#bounded arithmetic

[pv1isalsoimportant]

turns out that the reason why i couldn't come up with a proof was because the theorem didn't hold.

[cups-official]

the first rule of using pointers is have fun and be yourself

[flickering-chandelier]

Snow on the Beach

Pairing: Cassian x Reader

Summary: As soon as Cassian and Reader run into each other, she feels the bond snap into place. But feeling unworthy next to the famed Cassian, she keeps it to herself, assuming that he would never find out the truth. Before long, she finds that she was very wrong.

Based on this request! 🩷

Warnings: insecure reader

Word Count: 4k

While walking Nyx to school on an early autumn morning, Cassian was certainly glad that the Illyrian soldiers could keep themselves in line for a day or two, giving him a respite to come home to Velaris and spend time with his nephew. It seemed to him that the boy was growing up far too fast; he could hardly believe that he was already in school. 

“So kid,” Cassian said, gently steering Nyx away from the street with a hand on his back, “tell me about school. Have you made any friends?"

Little Nyx grinned. “Yeah!”

“I bet the other kids think it’s cool that your parents are the High Lord and Lady, huh?”

He nodded, “At first, they all thought it was really cool. I think they’re getting used to it now, though. Now everyone is talking about how Astrid’s mom opened that new restaurant by the rainbow.”

Cassian laughed, unable to resist ruffling Nyx’s hair. “Well, they do have good food.”

Nyx agreed, and Cassian took the tiny hand in his own once the boy started veering toward the street again. “Do you like your teacher?” 

“She’s really nice,” he said. “She barely even gives us any homework.”

“Now, that’s a good teacher,” Cassian agreed. 

Nyx chatted idly with Cassian until they got to school.

Cassian was surprised to see a beautiful young woman standing near the front door, greeting the kids as they filtered in through the building. 

You smiled at Nyx as he and Cassian approached. “Good morning, Nyx! Who did you bring with you today?”

“This is my uncle Cassian,” Nyx said proudly, before turning back to Cassian and informing him, “This is my teacher!”

Teachers did not look like that when I was in school, Cassian thought. 

“Hi,” Cassian smiled, extending his hand. You took it, your eyes widening slightly when your hand touched his. 

“Nice to meet you,” you said. Cassian thought you sounded a bit out of breath. “Are you picking Nyx up today?”

Cassian said, “I am,” and was suddenly very glad for it. 

You nodded, still looking somewhat dazed. “Okay, we’ll see you this afternoon then.”

Nyx said goodbye and bounded into the building, but Cassian couldn’t help but feel like something was off. “Are you okay?” he asked. “You look… a little pale, maybe?"

“I’m fine,” you said hurriedly. “Thank you.”

Cassian nodded. “Okay. Good.” Cauldron, why was he flustered now? “This afternoon, then. Have a good day. Good luck with the little ones.”

You laughed, thanking him, before Cassian turned away. 

---

You could hardly breathe. And he had known it too, which made everything so much worse. 

The last thing you were expecting today was to meet your mate three minutes before school started. 

Cauldron. How could the Cassian be your mate?

You were fairly sure you recognized him as he was walking up, but didn’t want to assume. The High Lord and his inner circle were well known here, of course, but you had never met any of them besides Rhysand and Feyre, and that was only because Nyx was in your class.

Cassian was unlike any male you’d ever seen before. His biceps were the size of your head. He could snap you in half like it was nothing. And yet he was holding little Nyx’s hand in his, walking him to school. 

Gods.

You were nervous enough about having the High Lord’s son in your class, but then a member of his trusted inner circle shows up, touches your hand, and the bond snaps into place?

And just like that, the commander of the High Lord’s army, the one that they’ve written entire books about, was your mate. 

Yet, here you were, teaching arithmetic to a classroom full of tiny children.

It didn’t make sense. You had never heard of the Mother making mistakes before, but surely this was one. Why would someone that powerful have a schoolteacher for a mate?

Cassian didn’t seem to feel it when you had. Surely you would have noticed if it had snapped for him at the same time. 

So, he didn’t know. And you figured it was better to keep it that way. You would sound foolish, telling this war hero that you were his mate. 

And it wasn’t likely that you would see him again after today, anyway. 

You just had to get through the day, and pretend that everything was normal. 

By the time parents were back picking up their kids, you had managed to put Cassian out of your mind. 

That is, until he showed up again, his massive wings tucked in closely behind him, his shirt spread tightly across his chest and his arms. Your heart leaped as you watched him make his way to the door. 

Your mate. He was yours. Or at least the Mother thought he should be. 

He smiled brightly when he saw you, and you immediately felt a pull on that invisible string that tied you to him. 

It was an effort to steady your breathing when he finally stood in front of you. 

Before you could say anything, one of your students ran into your legs from behind at full speed, causing you to stagger forward, into Cassian. He steadied you, his strong hands holding your waist, as your own hands involuntarily came to rest on his broad chest. 

Cassian looked into your eyes for a moment, the side of his mouth tugging up into a smile. You got lost in him for a moment, in those hazel eyes that you felt you could see the world in. 

Thankfully, you finally remembered where you were, and turned around and narrowed your eyes at your student. “Jan, buddy, we’ve talked about this.”

“Sorry,” he said sheepishly, taking a step back. “I forgot.”

“I know you did, it’s okay,” you said, gently patting his back. “Why don’t you go play until your dad gets here, okay?” 

You turned back to Cassian as he happily scampered off. Cassian’s eyes were bright as he smiled at you. “I don’t know how you do it.”

“Do what?” you asked, tilting your head slightly. 

He laughed, gesturing around, to the dozens of squealing children. “This. Stay so calm, being in charge of all these kids while they’re yelling and knocking into you.”

You shrugged, unable to keep the smile off your face, being so close to him. “It’s not that bad. It’s nothing like what you do.”

Cassian smiled, glancing around at the chaos that surrounded you both. “Well, you might be surprised.” His gaze landed back on you and you felt it like a shock down to your toes. “If you ever want to help me out with the Ilyrians, let me know. You might be better at keeping them in line than I am.”

“I highly doubt that,” you laughed. “With kids, you just need to give them a snack and they’ll love you forever.”

He pursed his lips slightly, seemingly thinking this over. “I might have to try that.”

You laughed. “Let me know how it goes.”

“Promise,” he smirked. 

In that moment, you knew you could spend all day looking at him and never get bored. It scared you, that thought. You couldn’t allow yourself to fall for him. There was no possible outcome that wouldn’t break your heart.

You cleared your throat, taking half a step back. “You’ll probably want to find Nyx.”

He nodded, as if he too had forgotten why he was actually there. 

As if on cue, Nyx bounded up, throwing his arms around Cassian’s legs. “Uncle Cassian!”

Cassian grinned down at him. “Hey, kid. Were you good for your teacher today?”

Nyx smiled shyly. “Yes.”

Cassian turned back to you, raising an eyebrow in question. 

You smiled, nodding. “He was.”

“Good,” Cassian said, ruffling Nyx’s hair, before turning his attention back to you. “I’ll see you around?”

You nodded, though you doubted you would see him again. You tried not to dwell on that, on the fact that you had finally, finally found your mate, and you knew deep down that you could never have him. 

Cassian glanced back over his shoulder, smiling at you once more after he had left and your heart missed a beat. 

You desperately hoped that you would be able to go back to your real life, to forget about him.

---

Cassian couldn’t get you out of his mind after that day. The way you were looking at him… 

Maybe it was just that you recognized him. You certainly weren’t the only woman who had looked up at him with big doe eyes. 

So, why couldn’t he stop thinking about you?

It wasn’t just that though, he knew, as you danced around his mind once again. You were clearly incredibly kind and sweet. And you were funny. 

He found himself smiling as he thought about you, and schooled his features back into his stern, commanding expression as he focused his attention back onto the warriors training in front of him. 

But before long, he was thinking of you again, wondering what kind of snack would make the stubborn, bull headed Illyrians slightly less annoying to work with. 

He would just have to focus, he told himself, until he could see you again. 

Rhysand had given him a questioning look when he had asked to take Nyx to school again, but agreed. 

It had only been a few days, but he was already itching to see you again.

He wondered if it was possible that you felt the same way. 

---

Before school, you always took it upon yourself to usher idle children into the building, making sure they got where they needed to be while parents were busy dropping them off and making plans for pick up. 

When you caught a glimpse of massive, outstretched wings, you felt a shock spark through your entire body.

Again? He was really dropping Nyx off again? 

It had only been a few days since you had last seen him. Not nearly enough time to catch your breath, to force thoughts of him out of your mind. 

That tug in your chest, that string urging you closer to him, was relentless. 

You understood now, how people had been driven to madness after their mating bond had been rejected. It took everything you had in you to stay put, to keep yourself from running to him. 

His bright smile as he approached, his eyes locked on yours, made your heart hurt. 

“Hey,” he said, ushering Nyx into the building with a gentle nudge. 

“Hi,” you said quietly, noticing now that nearly everyone had already gone inside or left. You were alone with Cassian. 

He cleared his throat, shifting his weight back and forth between his feet. Was he… nervous?

“So,” he said, then laughed lightly, turning his face to the mountains surrounding the city, scratching the back of his neck.

“Cassian?” You asked, confused, and honestly, slightly impatient. You only had a few minutes to get to your class.

He leveled his gaze back to you again, his eyes twinkling in the sun. “Do you want to get dinner?” 

That had not been what you were expecting, and you felt like your breath was completely caught in your throat. 

Cassian seemed to take your surprise for something else because he started talking again, quickly, like he couldn't get it out fast enough, “Unless you don't want to. It's not a big deal, I was just thinking --”

“Yes,” you cut him off, and he looked relieved, his taut shoulders visibly relaxing.

“You're sure?”

You couldn't keep the smile off your face. “Very sure.”

His face lit up with a grin then, and you had to bite your lip to keep from grinning yourself. You balled your hands into fists at your sides, worried that if you didn't, you would launch yourself at him.

The two of you agreed on a time and a place, and then he was off, and you once again, had to go teach a classroom full of kids as if nothing remarkable had just happened.

By the time dinner with Cassian rolled around, you had convinced yourself that it was definitely not a date. Probably. 

Just… casual dinner between two people who barely knew each other. Acquaintances went out for dinner all the time, right? 

You forced yourself to take a deep breath as you waited outside the restaurant, your fingers toying with the hem of your dress that you had finally decided on, after trying on nearly every piece of clothing that you owned. 

It was a habit of yours to always arrive early, and yet, you only waited a few minutes before Cassian sauntered up to you, like he didn’t have a care in the world. For about the millionth time, you wondered how it was possible that the two of you had been chosen as mates. You couldn’t be more different. 

Cassian’s smile was bright when he was finally towering over you, greeting you with the usual pleasantries before placing his hand on the small of your back and leading you inside. 

Your heart pounded in your chest at the contact. He hadn’t touched you since that first handshake, the touch that made the bond snap into place for you, like nothing you had ever felt before. 

You wondered if it would do the same for him someday. What would his reaction be if he knew?

Disappointed, surely. Confused. Upset?

It was one thing to ask someone to dinner, but to be mates? To be tethered to somebody for life? You suddenly couldn’t bear the thought of him finding out. 

Your mind was spinning by the time the two of you took your seats, and his brow furrowed in concern as his gaze settled on you. “Are you alright?”

You tried to brighten your expression, cursing your face for always being so easy to read. “Fine,” you said, and the smile became easier, more genuine, the longer that you looked at him. “I’ve just got a lot on my mind, I guess. It’s hard to shut it off.”

He nodded thoughtfully. “I understand. Do you want to talk about it?” 

You winced. “Not really. It’s not that interesting, anyway,” you said, waving a hand dismissively. “How are you?” you asked, before you could accidentally reveal something you would regret. 

The corner of his mouth turned up into a small smile, his gaze locked on you. “Why do you do that? Dismiss yourself so casually?”

“What do you mean?”

“You act like what you do isn’t important, or like your feelings don’t matter.” He leaned in closer, bracing his forearms on the table in front of him. “But it’s not true. You have one of the most important jobs… ever. And from what I can tell, you’re really good at it.” 

He relaxed his stance, leaning back in his chair again, his eyes never wavering from yours. “You don’t have to talk about whatever it is that’s bothering you. But, I won’t let you act like it doesn’t matter, because it does.”

