “I need to borrow a book,” she said. Robin raised an eyebrow, and she hurried to clarify. “Not from you. Master Breton told me to borrow a book from another student, but she won’t lend it to me. She won’t even talk to me. She slammed the door in my face!”

Robin snickered. Rose crossed her arms, pouting. It didn’t seem all that funny.

“You were probably being annoying,” Robin said.

“I was not.” Maybe she’d been a little annoying.

“It’s OK,” Robin said. “Believe me, I know annoying. You wouldn’t believe the things I’ve had to do to wheedle books out of people.”

“We might,” Alexandria said mildly. Robin shot her a look, and she held her hands up, not saying any more.

“She’s a student of the guy who told you to borrow the book?” Robin asked, turning back to Rose. She nodded. Robin smiled, and it looked a little off-putting on his otherwise surly face.