Stunned, you opened your mouth, completely unable to form a response. How could he read you so clearly? And why did he care so much?

Finally, all you could say was, “You don’t even know me.”

Cassian’s smile grew slowly. “I know enough. And I’d like to know more.”

The look on his face, the gravel in his voice, the words that he spoke, made heat rush to your cheeks. His smile only widened. 

“What do you want to know?”

---

Cassian couldn’t wipe the grin off his face after that date with you. You had talked for ages, the two of you, and he didn’t think he had ever been so enamored by somebody. You told him all about your family, your friends, how you grew up, your favorite hideout in Velaris, when you just needed to get away from it all. And he had done the same, admittedly, showing off a little with stories from being in Rhysand's trusted inner circle. 

But you didn’t fawn over him like some women did. You sat and listened, your eyes widening at all the right times, but it was like you really saw him. Not the version of him that people talk about in battle, but just… Cassian. 

And he really liked that. 

For weeks, you had consumed his every waking thought and, frankly, several of his dreams. The way you blushed when he smiled at you, the slight tilt of your head when you watched him animatedly tell stories, the way your eyes lit up when you saw him… all of it was driving him completely mad. He saw you as much as he could in those weeks, and though you seemed reluctant to show it, he could tell that you were excited to see him, too. 

Winter was almost upon Velaris, and the air was crisp, but the sun was bright as Cassian sauntered up to the school, hoping that your afternoon was free and he could steal you away for lunch. 

The sun's rays were shining on you like a beacon and he couldn't help but stare as you crouched down to be face to face with a kid, your smile bright as you undoubtedly said something encouraging to him. 

In that moment, he felt like the ground was swaying beneath him as the bond snapped into place.

The bond that tethered you to him, that confirmed what he had been feeling all these weeks, that proved you were meant to be his. 

For a moment, he was ecstatic, but that moment ended quickly as he suddenly remembered that first time you met, the way your eyes widened in shock as he touched you for the first time… you knew. You had known this whole time and you hadn't told him.

Did you not want to be his mate? Did you think it was a bad match? If that were the case, why on earth had you been spending so much time with him?

He stood frozen in place until the crowd of parents and kids had mostly cleared, his thoughts whirling.

Your eyes lit up when you noticed Cassian, but your face fell when you noticed his expression.

He could tell that you figured it out. That he knew. And that he knew you knew.

His heart broke as your eyes flooded with panic, and you turned from him, hurrying away without another glance.

Cassian followed, half debating flying above the city so he could see easier where you were going.

But before long, he knew your destination anyway.

You had told him weeks ago about the beach that you often went to when you needed to clear your head.

When he approached, you were sitting facing the river, your arms wrapped around yourself. The rocks beneath his boots cracked together and alerted you immediately to his presence, but you didn't turn around. 

He sat next to you, wincing a bit as a rock dug into his thigh, careful to tuck his wings in so they wouldn't brush against you.

Your eyes remained on the water for a few moments, a storm inside them. 

Finally, Cassian said, “Why didn't you tell me?”

Thinking, you bit your lip, and despite everything, it made his heart swell. 

“I didn't think you would want to know,” you said, your voice small.

It took an effort not to physically reel back like you had slapped him. He fought to keep his voice calm. “Why not?"

You let out a humorless laugh, still not so much as glancing in his direction. “Why do you think?”

Cassian furrowed his brow, wracking his brain for any indication he may have accidentally given you that would make you think he wouldn't want to be your mate, but he came up with nothing. “I don't know,” he finally said. “Did I do something?”

Your eyes finally met his then, and he felt the urge to cry for the first time in centuries. You looked so defeated, so pained. What had he done to make you react this way?

“No, it's not that, it's…” you bit your lip, your brow furrowed as you held his gaze. “Cassian, you're a warrior. You're in charge of armies, you're one of the most powerful Illyrians of all time, you've literally made history. And I'm…” you gestured to yourself, “I'm nothing compared to you.” You shook your head, facing the water again. “I hid it from you because it doesn't make sense. It must be a mistake.”

Cassian's heart pounded in his ears as he tried to make sense of it. “How could you think it's a mistake?” He said, his voice wavering, but he pushed past it. “How could you think that way about yourself?”

His heart broke when he heard you sniffle, still avoiding his gaze.

Gently, he took your chin between his fingers and urged you to look at him. “It's not a mistake,” he said quietly. “I haven't been able to get you out of my head since the moment I saw you. Do you know why?”

Your bottom lip trembled and you shook your head as much as you could while he was still holding your chin.

“You're incredible. You're endlessly kind and patient. You dedicate your life to helping kids and teaching them how to be who they're meant to be. There wouldn't be armies to lead or healers to fix us after battle, or anything else, if we didn't have teachers leading us along the way.”

Your eyes softened then, and Cassian nearly sighed with relief but he kept pushing, to be sure you would believe what he was telling you. “And gods, you're beautiful,” he smiled, stroking his thumb across your bottom lip. “You drive me crazy.”

A laugh bubbled from your throat and his heart leaped. He shifted his hands so they were cupping your face. “We're meant to be together, you and me. It's not a mistake. And I don't want to hear you talk about yourself like that again.”

You nodded, smiling, looking up at Cassian with stars in your eyes. “I just can't believe you want me.”

“Well, believe it, because it's true,” he murmured, leaning in so his lips brushed the shell of your ear, “I want you more than I've ever wanted anything.”

He felt your breath hitch, and just then it started to snow lightly, the small flakes sticking to your eyelashes and in your hair.

“I want you too, Cassian,” you said quietly before cupping your hand around the back of his neck and pulling him closer, bringing your lips to his.

Cassian couldn't stop his groan as he pulled you closer, kissing you the way he'd been wanting to for weeks.

When you finally parted to catch your breath, he pulled you to his side, and you rested your head on his shoulder as the two of you looked out across the water, the snow still gently drifting down.

It didn't feel real, that he had finally found his mate, the one he was meant to be with. And it was you, who had been consuming his thoughts since the moment you met.

He felt so, so unbelievably lucky and he prayed that you felt the same way about him.

@loving-and-dreaming @birdsflyhome @hanuh @sheblogs @iambored24601 @thalia-as-blog @ecliphttlunar @melmo567 @headacheseason @sillysillygoose444 @halibshepherd @cigvrette-dvydrevms @lilah-asteria @marina468 @evergreenlark @bookloverandalsocats @yourqueenlilith @mariamay02 @azrielshadows1nger @andreperez11

[losttheoldone-hereweareagain]

spoilers for whatever the hell steel did at the end of episode 47

Long moments pass, the western sky is hell itself. You see, glimmering for a moment, a point of green light appear. Concentric circles, lined with runes, stretching some hundred and twenty feet in all directions begin to spin in the manner of a gyroscope of light and figures, arithmetic and language arcane around the figure of the Wizard Slain, who appears in the sky.

Lowering a staff of the leader of the Citadel's war mages, he carves a line into the heart of one of the shahoran, destroying one of the sorcerers below. Raising up his staff, he begins to abjure, protecting the war mages around him as best he can. 

And the sun rises in the west. A sorcerer, so radiant as to blind you even some miles away, hangs in the air and extends a finger towards the Wizard Slain, beckoning him towards the light. And the Wizard Slain is unmade.

You look, and see, crowned in light and gold, a robe and cloak some forty feet in length, twisting nobly in the wind behind him - first one set of arms, a second, and a third, as Harmas Raunza, leader of House Raunza, appears, in visage over the battlefield.

As he appears, points of light begin teleporting and some hundred nobles of the House of Raunza appear on the battlefield, from across the wide world of Umora.

He points forward, towards Twelve Brooks. The dreadnoughts converge, and as he raises his hand, he opens a door in space. Hundreds of spirits, bound to the House of Raunza. The Bashaal - the spirits of those of his house that failed the trial of their ordeal to enter into sorcerous covenant with their noble lineage, who now bear the heads and wings of white eagles made of blinding light, wielding broadswords, doublehanded, curved at the end, fly forward, gushing onto the battlefield. A cheer goes up from the forces of Gaothmai. "For the Khanterranacht! For Raunza!"

As the lord of House Raunza holds his hands wide, "It is here we make an end to their tower!"

A white cape, streaking through the sky from the Epiphany. A flash of steel, a glint of a sword.

A bubble forms around the leader of House Raunza, and a white cloaked woman with auburn hair, who hangs in the air before him.

Time slows. She twists her wrist, turning her sword ninety degrees to the right. All of the world is mapped out, like a map of the stars. Your own body is simply lines, and the names of your joints and blood vessels. She twists to the left, raising the sword up in front of her in guard position, vertical, matching her straight spine. All the world is rendered in black and white, as though drawn in charcoal on fresh paper. She levels it.

Straight out, floating in air, written in the Lingua Arcana, is simply her namecloak - "The Wizard Steel". And before her, the symbol of House Raunza. With her offhand, she touches that symbol, undoing his true name in front of her. She points, draws her sword back. The image fades. It never happened, it was just a dream. How could she have changed the nature of the world itself?

The point of her sword, at his heart. "You shouldn't have brought so many of your grandchildren, old man." Pushes the sword through his heart. Blood bursts like a wave from his back, killing not only him but each and every one of his children, grandchildren and great grandchildren on this battlefield, who fall like rain from the sky, their light extinguished.

The Sword of the Citadel hangs in the air as the leader of a Great House falls before her blade. The cheer from Gaothmai dies, as quickly as it was born.

The Wizard, The Witch and the Wild One, episode 47

(Worlds Beyond Number podcast)

[lilith-hazel-mathematics]

Self-referencing functions

Hey mathblr, let me tell you about one of our favorite foundational systems for mathematics! It's designed to allow for unlimited self-reference, which is neat since self-reference is usually thought of as a big no-no in foundational systems. It turns out that it actually doesn't matter at all, because the power of self-reference is completely exhausted by the partial computable functions. The theory ends up being equivalent to Peano Arithmetic.

What are the axioms?

The theory is two-typed: the first type is for the natural numbers, and the second type is for functions between numbers. For convenience, numbers will be represented by lowercase variables, and uppercase variables represent functions. To prevent logical contradictions, we permit that some functions will fail to evaluate, so we include a non-number object ☒ called "null" for such cases. The axioms about numbers are basically what you'd expect, and we only need one axiom about functions.

The < relation is a strict total order between numbers.

Each nonempty class has a minimum: axiomatize the "min" operator with φ(n) ⇒ ∃m,(φ(m) ∧ min{k:φ(k)}=m≤n) for each predicate φ, and relatedly min{k:φ(k)}=☒ ⇔ ∀n, ¬φ(n).

Numbers exist: ∃n,n=n

There's no largest number: ∀n,∃k,n

There's no infinite number: ∀n,n=0 ∨ ∃k,n=S(k)

Every functional expression represents a function object that exists: ∃F, ∀(a,b,c), F(a,b,c)=Ψ for any function term Ψ. The term Ψ may mention F.

To clarify the fifth axiom, we define 0:=min{n : n=n}, and relatedly S(k):=min{n : k<n} is the successor function. The sixth axiom allows us to construct self-referencing functions using any "function term". Basically, a term is any expression which evaluates numerically. Formally, a "function term" is any well-formed formula generated from the following formation rules.

"n" is a term; any number variable.

"F(Θ,Φ,Ψ)" is a term, whenever Θ,Φ,Ψ are terms.

"Φ<Ψ" is a term, whenever Φ,Ψ are terms.

"min{n : Ψ}" is a term, whenever Ψ is a term.

In the third rule, we seem to be using the boolean relation < as if it were a numerical operator. To clarify this, we use the programmer convention that true=1 and false=0, hence (n<k)=1 whenever n<k is true, and otherwise it's zero. Similarly in the fourth rule, when we use the numerical function term Ψ as the argument to the "min" operator, we interpret Ψ as being false whenever it's 0, and true whenever it's positive. Formally, we can use the following definitions.

(n<k) = min{b : k=0 ∨ ((n<k ⇔ b=1) ∧ n≠☒≠k)} min{n : Ψ(n)} = min{n : 0<Ψ(n) ∧ ∀(k<n),Ψ(k)=0}

Okay, what can it do?

The formation rules on functions actually gives us a TON of versatility. For example, the "<" relation can be used to encode literally all boolean logic. Here's how you might do that.

¬x = (x<1) (x≤y) = ¬(y<x) x⇒y = (¬¬x ≤ ¬¬y) x∨y = (¬x ⇒ y) x∧y = ¬(¬x ∨ ¬y) (x=y) = ((x≤y)∧(y≤x)) [p?x:y] = min{z : (p∧(z=x))∨(¬p∧(z=y))}

That last one is the ternary conditional operator, which can be used to implement casewise definitions. If you wanna get really creative, you can implement bounded quantification as an operator, which can then be used to define the supremum/maximum operator!

∃[t<x, F(t)] = (min{t : t=x ∨ ¬F(t)}<x) ∀[t<x, F(t)] = ¬∃[t<x, ¬F(t)] sup{F(t) : t<x} = min{y : ∀[t<x, F(t)≤y]}

Of course, none of this is even taking advantage of the self-reference that our rules permit. For example, we could implement addition and multiplication using their recursive definitions, provided we define the predecessor operation first. Alternatively, we can use the supremum operator as a little shortcut.

x+y = [y ? sup{succ(x+t) : t<y} : x] x*y = sup{(x*t)+x : t<x} x^y = [y ? sup{(x^t)*x : t<y} : 1]

Using the axioms we established, basically as a simple induction, it can be proved that these operations are total and obey their ordinary recursive definitions. So, our theory is at least as strong as Peano Arithmetic. It's not hard to believe that our functions can represent any partial computable function, and it's only a little harder to prove it formally. Conversely, all our axioms are true when restricted to the domain of partial computable functions, so it's consistent that all our functions are computable. In particular, there's a straightforward way to interpret each function term as a computer program. Since PA can quantify over computable functions, our theory is exactly as strong as PA. In fact, it's basically just a definitorial extension of PA. Pretty neat, right?

Set theory jumpscare

Hey didn't you think it was weird how we never asserted the axiom of induction? We asserted wellfoundedness with the minimization operator, which is basically equivalent, but we also had to deny infinite numbers for induction to work. What if we didn't do that? What if we did the opposite? Axiom of finity unfriended, our domain of discourse is now the ordinal numbers. New axioms just dropped.

There's an infinite number: ∃w, 0≠w ∧ ∀k, S(k)≠w

Supremums: (∀(x≤a),∃y,φ(x,y)) ⇒ ∃b,∀(x≤a),∃(y≤b),φ(x,y)

Unlimited Cardinals: ∀a, ∃b, #(a)<#(b), where #(n) denotes the cardinality operation.

Each of the above axioms basically just assert the existence of larger and larger ordinal numbers, continuing the pattern set out by the third and fourth axioms from before. Similar to how the previous theory could represent all computable functions, this theory can represent all the ordinal recursive functions. These are the functions which are representable using an Ordinal Turing Machine (OTM). Conversely, it's consistent that all functions are ordinal recursive, since each function term can be interpreted as a program that's executable by an OTM. Moreover, just like how the previous theory was exactly as strong as PA, this theory is exactly as strong as ZFC.

It takes a lot of work to interpret ZFC, but basically, a set can be represented by its wellfounded and extensional membership graph. The membership graphs can, in turn, be encoded by our ordinal recursive functions. Using the Supremums axiom, it can be shown that the resulting universe of sets obeys a version of the Axiom of Replacement, which can be used to prove the Reflection Theorems, ultimately leading to the Specification Axiom. By adapting similar techniques relative to some regular cardinal, it can then be shown that every set admits a powerset. Lastly, since our functions are basically generated from infinitary computer code, they can be encoded by finite strings having ordinal numbers as symbols. Those finite strings are wellorderable, which induces a global choice function, proving the Axiom of Choice. Excluding a few loose ends, this covers all the ZFC axioms, giving the desired interpretation.

In the finitistic version of this theory, we made the observation that the theory was basically just a definitorial expansion of PA. In the infinitary case however, we unfortunately cannot say the same about ZFC. This ultimately comes down to the fact that our theory provides explicit and definable choice functions, meanwhile ZFC cannot. Although ZFC guarantees that choice functions exist, it cannot prove the existence of a definable choice function. This is because ZFC is an inferior theory has no clue where its sets come from, or what they really look like. Our theory, built from unlimited self-reference, and interpreted under the banner of ordinal recursive functions, is instead equivalent to the theory ZFC+"V=L".

[sleepymoonlady]

Roevember Day 19: Temper

"This must be important business indeed. Though if it concerns anything so underhand as an assassination, I fear I can be of little help."

Even as the negotiations were well underway, Rose still repeated Lolorito's glib little jab to herself over and over again, gritting her teeth all the while. It had been two years since that fateful night--the Bloody Banquet, in all its infamy. Two years since the Scions were disgraced and very nearly wiped out, all unwitting and unwilling pawns in a game of chess played by two warring factions of Ul'dah's Syndicate.

She had accepted--bitterly--that Lolorito's bargain was better for the stability of Ul'dah than the vengeance she had been craving since that night. Well, that wasn't entirely true: she had accepted that Raubahn, Alphinaud, and Nanamo all arrived at that conclusion. And out of respect for the wishes of her friends and closest allies, she stood down. But it never sat right with her. Even after he--through Hancock--furnished the Scions' efforts in Othard with a base of operations and more gil than they could spend. Even after Nanamo had come to Rose and told her that they needed his help, for the sake of Ala Mhigo and Ul'dah both. Rose was a woman of many talents, but neither forgiving nor forgetting were chief among them.

Ever since she was a child--even before she lied about her age to debut as a gladiator on the Bloodsands--she had been a person of action. She loathed passivity, couldn't stand to sit by and watch, and had never been good at forgiving--or at forgetting. She solved her problems, more often than not, by beating them into submission. But the problem of Lolorito--that opportunistic little shite--was off-limits. NOBODY should be above justice. But somehow, he kept managing to be just that. Even Thordan and his lackeys weren't.

She needed something. Anything. Some kind of closure. So when the meeting came to a close, she said she needed to speak with him in private. She concocted some kind of lie that felt right in the moment--damn if she remembered what it was. Something about discussing further contributions to the East Aldenard Trading Company no doubt. As Nanamo left the room, Rose kneeled down to be... closer to Lolorito's eye-level, at least. The man turned on his stool to face her.

"I must admit, champion, I'm curious to hear your idea," he said, with that smug half-smile that never seemed to leave his face--or his voice. "I didn't think you had much of a mind for business." Rose felt the anger that had been festering in her chest rising--gods, how did she expect to talk to this little fucker? She had forgotten how infuriating it was--he spoke at you, not to you. You were never his bloody equal. Did he even know that she had helped run her mums' shop growing up? That she had to learn arithmetic just to help them make ends meet? Not much of a mind for business, indeed. If fuckers like him weren't so greedy, maybe things would have been less tight growing up--THEN she wouldn't need a "mind for business."

"Honestly I rather thought it was too complicated a topic--"

Lolorito's next backhanded observation was ended--rather abruptly, too--by Rose's gauntleted fist crashing into his jaw with a sickening crack, sending him flying off the stool and across the room. Before she knew what she was doing--before she could even consider the consequences--she bounded over the table and pinned him to the ground with her left arm, before raising her right in preparation for another blow.

"I am SICK and BLOODY TIRED of this GODS-DAMNED CHARADE, LOLORITO," she snarled through gritted teeth.

"Have you LOST your MIND?" Came the retort from the merchant, spoken laboriously through a broken jaw. "Have you not thought of the CONSEQUENCES of assaulting a member of the Syndicate!? I'll have you--"

"SHUT UP!" Rose punctuated her demand with a raise of her fist. Her mind spun as she stared down Lolorito. This man KNEW what was going to happen that night. He could have stopped it, showed his hand earlier, anything. But he didn't. He didn't. Did he have ANY idea what he did? What that night had cost!?

Thancred couldn't use magic anymore.

Shtola lost her sight.

Min...

Rose's fist began to shake as she remembered. As she turned the sentence over and over again in her head, still afraid to say it to herself after all this time.

Why her? Why couldn't it have been someone else?

Why not HIM?

Shakily, she finally spoke again. "Her Grace has decided that you're better off to her--to us--alive, Lolorito. Out of respect for her, I've kept my peace all this time."

"But make no mistake, you miserable little shite:" As Rose spoke these next words, the fury in her voice could have shattered stone, and the hatred in her eyes--a hatred only the likes of Gaius, Thordan, or Zenos had seen before--shone brightly enough to melt through steel.

"The second you outlive your usefulness to her? The bloody MOMENT I even BEGIN to suspect that you're harboring any foolish delusions beyond your station?

I will personally deliver you to Thal."

-----------------

Hi hey if you made it all the way here uhhhh have a funny:

[guacamolleee]

Among The Multitudes

(Read on AO3)

Written for the Dragon Age Big Bang 2025. Illustrated by @the-font-bandit

M. 41,111 words.

Summary:

The first word Emmrich learnt to read was Johanna.

His eyes followed the sharp edges of each letter, cutting across his right wrist, staking some wordless claim, the ink as dark as blood. Each edge was distinct from the other, downward strokes hard and impressive, straightforward. Emmrich traced each letter — wrote it out, charcoal on paper, on leaves, fingers in the dirt, until they were identical to his skin, until he knew Johanna by heart.

Then a second name came after, months later, much more surprising than the first.

On his left wrist, all curving swirls, rounded letters, and sweeping lines, much more difficult for his young eyes to follow. The H molded into the A, pressed even closer to the N, as if written in a hurry, ink so light, the word untethered to its writer. Mummy had to help him decipher it, holding him close, her long dark hair plaited, the tips of it tickling his nose. She laughed, bright and tinkling — “Your soulmate has terrible handwriting, my love,” — before settling on Thana. Death.

—

Or, one Emmrich Volkarin, bearer of two soul marks, and a lifetime's exploration of the different faces of love and heartbreak.

Preview under the cut

Emmrich wondered — not for the first time in the last few years — what his soulmates were like.

His thoughts often strayed to them when he accompanied his mother to one of the manors she worked at. Early mornings kneading dough, late evenings cooking for some noble's party, sweat on her brow from the heat of the kitchens. Or when he would stay with his father at the shop, the scent of meat in the air, the rhythmic sound of a knife slicing through flesh, through bone, on a wooden block, the occasional greeting to a customer.

Were their parents like his? Did they go to market days together — spices and fruits and vegetables at every stall? Were there quiet smiles, lingering touches when passing by, eyes that lit up whenever they saw each other? Days off and summer picnics, shaky legs skating on the Minanter in the winter?

(Would there be with him when they grow up? Hands in his, laughter that rang through streets and love that woke with the sun and reminded him of his parents. He imagined Johanna with a grin as sharp as their name on his wrist, and Thana with soft, light hands, fingers making swirling patterns in the air.)

Did they like to read as much as he did?

The Chantry near his home was a tiny, modest thing — very different from the one closer to the heart of Nevarra City, with its tall towers and gleaming windows, always smelling like incense and myrrh — and Mother Dellah said he was turning into quite a studious learner, mind expanding in leaps and bounds. The Chantry opened their doors to the neighborhood children on Sundays, providing lessons on arithmetic, history, religion, and all sorts of other things. Emmrich soaked it all in like a sponge.

(Would they sit and read with him? He hoped they would, pointing to their favorite passages, legs knocking together. Perhaps in the Chantry library, right where he was now, whispering and giggling until Mother Dellah scolded them and kicked them out. He wouldn’t mind it that much as long as they were with him — the three of them would find something else to do together — together — always together.)

(Read on AO3)

[monstercampus]

i would loooove to learn more about the Minotaur Art Prof!!

hehe! your wish is my command <33

Professor Arjun Minot is quite the imposing monster amongst his peers. There's a reason he doesn't speak much, and often prefers to stick to his office rather than roam around campus. He's not bound to a contract like Coach Dio or the product of a school experiment like Tentacles, he's a born-and-bred minotaur who comes from the island of Kefalonia and has lived quite the storied life.

In his world, people tend to view minotaurs--as rare as they are--as either fascinating, ancient beasts or as creatures who shouldn't share civilized spaces with other monsters or with humans. There are only a handful of them that exist and plenty believe that it's for good reason, if not for the mythos that depict minotaurs as bloodthirsty and bestial, then for the simple fact of their unnatural births between a woman and an animal. Arjun himself doesn't know the circumstances of his birth, as he was abandoned in a cave shortly after he was born. But he was rescued by a human couple who heard his cries and they decided to raise him in secret; the husband dug a tunnel under their farm and built a place for them to care for him, and so they could hide him from anyone who might react with violence at the idea of the monstrous creature residing nearby. His birthplace was and is home to mostly humans, so his elderly parents took great pains to raise him away from the rest of their community.

Since they couldn't be too lax with letting him roam freely, the couple taught him how to read and do arithmetic first, before slowly letting him help with the farm work in the early mornings before most others would be out of bed. They found he had a talent for sketching on the walls of his underground room, so they brought him paints and tools to sculpt with. Over time, he grew to be a skilled artist and idolized the master artisans he read about in books, hoping that one day he might be accepted into human society if he showed his mastery of their most beloved cultures. He even fell madly in love with a girl from his village who he would see glimpses of when he snuck to and from the basement, and painted gorgeous portraits of her that he intended to submit anonymously to a gallery, so her beauty would remain immortal forever.

However, when he finally made the decision to slip out in the early dawn and approach her as his true self, he was met with terror and disgust the likes of which he'd been sheltered from all his life. The village soon came down on him in a frenzy, and in a hurry, his mother and father packed his few things and sent him away on a ship to the mainland to escape certain death. Since then, he's never seen or heard from them again. After reaching the nearest port, he was swiftly arrested and spent ten years enslaved in service to the kingdom, doing hard labour and living in cramped quarters until he was finally released--not by serving his time, but by a transferrance of ownership to a private buyer overseas.

That buyer happened to be Lysandre Drākon, who had heard from a contact about the rare monster and his circumstances and paid for his travel to come to Runerhea. One can imagine how startled he was to realize that his new master wasn't a human, and that he had no intention of being a master at all. After hearing Arjun's story, Lysandre got him a scholarship to put him through art college and subsequently invited him to work for Monster Campus as a professor, which he readily accepted.

Even so, being surrounded by people like him, he still feels the stares and knows that he remains an outlier even in monster society. He prefers to remain on the outskirts of life and keeps to himself almost entirely, even though he's lauded as a magnificent artist and his students count themselves lucky to have the chance to learn from someone so talented. Despite the brutal treatment he's received and the scars and pain he bears, he hasn't become so jaded as to feel angry at the world, though--it's too beautiful to do so. Although he's not much for words, he expresses himself vividly through paintings and sculptures, and is always searching for a muse to further inspire his works. Someone lovely, someone with a pure heart...someone who won't shy away when they see him, and who might even manage to smile at him. That's his dream.

[factorialsotherfandoms]

Silly pre-canon Xande and Guizo. As it's their birthdays.

The door to their hideout slams open, startling Xande from his nap. A textbook still lies open under his arms, just as meaningless as it was when he started studying a month ago. Xande is stupid, he knows that, but if he doesn't pass…

Guizo tells him not to worry about it, they'll still have each other.

Xande just isn't convinced it is true.

He looks around, and finds that same best friend of his in the doorway, a slightly manic grin on his face. There's blood on him - not his blood, ritual blood, Xande quickly ascertains - chalk, and candlewax. Even from the other side of their room he smells faintly of incense, in that way that means it is not incense at all.

Guizo was doing rituals without Xande. Worse, while Xande was busy studying - he raises his head a little more, ready to complain about the situation.

"I've worked it out!" Guizo says, not allowing him a word in edgeways. "A bit of- doesn't matter. You said you just need to pass? And don't care about cheating, right?"

Well, yeah?

He nods.

At this point, Xande will take anything to scrape a pass in just enough classes that he can leave school without consequences, and damned be the rest of it. It is still a tall order; words don't always make sense, letters wriggle on the page, and at some point or another he must have hit his head hard skateboarding because remembering new info is hard.

Aliens, sure, he can remember about aliens.

But arithmetic? Molecular structures? Electrical diagrams?

No.

"Well," Guizo seems pleased with himself at least. "What about if I can give you answers?"

"Won't work," they've tried the disguise ritual before; Guizo is great at changing himself, but not at copying someone else.

"Not like before," Guizo replies. "I've got a new ritual! Was working on it for a while, but didn't want to say anything until I was sure it'd work."

"What ritual?"

Guizo bounds over, before handing Xande a ring. He puts it on, only for it to do… nothing.

Maybe Guizo needs a nap. Everyone needs more naps.

But then Guizo puts a ring on his own finger, too, and from each ring a golden tattoo winds up their arms, and to their ears.

"Its pretty obvious," Guizo frowns a bit. "But there's no way they'll know what it is. Even if the teacher notices, it just looks like a tattoo. And who cares if you got a tattoo, right?"

Xande has plenty of tattoos, he just usually keeps them hidden.

"What's it do?" He asks.

Guizo's grin grows wider.

His lips do not move, and yet Xande can very clearly hear him say /"this".

Xande looks, but there's no speaker or anything. Removing the ring leaves a golden mark on his finger, but the marks stay.

"This?"

"/Yeah this!/" Guizo's lips still won't move. "/24 hour, unlimited range telepathy! I didn't do great on my exams, either, but if you just think really hard about the exam paper, I'll hear it, and then I can think the answers hard back at you. Or what I think the answers are, anyway./"

… that explains the wax and incense.

Some stupid part of Xande wants to cry; he knows how long developing new rituals takes. There are bags under Guizo's eyes and he buzzes slightly with too much caffeine. How many nights has he stayed up on this, to get that much wax on his shirt?

Xande doesn't want to know.

He reaches out, and pulls Guizo into a very quick hug instead.

Guizo pats his back, but does not stop him from pulling away half a second later.

"It's my turn to nap, though," Guizo uses his mouth voice again. "Wake me up before you head to class, and I'll skim the textbook while you get ready and that."

He can do that. He can do that a lot easier than passing a science exam by himself. The occult? The occult makes sense, and puzzles, and games, and how he needs to shift his weight to do a cool flip on his skateboard. Science equations, though? They don't work like occult ones, no matter what anyone says. Guizo agrees that they're different, and Guizo is significantly more clever.

… Guizo has also passed out in Xande's favourite spot.

Xande supposes, given this, given that he finally has hope of leaving school alive, he can be forgiven just this once.

[ghost-with-a-teacup]

𝐠𝐨𝐥𝐝𝐞𝐧 𝐞𝐲𝐞𝐬 | 𝐫𝐨𝐲𝐚𝐥!𝐚𝐮

𝐩𝐚𝐢𝐫𝐢𝐧𝐠: Simon "Ghost" Riley x f!Reader 𝐬𝐮𝐦𝐦𝐚𝐫𝐲: War rages on within the Empire, and you might just be the fool to traipse into enemy territory with an oath of fealty on your lips. 𝐰𝐨𝐫𝐝 𝐜𝐨𝐮𝐧𝐭: 1.5k 𝐭𝐚𝐠𝐬: War, fires, mentions of death, enemies to lovers, slow burn :)) 𝐚/𝐧: i've been craving to write a slow burn fic ever since i started reading 'bleeding blue' by @nsharks because it destroys me inside in the best way. and also because i haven't written a long fic in ages, so here you go! enjoy~

𝐜𝐡𝐚𝐩𝐭𝐞𝐫 𝐈

“What does it mean to be human, my prince?”

The young prince looks up at his tutor confusedly. They had just been speaking about arithmetics, had they not? Of course, he wouldn’t know, his head was perpetually lost in thought about anything other than his lessons.

“I’m not sure I understand what you mean, professor,” the boy says in turn.

“Just...think about it for a moment, there is no correct or incorrect answer, young princeling,” the old tutor says, his eyes inquisitive as to what the boy would say.

How do the ones who have life handed to them on a silver platter perceive the vast world around them?

The boy ponders for a moment, his head swirling around potential answers. It was surely not as simple as eating his favourite cake or riding horses with his brother along the shoreline. Nor was it the bad things, like his mother tugging his ear when he misbehaved, or his father’s angry eyes. But anything more than superficial was inconceivable.

Perhaps it was the collection of all those things, the good and the bad that made up the answer to that tricky question. It was a start, he thought. But despite his tutor’s push to answer he was still unsure.

“I…I don’t know, professor,” he admits after a few minutes of silence as he thought. But the tutor didn’t seem to be upset, a kind smile gracing his lips.

“No one does. It is the question of the universe, and no one quite has the answer. Or perhaps they all do, since human life is vast and no life is quite the same as the next. Do not be discouraged, but question it as you go to sleep tonight. That is your only task for today, understood?”

The young boy's face lights up at his tutor’s words, this meant he could play.

“Yes, professor!” he says with a giddy smile, leaping out of his chair before bounding towards the door. As he turned the doorknob he suddenly remembered his manners, nodding his head politely once before running out the door.

He would not be getting any sleep tonight.

“MOTHER, FATHER,” the young prince cries out, the fires blazing in a deadly inferno. Like hands, the wisps of flames reached out, ready to drag him to the depths of hell themselves.

With each inch he crawled back, two more were devoured by the blaze.

“BROTHER, PLEASE,” he begged as feet scuffled by his door, shouts to put out the flames throughout the rest of the castle, all deaf to his cries.

The prince was young, but he was smart. He quickly realized there wouldn’t be anyone coming for him, not when the Emperor, Empress and Crown Prince were still in danger. So he searched for options.

The flames had already engulfed their way toward his door, where more would be waiting outside, so that wasn’t an option.

There was nowhere else to hide. Where could you possibly hide from the fury of fire?

His only option was the balcony, but to open the doors meant to provide oxygen to feed the flames. But it was an escape and he had no choice.

Through the flames, he ran. The distance that once felt so short to him now felt like an eternity. The heat of the rocky floor burned his feet, the wisps of the blaze scorched his face, the acrid smoke stinging his lungs.

But he pressed forward.

Closer.

Closer.

Closer to the only salvation in his sight. The only freedom he could grasp.

He bursts through the balcony doors and the flames behind him explode with a newfound fervour only oxygen could bring. But he was free, if only for a little while.

Looking around frantically, he spots the safety net. A thick entanglement of vines that climbed from the base of the castle to his balcony door.

The young prince was unafraid of heights, but despite that his fingers shook as he grasped onto the thick gnarled wall, the implication of a fall from this height not escaping him.

But there was no choice, the fire still chasing behind him. So with a steadying breath, he leaped over the balcony fence, praying to any gods above that the vines would hold.

To his luck they did, and step by step he climbed down, each more sure than the last as he got closer to the ground.

But even still, the climb felt like an eternity. The thorns dug into his hands and feet, dead vines were interlaced with living ones. One wrong step and he would die.

But at long last, there were only perhaps 10 metres of climbing left.

He was going to make it! He was going to survive.

But then all of a sudden a hiss sounds from behind him, a rush of air flying by his ear before a flaming arrow embeds itself into the vines.

His feet slip first in the shock.

His arms follow, unable to support his weight.

Then all of a sudden he was falling.

Falling.

The last thing he could see was the viridian fletching of the arrow as he fell, and with it a darkness creeping around his heart that was never there before.

And then,

darkness.

Heavy boots echo through the throne room, a cloak of obsidian and scarlet flowing as the man walks.

The weight of the world sat heavy upon his shoulders, the crown of an empire heavy on his head. And yet his posture stood tall, never faltering. On his face sat a mask, piercing white and carved of bone into the shape of a skull. Perhaps even a skull itself.

Flanking his sides were the Lord Commander and the Advisor of the empire, while servants bowed to pay their respects on the sides of the room.

With a heavy sigh, the man sat down on the throne. The same one he saw his father sit on in the past.

‘What does it mean to be human’, someone had once asked him long ago. When he was just a boy, where the world around him was still warm and full of splendour, and the light of the sun still shone on the golden prince of the empire.

He had that answer now.

To be human was to survive. To claw your way through the dirt, past the cold grasp of death, past the infernal chains that wished to drag him through the hells. And to survive was to kill, for the world was cruel and treacherous, and there was no other way.

This is why people were whispering today as they looked at the crumpled form of a young woman in a scarlet robe lying at the base of the dais the Emperor’s throne sat upon.

Outsiders were unwelcome here, the borders closed long ago at the start of the war. Yet the young woman was seen crossing the border by the Emperor himself no less as he accompanied a patrol. In any other circumstance, she would be dead before a foot was within Empire lines.

But here she lay, face unfamiliar and thoroughly unwelcome. The air was thick in anticipation as the twitches of consciousness began to take hold.

She would have the answers the Emperor seeks. If they were inadequate, her blood would stain the carpet as so many others have done before.

You groan softly as you begin to wake up, the ground beneath you harder than any forest floor you had slept on in the trek to your destination.

Your head throbs as you sit up, the impact from the hilt of a sword still lingering.

Your eyes are blurry, likely from the injury but you try your best to blink it away blearily.

Before long your eyes clear, and they dart around in the unfamiliar setting. Stone walls that were charred black, denser near the floor that crawled up toward the sealing.

Tapestries that you had once read to be gold were now the scarlet red of blood, perhaps stained by your people's fallen.

No.

They were no longer your people, that you were sure of.

Finally, your eyes fall upon the imposing figure at the head of the room, his expression unreadable under that striking mask.

“Ah, she wakes at last,” the Emperor’s thunderous voice echoed. “Welcome to the Sol Aurelian Empire, give me one reason to not kill you where you lie.”

[pv1isalsoimportant]

(Semi-regularly updated) list of resources for (not only) young mathematicians interested in logic and all things related:

Igor Oliveira's survey article on the main results from complexity theory and bounded arithmetic is a good starting point if you're interested in these topics.

The Complexity Zoo for information on complexity classes.

The Proof Complexity Zoo for information on proof systems and relationships between them.

Computational Complexity blog for opinions and interesting blog posts about computational complexity and bunch of other stuff.

Student logic seminar's home page for worksheets on proof complexity, bounded arithmetic and forcing with random variables (great introduction for beginners).

Eitetsu Ken's list for resources on proof complexity, computational complexity, logic, graph theory, finite model theory, combinatorial game theory and type theory.

Jan Krajíček's page is full of old teaching materials and resources for students (click past teaching) concernig logic, model theory and bounded arithmetic. I also recommend checking out his books. They are basically the equivalent of a bible for this stuff, although they are a bit difficult to read.

I also recommend the page of Sam Buss, there are downloadable versions of most of his articles and books and archive of old courses including resources on logic, set theory and some misc computer science. I especially recommend his chapters in Hnadbook of Proof Theory.

Amir Akbar Tabatabai's page for materials on topos theory and categories including lecture notes and recordings of lectures.

Andrej Bauer's article "Five stages of accepting constructive mathematics" for a funny and well-written introduction into constructive mathematics.

Lean Game Server for learning the proof assistant Lean by playing fun games.

[catgirl-lucy]

Linear Orders

Meow meow :3

Maybe I'll create a pfp later ^^ But for now, linear orders!

Today, I'll be talking- typing about linear orders, I'll abbreviate this to simply LO. These are mathematical structures that look like you can put them on a line :3

Here's what we'll do today:

In the introduction, I'll explain what a linear order is and I'll explain what ω, ζ, η and θ are.

In chapter II, I'll explain the category of linear orders: what morphisms are, what embeddings are, and I'll define a relation ≼ on LO's.

We'll look at arithmetic of linear orders and show some basic facts about them.

In chapter IV, we'll take a closer look at η and explore dense orders.

In chapter II part II, a.k.a chapter V, We'll look at automorphisms of some LO's. In particular, we'll take a closer look at Aut(ζ).

Ordinal numbers! :D

In this chapter, we'll look at the topology of LO's.

We'll end with connected orders. This hopefully completes the basic picture of the LO's introduced in chapter I.

I. Introduction

A linear order is a structure (A,≤) with a set A and a binary relation ≤ on A such that:

≤ is reflexive: x ≤ x for all x;

≤ is transitive: if x ≤ y and y ≤ z, then x ≤ z;

≤ is antisymmetric: if x ≤ y and y ≤ x, then x = y;

≤ is total: x ≤ y or y ≤ x for all x and y.

Intuitively, you can put all points of a linear order on a line, and one point x is less than another point y if it's to the left of it.

Let's look at some examples! For every finite number n, there is exactly one (up to isomorphism) linear order with n points. ω is the order type of ℕ. (An order type is basically what a structure looks like when only looking at the order.) ζ is the LO of ℤ, η is the order of ℚ and θ is that of ℝ. No one can really agree on what letter to use for ot(ℝ), I've also seen λ and ρ used, but I'll use θ throughout this blog-post. Here is a fun picture depicting these:

I call a linear order left-bounded if it has a smallest element, right-bounded if it has a greatest element, left-unbounded if it has no smallest element, right-unbounded if it has no greatest element, bounded if it's both left- and right-bounded and unbounded if it's both left- and right-unbounded.

We can see that 0 is unbounded, 1 is bounded, ω is left-bounded (0 is the smallest element) but right-unbounded (for every n ∈ ω, we have n + 1 > n) and ζ, η and θ are all unbounded.

Linear orders of any size exist. ω, ζ and η are all countable, meaning that we can enumerate the points of them in a list. However, θ is uncountable. You can read my blog post about cardinal numbers if you want to understand infinite sizes better.

II. Morphisms

A morphism from a linear order (A,≤A) to a linear order (B,≤B) is a function f: A → B such that, for all x,y ∈ A, if x ≤A y, then f(x) ≤B f(y). You can think of a morphism as a function that moves the points around, but never "swaps" the order of two points. Though it may not swap two points, it can put two points in the same place. For example, n ↦ ⌊n/2⌋ is a morphism from ω to itself.

In this category, 0 is the initial object and 1 is the terminal object. This means that, for any α, there is a unique morphism from 0 to α (the empty function), and a unique morphism from α to 1 (it sends every x ∈ α to the unique element of 1).

In category theory, a monomorphism is a morphism f: A → B such that, for any object C and any two g₁,g₂: C → A, if f ○ g₁ = f ○ g₂, then g₁ = g₂. This might seem complicated, but in the category of linear orders, this just means that f is injective. Equivalently, x ≤ y if and only if f(x) ≤ f(y). I'll call monomorphisms embeddings from now on. Embeddings are a way one LO can sit inside another. I'll write f: α ↪ β to mean that f is an embedding from α into β.

An epimorphism is a morphism f: A → B such that, for any object C and any two g₁,g₂: B → C, if g₁ ○ f = g₂ ○ f, then g₁ = g₂. In LO, epimorphisms are exactly the surjective morphisms. Thus, f: α → β is an epimorphism if the image of f is β. I'll write f: α ↠ β to mean that f is an epimorphism.

An isomorphism is a morphism f: A → B for which there exists an inverse morphism f⁻¹: B → A s.t. f⁻¹ ○ f = id_A is the identity morphism on A and f ○ f⁻¹ = id_B is the identity morphism on B. In LO, this means that f is bijective. I'll write f: α ≅ β to mean that f is an isomorphism.

If there exists an isomorphism between α and β, then α and β are isomorphic. I'll treat isomorphic linear orders as the same linear order. Thus, I'll write α = β for ‘α and β are isomorphic’.

I'll write α ≼ β (‘α embeds into β’) to mean that there exists an embedding j: α ↪ β. We can see that ≼ is a pre-order:

≼ is reflexive: α ≼ α for all LO's α;

≼ is transitive: if α ≼ β and β ≼ γ, then α ≼ γ.

However, it is not antisymmetric or total. Try to find counterexamples to this! I.e., try to find some α and β so that α and β are not isomorphic, yet α embeds into β (α ≼ β) and β embeds into α (β ≼ α). And try to find γ and δ such that neither γ embeds into δ (γ ⋠ δ) nor δ embeds into γ (δ ⋠ γ).

Since ≼ is not antisymmetric, we can have α and β such that α ≼ β and β ≼ α, yet α ≠ β. α and β that embed into each other I'll call order equivalent, denoted α ≡ β. This means that they're sort-of equal, but not really.

If α ≼ β and β ⋠ α, then I'll write α ≺ β (this is not the same as α ≼ b ∧ α ≠ β). We have ω ≺ ζ ≺ η ≺ θ.

We'll look more at isomorphisms and automorphisms (isomorphisms f: α ≅ α from an object to itself) in chapter V.

III. Arithmetic

Mrrowr :3

In this chapter, we'll look at the three basic operators *, + and × on linear orders. We'll start with the simplest one, *!

For a linear order α, α* is the dual order of α. α* has the same points as α, but the order is reversed: x ≤ y is true in α* iff y ≤ x is true in α.

We can see that if we reverse the order of any finite LO n, we'll just get n back. I.e. n* = n. Some infinite α are also equal to its dual, e.g. ζ* = ζ, η* = η and θ* = θ.

If we take the dual of the dual (thus, we flip α twice), we just get the same LO back. I.e. α** = α.

For two linear orders α = (A,≤A) and β = (B,≤B), we can add them together to create a new linear order (A,≤A) + (B,≤B) = (A+B,≤). A+B is the disjoint union of A and B, meaning that points in α+β are of the form (a,0) and (b,1) for a ∈ A and b ∈ B. We have the usual order of α and β in α+β: (x,0) ≤ (y,0) iff x ≤A y and (x,1) ≤ (y,1) iff x ≤B y. In α+β, everything in A is to the left of everything in β, thus (x,0) ≤ (y,1) for all x ∈ A and all y ∈ B.

You can view α+β as taking α and adjoining β to the right of it (or taking β and adjoining α to the left of it).

Here are some basic facts about addition:

0 is the identity for addition, i.e. α+0 = 0+α = α;

Addition is associative, i.e. (α+β)+γ = α+(β+γ) = α+β+γ;

α ≼ α+β and β ≼ α+β.

However, as it turns out, addition is not commutative! OwO Try to find α and β for which α+β ≠ β+α!

We can see that ζ = ω* + ω, 6+ω = ω, η+η = η and θ+θ ≠ θ, but θ+1+θ = θ.

We can see that addition interacts with duality in an interesting way: (α+β)* = β*+α*. Thus, taking the dual of a sum is the same as summing up the duals, but in reverse order :P.

The most complicated basic operation on linear orders is multiplication. For linear orders α = (A,≤A) and β = (B,≤B), points in αβ are pairs (a,b) of a point a in A and a point b in B. In αβ, (a,b) ≤ (c,d) iff b < d or [b = d and a ≤ c]. Intuitively, you take the order β and replace each point with a copy of α.

Multiplication is associative, (αβ)γ = α(βγ), * distributes over multiplication, (αβ)* = α* · β*, and multiplication is left-distributive over addition, α(β+γ) = αβ+αγ. Of course, you can try proving these basic facts if you want to. Just like addition, multiplication isn't commutative. Finding α and β for which αβ ≠ βα is left as an exercise. Here is something funny: although multiplication is left-distributive over addition, it isn't right distributive! Thus, for some α, β and γ, we have (α+β)γ ≠ αγ+βγ.

I'll often write α^n for α multiplied with itself n times. It isn't really possible to exponentiate with infinite linear order powers. If we have linear orders α and β, where β is infinite, we need some "center" or "zero" 0 ∈ α if we want to define α^β. If we have chosen such a 0, we can define α^β to be the order-type of finite support functions f from β to α, where ‘finite support’ means that {x ∈ β | f(x) ≠ 0} is finite ({x ∈ β | f(x) ≠ 0} is called the support of f). If we don't require f to have finite support, then lexicographical ordering might not be possible.

I'll stop talking about exponentiation and centers of linear orders now, so you can explore more of this on your own. There might also be different ways to exponentiate linear orders.

IV. Dense Orders

[Definition] A linear order α is dense iff for all x,y ∈ α, if x < y, then there is some z ∈ α so that x < z < y.

Thus, between any two points, there is another point. A dense order can alternatively be defined as an order in which every point is a limit point, we'll talk more about limit points and discrete orders in chapter VII.

Trivial examples of dense orders are 0 and 1. These are dense because there aren't enough points for them to have x < y somewhere, so they're vacuously dense. I'll call an order that isn't 0 or 1 a non-trivial linear order. Any finite LO beyond that (2, 3, 4, etc) isn't dense. ω and ζ also both aren't dense, while η and θ are dense. θ is a bit more than dense: it is connected, which I'll talk more about in chapter VIII.

In some way, η is the simplest (non-trivial) dense order, as it embeds into every other dense linear order. Simultaneously, it is the most complex countable order, as every countable order embeds into it. Both follow from the theorem below:

[Theorem] Every countable linear order embeds into every non-trivial dense linear order.

Please try to prove this theorem yourself before reading my proof.

[Proof] Let α be a countable linear order and let β be a dense linear order. And assume, without loss of generality, that β is unbounded: if β is left- and/or right-bounded, then we can simply cut off the ends, making it unbounded by our assumption that it is dense. By the assumption that α is countable, we have some enumeration a₀,a₁,a₂,... of points in α. We can define an embedding f: α ↪ β by induction. Basically, we put more and more points from α in β, making sure each time that they're in the right spot. First, let f(a₀) be any point in β. Suppose f(aₘ) is already defined for all m < n, we'll now define f(aₙ). We have the set L = {f(aₘ) | m < n; aₘ < aₙ} of points to the left of f(aₙ) (or, well, where f(aₙ) should be) and R = {f(aₘ) | m < n; aₘ > aₙ} of points to the right of where f(aₙ) should be. Since L is a finite set, it must have some maximal element l = max(L). And since R is finite as well, it has some minimal element r = min(R). If L is empty (and thus, l does not exist), we can take f(aₙ) to be some number below r, which exists as β is left-unbounded. Dually, if R is empty, we can take f(aₙ) > l. If both l and r exist, we can take f(aₙ) to be some point such that l < f(aₙ) < r, which exists as β is a dense order. ∎

Since η is countable, it embeds into every non-trivial dense order, and since η is dense, every countable order embeds into it. We thus have that all countably infinite dense orders are order equivalent. It turns out that η ≡ η+1 ≡ 1+η ≡ 1+η+1 are the only countably infinite dense linear orders, I leave a proof of this as an exercise to the reader.

I'll end this chapter with a list facts about how dense orders interact with arithmetic:

α is dense iff α* is dense.

α+β is dense iff α is dense, β is dense and at least one of the following holds: α is right-unbounded, or β is left-unbounded.

αβ is dense iff α is dense and [α is not bounded or β is dense].

These are all pretty easy exercises.

V. Automorphisms

In group theory, a group is a mathematical structure (G,·) with a set G and a binary operator · such that:

There is an identity element e ∈ G: e·x = x·e for all x ∈ G;

Every x ∈ G has a unique inverse x⁻¹ ∈ G, x·x⁻¹ = x⁻¹·x = e;

· is associative, i.e. (x·y)·z = x·(y·z) for all x,y,z ∈ G.

One type of group is an automorphism group. Given an object A, the automorphism group of A, denoted Aut(A), is the set of all automorphisms f: A ≅ A. In this group, we take morphism composition (written ○) as our binary operator. In this group, the identity element is the identity morphism and the inverse element is the inverse morphism.

The trivial group is the group with a single element, which is the identity element. I'll write the trivial group with a bold 1. Some linear orders have the trivial group as automorphism group, for example Aut(2) = Aut(ω) = Aut(ω2) = 1. There is no way to move the elements of ω around other then leaving them all where they started.

Some linear orders have a more interesting automorphism group. For example, Aut(ζ) = (ℤ,+) (the cyclic group of order infinity) and Aut(η) and Aut(θ) are kinda complicated.

To explain why Aut(ζ) = (ℤ,+): an automorphism of ζ shifts the elements to the left or right by some amount x. First shifting by x amount and then shifting by y is the same as shifting by x+y. We thus have that the automorphism group of ζ is the integers under addition.

The automorphism group of θ corresponds to strictly increasing continuous functions on the real number line. It has 𝔠 many elements. I don't know if this group has been researched a lot, tell me if you find anything interesting about it!

One natural question to ask is: what groups can be the automorphism group of a linear order?

I'll give you part of the answer to this question. A subgroup of a group (G,·) is a set H ⊂ G such that (H,·) is itself a group: the identity element of G must be in H, the inverse element of any x ∈ H must be in H and, for any two x,y ∈ H, we have x·y ∈ H. Given a set X ⊂ G, we write ⟨X⟩ for the subgroup of G generated by X. This is the smallest subgroup of G that includes X. Given a single element a ∈ G, we can also have ⟨a⟩, which is the smallest subgroup of G that contains a. If a = e is the identity element, then ⟨e⟩ = {e} is just the trivial subgroup. For other a, we have ⟨a⟩ = {..., a⁻², a⁻¹, e, a, a², ...}. In group theory, we often write a^n for a · ... · a w/ n copies of a. We might have something like a³ = e, in which case, {..., a⁻², a⁻¹, e, a, a², ...} = {e, a, a²}. However, we can also have a, a², a³, a⁴, etc, be all different elements of G. In which case, we have ⟨a⟩ ≅ (ℤ,+)

It turns out that, in the automorphism groups of linear orders, if f ∈ Aut(α) is an automorphism that is not the identity, then ⟨f⟩ must be isomorphic to (ℤ,+). We can see this pretty easily: if f moves some x ∈ α to the right, i.e. f(x) > x, then it must also move f(x) to the right, and f(f(x)) = f²(x), and f³(x), etc. Meaning that f(..f(x)..), no matter how many applications of f you have, can never be x again. Thus, f, f², f³, f⁴, etc, must all be different automorphisms. This is only one restriction groups induced by linear orders must have, and I'm sure you can find more.

(ℤ,+) is in some sense the simplest non-trivial group that can be induced by a linear order. There are a lot of linear orders that induce (ℤ,+) (that have (ℤ,+) as automorphism groups). As mentioned above, Aut(ζ) = (ℤ,+), but this is also the automorphism group of ω+ζ, ζ+2, etc.

In the same way that η is the simplest dense LO, ζ is the simplest order with a non-trivial automorphism group:

[Theorem] ζ embeds into every linear order with a non-trivial automogrphism group.

Unlike η, where η+ω, 1+η+1, etc, also all embed into all dense LO's (and all dense LO's embed into them), ζ is the unique simplest linear order with a non-trivial automorphism group:

[Theorem] If α has a non-trivial automorphis group and embeds into every linear order with a non-trivial automorphism group, then α = ζ.

Another fun fact: we know when a linear order has a non-trivial automorphism group when ζ embeds into that LO.

[Theorem] ζ embeds into α iff α has a non-trivial automorphism group.

Proofs of these theorems are left as an exercise.

VI. Ordinal Numbers

In mathematics, a well order is a specific kind of linear order. A LO (A,≤) is defined to be a well-order if:

For all non-empty S ⊂ A, S has some minimal element x, i.e. for all y in S, we have x ≤ y.

Every finite LO n is a well order. ω is a well order as well but, e.g., ζ is not well-ordered: ℤ⁻ ⊂ ζ, the set of negative integers, does not have a least element. Order types of well orders are called ordinals. They are an important concept in set theory as they describe the heights of trees and sets, and because of transfinite induction.

[Theorem] For a linear order α, the following are equivalent:

α is an ordinal;

every strictly decreasing sequence in α is finite;

ω* does not embed into α.

[Definition] A set X ⊂ α is inductive if for all x ∈ α, if for all y < x, we have y ∈ X, then we have x ∈ X as well.

Ex. The set of all rational numbers below the square root of 2 is inductive in η.

[Theorem] If α is an ordinal and X ⊂ α is inductive, then X = α.

Both of these theorems are left as an exercise to the reader.

Ordinals have a lot of nice properties. For example, α+β and αβ for any two ordinals α and β are ordinals as well. Also, every infinite ordinal has a smallest element, which we can take as our center in exponentiation, meaning that ordinal exponentiation is well-defined. We also have that ≼ is itself a well-order on ordinals:

≼ is antisymmetric on ordinals: if α and β are ordinals, α ≼ β and β ≼ α, then α = β;

≼ is total on ordinals: for ordinals α and β, we have α ≼ β or β ≼ α;

≼ is well-founded: all sets of ordinals have a ≼-minimal element.

This means that the theorem of induction (X is inductive → X = α) also applies to Ord, the class of ordinals. We can also view each point in an ordinal as its own ordinal: for x ∈ α, we can define (x) = {y ∈ α | y < x}, and this set with the usual order of α is an ordinal (x) < α.

A von Neumann ordinal is a specific representation of an ordinal. It is a transitive set of transitive sets. For von Neumann ordinals α and β, α < β is defined as α ∈ β. Von Neumann ordinals are often used in set theory.

Given a set of ordinals S, the supremum of S, written sup(S), is the smallest ordinal α so that β ≤ α for all β ∈ S.

Here are some more examples of ordinals:

ε₀ (epsilon-nought) is defined as the smallest ordinal for which ε₀ = ω^ε₀;

ω₁ck (Church-Kleene ordinal) is defined as the smallest ordinal for which there is no Turing machine that defines an order that is isomorphic to ω₁ck;

ω₁ is defined as the smallest uncountable ordinal.

[Definition] Given a linear order α and a set S ⊂ α, S is cofinal in α iff for all x ∈ α, there is some y ∈ S so that x ≤ y. The cofinality of α, written cof(α) or cf(α), is the smallest cardinality (i.e. size) of a cofinal subset S ⊂ α.

For example, cf(0) = 0, cf(1) = cf(α+1) = 1 and cf(ω) = cf(ζ) = cf(η) = cf(θ) = ℵ₀. ω₁ has uncountable cofinality (it has cofinality ℵ₁), meaning that, for all countable subsets S ⊂ ω₁, there is some y ∈ ω₁ so that x < y for all x ∈ S. If it'd've'd countable cofinality, then we could take some countable cofinal S ⊂ ω₁ and some enumeration aₙ of S. Then, because ω₁ is the smallest uncountable ordinal, each ordinal (aₙ) must be countable. Thus, we can take injections fₙ: (aₙ) → ℵ₀. But then we can define an injection g: ω₁ → ℵ₀² by setting g(x) = (fₙ(x),n) for the smallest n for which x < aₙ. However, everyone knows that ℵ₀² = ℵ₀, so we have an injection g: ω₁ → ℵ₀ witnessing ω₁ is countable, thus a contradiction.

Please tell me if that was too hard to follow...

I'm going to sleep now. I'll write the next chapters tomorrow.

VII. Topology

It's midnight. Technically the next day.

In maphs, a topology is defined as a structure (X,τ) with a set of points X and a family τ ⊂ P(X) of subsets of X, such that:

The union of any number of sets in τ is in τ;

The intersection of any finite number of sets in τ is in τ.

The empty union is the empty set and the empty intersection is the full space X itself, so ∅ and X must both be in τ. In a topology, members of τ are called open sets. A set is closed if its complement is open. It is clopen if it's both open and closed. We can see that ∅ and X are always clopen.

Every linear order has an order topology. Given a linear order α and some point x ∈ α, (-∞,x) = {y ∈ α | y < x} is the set of points below x and (x,∞) = {y ∈ α | x < y} is the set of points above x. For x,z ∈ α with x < z, (x,z) = (x,∞) ∩ (-∞,z) = {y ∈ α | x < y < z} is the set of points between x and z. (x,z) is called the open interval from x to z. A set in the order topology on α is open iff it is a union of open intervals. You can verify that this indeed defines a topology. Another equivalent definition is: O ⊂ α is open iff ∀y ∈ O ∃x,z ∈ α ∪ {-∞,∞} x < z ∧ (x,z) ⊂ O.

A topological space in which every set is an open set is called a discrete space. A discrete space can alternatively be defined as a topology where every singleton (every set with a single element) is open. A limit point is a point x ∈ X for which {x} is not open. A discrete space thus is a space with no limit points. A limit point in a linear order α is a point x for which (1) for all y < x, there is some z < x so that y < z, or (2) for all y > x, there is some z > x so that z < y, the reader may verify that this is correct.

The order topology of any finite LO is discrete. ω and ζ both also have a discrete topology. However, the topology of η and θ are not discrete. In fact, η and θ are dense linear orders: all points in η and θ are limit points. ω+1 is neither discrete nor dense: it only has one limit point. (Assuming the axiom of choice) a discrete linear order of any size exists, a proof of this is left as an exercise to the reader.

In topology, a dense set (not to be confused with dense orders) is some set D for which, for all non-empty open O, D ∩ O is non-empty. For example, the set of rationals is dense in θ, while the set of integers is not (it does not intersect the open set (2.6, 2.74) ∪ (12.2, 12.2002)). Equivalently, D is dense iff D¯, the closure of D, is the whole space X. The closure of a set A is defined as the (inclussion-)smallest closed set that includes A, i.e. A¯ = ⋂{C | A ⊂ C ∧ C is closed} = {x | ∀O O is open ∧ A ⊂ O → x ∈ O}. I'll say a LO α is dense in another LO β if there is an embedding f: α ↪ β for which its range is dense in β. For example, η is dense in θ but not in θω₁. Usually, the bar is placed on top of the set A to denote its closure, but I can't do that here, so I'll write it next to it instead :P

For a set A, the interior of A, denoted int(A), is the largest open set included in A. I.e. int(A) = ⋃{O | O ⊂ A ∧ O is open} = {x | ∃O O is open ∧ O ⊂ A ∧ x ∈ O}. int(A^c) = (A^c)¯.

In topology, we often talk about local properties of a space. Given some point x in a topological space, a neighboorhood of x is a set U that includes an open set that contains x. Thus, x ∈ O ⊂ U for some open O. Given some A ⊂ X, we can define a topology on A as follows: τ_A = {O ∩ A | O ∈ τ_X}. (A,τ_A) is a subspace of (X,τ_X). A topological space (X,τ) has a property P locally iff for every point x ∈ X, there is some neighboorhood U of x with that property P. For example, αβ is locally isomorphic to α and ω is locally compact.

For two points x and y in a topological space (X,τ), I say x and y are connected if there is no clopen set A so that x ∈ A and y ∉ A. Equivalently, there are no open U and V such that x ∈ U, y ∈ V, U ∩ V = ∅ and U ∪ V = X. Trivially, every point is connected to itself. A topology is connected if all points in the topology are pairwise connected. It is completely disconnected if no two distinct points are connected. In a linear order α, two points x < y are connected if there is no "gap" between x and y. η is completely disconnected, as between any two rational numbers, there is an irrational. However, θ is connected. The reader can verify that a linear order is locally connected iff every limit point is connected to some other point.

A topology is compact if every open cover (that is, every family of open sets F such that ⋃F is the whole space) has a finite subcover (some finite F₀ ⊂ F that still covers the space). Every finite space is compact as the only open covers are already finite. ω + ω* is not compact as {{x} | x ∈ ω + ω*} is an open cover with no finite subcover, but ω + 1 + ω* is compact as any open set that includes the middle element (which I'll call ‘X’) must also include (n,X) and (X,m) for some finite n and m. Intuitively, a compact space is bounded and has no small gaps.

VIII. Connected Orders

Intuitively, a connected order is a linear order with no gaps or holes. Thus, θ is connected, but η is not as every irrational number forms a hole. In other words, it is very dense (the densetest it can be). 0 and 1 are trivial connected orders. The simplest non-trivial case is θ, as it embeds into every other connected linear order.

[Theorem] If α is a non-trivial connected LO, then θ ≼ α.

θ+1, 1+θ and 1+θ+1 are also connected linear orders with this property (thus, like with the η case, we have θ ≡ 1+θ ≡ θ+1 ≡ 1+θ+1). θ+1+θ = θ, however, θ2 ≠ θ as θ2 has a disconnect between the first and second copy of θ.

Not every non-trivial connected LO is order-equivalent to θ. For example, the long line, (1+θ)ω₁, is a connected linear order that is too long to be squashed into θ. The reader can verify that (1+θ)ω₁ ⋠ θ. Sometimes, (1+θ)ω₁ is referred to as the right side of the long line, ((1+θ)ω₁)* = (θ+1)ω₁* is the left side and the full long line is made by gluing the left and right side together, removing the greatest element of the left side to make it connected. The long line can also refer to the right side of the long line, with the least element removed to make it unbounded. I'll use "long line" to refer to the last one from now on.

If α is an infinite countable ordinal, then (1+θ)α = 1+θ. If α > ω₁, then (1+θ)α with the smallest point removed is no longer a homogeneous linear order (I call an LO α homogeneous iff ∀x,y ∈ α ∃f ∈ Aut(α) f(x) = y). ω₁ is thus purrfect for making a long line.

Here is a funni connected order I came up with: (1+θ+1)θ.

Here are some theorems that state more generally how arithmetic works with connected linear orders:

α+β is connected iff α is connected, β is connected, and either α is right-bounded and β is left-unbounded, or α is right-unbounded and β is left-bounded;

α* is connected iff α is connnected;

For unbounded α, αβ is connected iff α is connected and [β = 0 or β = 1];

For bounded α, αβ is connected iff α and β are connected;

For left-bounded right-unbounded α, αβ is connected iff α is connected, every S ⊂ β with an upper bound has a supremum (least upper bound), and every point x ∈ β has a direct next point y ∈ β (∄z ∈ β x < z < y).

I like the last one :3 You can try proving these theorems if you want to.

Bye!~

I hope I've given you good intuition on the most common linear order types ω, ζ, η and θ ^^ If you spot any mistakes in my post, please tell me!

I'm planning to write an introduction to set theory next :3

[udoai]

An ancient, leather-bound book inspired by Luca Pacioli's Summa de arithmetica, open on a wooden table in a dimly lit study. Numbers, symbols, and complex arithmetic formulas begin to rise gracefully from the pages, glowing faintly with a mystical aura. The symbols float upward in a gentle, spiraling motion, as if the knowledge is coming to life. The scene has a Renaissance atmosphere, with soft, warm lighting casting shadows across the book and the table, emphasizing the historical importance of the text. The entire scene is highly detailed, capturing the ornate design of the book cover and the worn, aged texture of the pages --chaos 20 --ar 9:16 --style raw --stylize 1000 --v 6.1

[sharingmystoriesetc]

Faber suae fortunae

Or Maider's love story towards freedom.

81 A.D.

Maider's life had never been easy.

Firstly, because of her family.

Her mother, an authentic oriental beauty, according to her father, had died giving birth to her.

And her father, a jewish nobleman and intellectual fallen into disgrace, had entrusted her to the Ben-Hur family, in the beautiful Jerusalem.

There, Maider had grown up as a sister to the family's heirs, learning to read, write, and do arithmetic. She had been educated to be Sarah Ben-Hur's maid of honor, who was her age. The two had become friends, and Sarah and the rest of the family loved Maider dearly.

They loved her sense of humor, her playfulness, and her voice.

Maider, in fact, had a wonderful singing voice. Clear, powerful. Juda Ben-Hur told her she was the most talented singer in all Jerusalem. Maider didn’t think it was true, but the praise never failed to make her blush.

Her father Jeremiah visited her from time to time, and every single time he taught her a new song. When Maider sang, time seemed to stop. And Maider could feel a strong connection to the divine, with whom she had never had, in her opinion, a good relationship.

Then, in the blink of an eye, everything changed.

The Romans oppression to Jewish people was growing stronger, and one day, a Roman officer visiting Jerusalem saw Maider at the market and ordered his soldiers to take her as a slave. As a possession of the Empire.

She, a young woman with thin light eyes and straight dark hair, would fetch a lot of money as a slave for a wealthy Roman family, and would be a perfect lover for him during the voyage back to Rome.

Maider, blessed with an extraordinarily strong personality and resilience, couldn’t help but burst into tears.

The Ben-Hur family couldn’t do anything but try to persuade the Roman officer. But it was all useless.

Maider was chained and taken to Jaffa’s port.

Her father desperately ran after her, shouting to try to buy her freedom, but a legionnaire struck him with the hilt of a dagger, leaving him unconscious on the street.

Maider started screaming, but the legionnaire gave her the same treatment as her father.

And everything went black.

***

Maider woke up in chains, being pushed to board a ship.

In fact, a merchant ship bound for Rome, as she had learned.

She tried to wriggle out of the grasp of the legionnaire who held her chains tightly, but it was all in vain.

As soon as the Roman officer saw her, despite the huge bruise on her temple, he found her very exotic and beautiful, and with a smirk, ordered her to be taken to his cabin.

Maider was thrown into the officer’s cabin as the ship set sail.

She and the man looked at each other. He was handsome, but his dark eyes transmitted coldness and toughness. He looked at her like a butcher watches a piece of meat.

Maider shivered.

He had a defined physique. If she rebelled, he would overpower her in seconds. So, instinctively, she held out her hands.

"Wait, slow down. Let's be reasonable for a second”.

The officer gave her a puzzled look.

"I was told you were educated, but not that you spoke my language so well”.

"I speak several languages. And I also know that my value as a slave would decrease if you got me pregnant”.

He smirked.

"We'll take that risk”.

Maider shivered again, but did not show it.

"Listen, I beg you. I can make it enjoyable. I'm a virgin; you'll sell me better in Rome with that characteristic. But I've heard stories. Ways I can satisfy you without you taking my virtue”.

He laughed.

"I know what you're talking about. But it's not as satisfying”.

"I can imagine. But let me try. I'm not the only slave on this ship, and I'm sure there are other women just as beautiful whose virtue is less important than mine”.

Maider felt like sinking for what she had said, but she had to stay alert. At that moment, it was life or death.

She didn’t know that, but she had been lucky. The officer was smarter than the average. He quickly realized that with some adjustments, he could get the best out of the situation. He would humiliate the girl but preserve her virginity. And he would still sleep with any other woman he wanted. It seemed inviting to him. Something new.

"All right, young slave. Show me what you can do”.

Maider tried to isolate her mind. She would have liked to cry, scream, and struggle.

But it would have been useless.

The officer had to think she was strong, intelligent, and cunning, at least as much as he was.

No matter how much he humiliated her, he had to consider her his equal.

Only then would she arrive in Rome unharmed.

She gave him a lascivious look and approached him.

***

Twenty days of hell.

Twenty days in which Maider managed not to be violated but humiliated herself in every possible way.

Twenty days in which she slept in the officer’s cabin, on the floor, letting herself cry only when he was asleep.

Twenty days in which she almost lost herself. Almost. Because in moments of solitude, she sang her parents’ songs to herself and felt she was not alone.

Once arrived in Ostia, the passage to Rome was without particular problems.

The other slaves on the ship called her the “officer’s whore," and whenever they saw her, they did not spare her insults.

Fortunately, at Ostia they were divided. She, as she would later understand, was among the high-ranking slaves destined for noble Roman families.

She was cleaned and well-dressed.

In Rome, they put a sign around her neck and pushed her onto a small stage.

She was in the Esquiline, a noble area full of villas and beautiful buildings.

Despite this, she observed the faces of her potential future masters, terrified.

A tear ran down her cheek. Fighting on that ship had been useless: she would become the slave of a ruthless Roman, and who knew what he would do to her.

She thought of Jerusalem. The Ben-Hur's house, her friends, the life she had considered difficult, which now seemed like paradise.

"Ladies and gentlemen, we have a real gem here! A 25-year-old slave from Jerusalem. No, she's not entirely Jewish: look at her thin eyes. Her origins are from the far East. An exotic and educated virgin jewel, a girl everyone would want in their home!"

Sneers and smirks among the bystanders. The noblewomen observed her impassively.

Maider held back a sob.

"The auction starts at one thousand sesterces!"

And, for no apparent reason, she began to sing:

"You can't take my past

You can't take my history

You could take my pa, but his name's a mystery..."

At that very moment, the Aedile Ludi was passing through the market. He was a thirty-three-year-old man, not too tall, with dark hair and blue eyes. Once the king of the Suburra for his shady dealings, he had recently left the neighborhood to live in the Esquiline. That morning, he was leaving home to go to his betting tavern, which still earned him a certain income and from which, for some reason, he could not separate himself.

That man was named Tenax.

And hearing her singing, he stopped abruptly.

He then met her gaze.

Green eyes into blue eyes.

For a moment, time stood still.

Maider stopped singing, looking at him.

The auctioneer took the opportunity to grab her chin.

"What did I tell you? A constant surprise, this exotic jewel!”

Maider looked at him with disgust. Struggling.

"Get your hands off me!"

The man slapped her.

Usually, Tenax would not intervene. He was a man of gray morality, but for some reason, this time he felt he should act.

He met the girl's gaze again.

He was captivated.

He stepped forward and said:

"I offer three thousand sesterces!"

The auctioneer looked at him in silence.

Maider looked at him in silence.

The bystanders looked at him in silence.

"Oh, Aedile Ludi..." murmured the man, "All right. She's yours, if you want her”.

"Yes”.

And he signaled for the girl to come down from the stage.

Maider obeyed, eager to escape the grasp of that horrible man.

Tenax grabbed her by the arm, handed the man a bag of money, and together they walked towards home.

***

Maider looked at him with a mix of fear and fascination.

Fascinated because she found him to be a handsome man.

Afraid because she feared that behind those big blue eyes hid a violent man who would take advantage of her without pity.

After all, he hadn't even untied her.

Among the thousand doubts occupying her mind, an uncertain question escaped her lips:

"Who are you?"

He was surprised to hear her speak.

"My name is Tenax”.

"I am Maider. What does it mean that you are an… ehm, Aedile Ludi?"

Tenax gave her an even more surprised look. Was she a talker, or was it anxiety making her babble?

"That I oversee the games at the Flavian Amphitheater and the Circus Maximus”.

"Oh. So you're an engineer”.

"Are you studying me?"

Maider met his gaze. He understood her immediately. She appreciated his sharpness with a bit of fear.

"When I'm nervous, I tend to babble. Not knowing what you want to do with me makes me nervous”.

Tenax felt exposed. The girl was right: what the hell did he want to do with her? She was too clever to relegate her to washing the floors of his large villa.

Indeed, and he hated to admit it, he found her far too interesting to make her a simple slave.

"You will stay with me. I have a housekeeper who will assign you tasks. Done with the questions?"

"Actually, no. But if you command me to be silent, I will”.

Tenax sighed.

"Actually, I don't know what I prefer. Slaves are usually few of words”.

"Until a couple of weeks ago, I was a free person. I was educated in a great Jewish family. I can read, write, do arithmetic..."

"And sing”.

They met each other's gaze again. Maider blushed.

"Only for a selected few”.

Tenax wanted to laugh but did not show it. He saw her open and close her hands: her wrists were still tied.

"If I untie you, will you run away?"

"I would get lost after five minutes”.

"Probably”.

He paused for a moment and cut the ropes tying her wrists with a dagger. Maider sighed in relief, massaging the abrasions.

"Thank you, Tenax”.

He gave her a nod of acknowledgment.

Once they arrived home, Tenax knocked on the door.

"Claudia! It's me”.

After a few seconds, a middle-aged woman moved the heavy door aside.

"Tenax. I wasn't expecting you so early”.

Then her eyes fell on the girl. With an eloquent look, she asked her master who she was.

"Her name is Maider. I want you to show her the house. If you have any particularly heavy tasks, get her to help. Get to know each other”.

With that, and with a nod of his head, he left.

Claudia extended her hand.

"Come, dear”.

Maider smiled softly at her, taking it.

***

Hey! It's Eli here. Thank you for reading! Let me know in the comments if you liked this chapter and if you want to read more ❤️

Here you have Maider and Tenax ❤️

[lilith-hazel-mathematics]

Cardinal arithmetic is easy

One of the most satisfying things about cardinal arithmetic is when you're doing a counting problem with infinite cardinals, and despite the massive complexity of the problem, the answer turns out to be exactly what you'd expect it to be. Like, there'll be some really easy to prove bound, and then a ridiculously complicated proof demonstrates that the obvious bound was the exact correct answer the whole time. It happens so often, that at some point you stop being surprised and just start expecting it.

Assuming #(Z)≤#(S), you might conjecture that S is so much larger than Z that #(S∪Z)=#S. That's correct.

You might similarly conjecture that #(S×Z)=#S. That's correct.

A bijective function S→S is a special type of subset of S^2. Given that #(S)=#(S^2), you might conjecture that the number of such bijections is #(2^S). That's correct.

A topology on S is a special type of subset of 2^S, so you might conjecture that the number of topologies on S, modulo homeomorphism, is #(2^(2^S)). That's correct.

A group structure on S is described in terms of a binary operation (S×S)→S, and the number of such functions is upper bounded to #(2^S). You might conjecture that's the exact number of non-isomorphic group structures on S. That's correct.

A total order on S is a special type of subset of S^2, so you might conjecture that the number of total orderings of S, modulo isomorphism, is #(2^S). That's correct.

The above observation immediately extends to counting the number of partial orders. Many other corollaries are possible.

A wellfounded partial order is a more restrictive type of partial order. You might conjecture that the number of such relations on S, modulo isomorphism, is exactly #(2^S). That's correct.

A wellordering is a wellfounded total order. If you don't collapse via isomorphism, you might conjecture that the number of wellorders on S is exactly #(2^S). That's correct.

You can similarly show that the number of non-isomorphic wellorders on S is strictly larger than #S. You might conjecture that it's exactly #(2^S). This is the Generalized Continuum hypothesis, which is undecidable over ZFC :)

[cupidologys]

⸻ CHAPTER FIVE; ALL MEN ARE EQUAL

pairing: dazai x f!reader (fantasy au)

warnings: mentions/themes of depression

chapter list: this is CHAPTER FIVE of a multi-chapter fic series. PLEASE read the chapters below (in order) before this one or you will be very lost!!

prologue

one

two

three

four

word count: 3.5k

+ + + + + + + + + + + + +

Back in your homeland, at the Imperial Palace, the largest constructed facilities are ones of sport and training. Sharpened swords and polished armour take the place of bookshelves on bedroom walls, and the practice of scripture is seldom found. Higher education, though no less important than warfare, is strictly limited to scriveners, court officials, and the professional erudites of your father’s choosing. In the face of current conflicts, most of your father’s people are far more absorbed in military affairs and bureaucracy than arithmetics, the sciences and the humanities.

Although, when it came to you, it was like a switch went off and all those sentiments were turned upside down. 

By a certain age, your tutelage switched from scholarly knowledge to that of etiquette and what he referred to as ‘womanly affairs’. Those usually consisted of things like sewing, music, and art classes. The only one you ever enjoyed was the horseback lessons. 

But thankfully, your father’s one track mind meant you were never discovered for—or suspected of—possessing further-education books and studying politics, diplomacy, and military tactics on the days general schooling lessons were cancelled. It is why you find yourself in the royal library, hours before you are due to meet Dazai for dinner. 

Hundreds, if not thousands, of marble shelves line the walls from floor to ceiling. Each one is stacked, end-to-end, with leather bound tomes and tea-stained manuscripts. There is a fireplace in the right corner, carved from blackened stone and crackling with warmth. Around it sits a pair of dark-green, thickly-cushioned armchairs, along with a matching sofa that is wide enough to fit at least four people. 

You walk further in and are greeted with four arched windows spanning the length and height of the space, each one clear as the summer sea. You squint, momentarily blinded by a sudden passing ray of sunlight. Birds are chirping underneath the morning sky, and branches of a looming willow tree sway in front of the left-most window. You take in the sprawling garden view; a labyrinthine maze of hedges take up the centre, and a large assortment of decorations speckle the grounds. Smaller fountains, rainbow flower beds, and iron-wrought benches are only a few of what you can see. 

You look around a bit more, noting the study tables anchored to the floor and the winding staircase that leads to the open-plan second floor. The library is well-kept, as shown by the pots holding blooming flowers along the window sills, but the dust lining the shelves indicates that no one has used the archives in a long time. You wonder why—it is the first and only comforting place that you have found in the cold, lonely palace. 

You make your way down the stacks before a section catches your eye.

A Comprehensive Guide on Abilities and a Meta Analysis on their Structural Archetypes; 

The Scholar’s Circle’s Codex on Yokohama’s Political Affairs;

North vs. South: A Dynastic Tale of Continental History. 

You grab all three and almost lose your balance from the weight of each text. More and more books are added to the pile in your arms until you can no longer see straight ahead. 

With a huff, you drop the mountain of pending research onto an oak-stained study table and quickly get to work. 

Hours pass, the concept of time long faded as you lose yourself in the world of preternatural powers, warring states, and the cluttered institutions that make up the Kingdom in its most present form. 

The striking differences between Yokohama and the Northern Empire are more vast than you had ever imagined. It's a stark contrast—governance, industry, arts, religion and everything else you've come across so far. Not a single commonality to be found.

“How has…? But wouldn’t the roots originate from the dark ages? Let’s see…” you mumble, talking to no one in particular. 

“Have you found a specially interesting read?” A particular person asks. 

You fall out of your seat in surprise. 

“General!” You squeak, reeling from his sudden appearance. 

The mild-mannered Fukuzawa gives you a gentle smile and moves to help you up. He hooks two large arms under your own and lifts you back onto your chair. The scene reminds you of a mother cat picking its kitten up by the scruff of its neck.

You drop your head onto the table in embarrassment, refusing to make eye contact until, hopefully, a meteor comes falling onto earth and crushes you to death. 

“Good morning, General,” you mutter. 

“Hmm.”

You peek up at him with one eye. “What?”

“It is five in the evening,” he replies, bemused. 

“What?!” You bolt up, shame long forgotten. 

It takes you a second to realize how orange the library is, cast in the hues from the setting sun. 

You drag a hand over your face, rubbing the fatigue from your eyes. “Shit, I didn’t realize how late it had gotten.”

Fukuzawa raises a brow. 

“What? You’ve never heard a noble cuss before?” 

He taps his chin. “I can’t say I have. You truly are a breath of fresh air, Your Highness.”

You grin. “As are you, General. And please…”

He listens, head tilting in curiosity. 

“It is [name]. We are friends, are we not?” Your false sincerity coats your words like a second skin.  

The sun dips far below the horizon, robbing the world of its light. You take in the storm clouds in the distance, absentmindedly wondering if the Empire would experience the same downpour later in the night. 

Fukuzawa ponders your question for a moment longer before answering. “We are, but I am also your subordinate, so I am afraid I must decline.”

“And if it is an order?”

Fukuzawa’s eyes sparkle. “Then I am under aristocratic obligation to comply.”

In a tone laced with authority and bemusement, you proclaim: “I, acting Monarch of Yokohama, hereby order General Yukichi Fukuzawa to act beyond propriety and address me by given name only. No titles, no fancy designations. Just [name].” 

“As long as you are willing to grant me that same honor, [name].”

You grin. “See? Isn’t that so much better, Yukichi?”

The General only laughs and turns to take a seat across from you. The armour he dons makes a clanging noise as he settles himself. Patches of dirt litter the surface of the metal while other areas sport minor indents—likely from the force of a blade's flat or hilt. 

“Did that hurt?” You nod towards the largest dip in the steel. 

He looks down at his left side, around the area between his upper ribs. “Couldn’t even feel it.”

“Of course not,” you wave, returning your attention back to the pages. 

“I see you are interested in…” Fukuzawa leans over the table, peering at the emboldened titles of each tome. “Yokohama politics, history, and culture?”

“The pen is mightier than the sword, as they say,” you muse. “And a bright mind is far mightier than those stumbling blind in the darkness of their own ignorance.”

“I do wish more members of the court shared that sentiment. It would certainly make my migraines less frequent.” 

You faintly recall the term from a book you finished earlier. “The… inner court?”

“The very same. A parliamentary round table of aristocrats and representatives, headed by the Four Noble Houses.”

“The Four Noble Houses? You mean…” You cringe, an unpleasant memory resurfacing. 

Fukuzawa’s eyes gleam with amusement. “Ah, yes. I recall a certain purple-faced duke drenched in the colours of His Majesty’s most favoured cabernet sauvignon.”

You smile sheepishly. “I messed up, didn’t I?”

“Formally? Yes.”

You groan and drop your head in your hands.

Fukuzawa lays a palm on your shoulder and gives you a gentle pat. 

“But reasonably? Absolutely not. He deserved ten times worse than what he got.”

“Someone needed to stand up to him,” you point out. 

“Sadly, there are not many people who can.”

You sigh at that and go back to your research. The moment you set your eyes back on the book, the pages in front of you begin to blur and mesh into a whirlpool of ink. 

“Maybe it is time for a break…” you murmur. 

Fukuzawa leans forward and studies your fatigued expression. 

“What have you learned so far?”

You snort. “You mean other than our sordid history? The decades of hatred and conflict brewing between our countries?”

“Ah, yes. Besides that fun little facet of our politics.”

You run through the miles of information you had just absorbed, each little bit coming together piece by piece to paint a very clear picture of the modern world—one where mystic abilities, gods of old, and monsters coexist in disharmony. 

‘Abilities’ as you have come to know them, are practically non-existent among the lower caste in the Northern Empire. The only ones who wield them are of noble blood, aside from the rare few commoners—unfortunate individuals who would be executed for merely holding power outside of their status. Even then, barely anyone manifests one. In recent years, the only ability-user you know of is Chuuya.  

In Yokohama, these powers are respected, admired, and much more plentiful. In your textual observations, it is noted that the military and governing leaders are chosen for their abilities. 

“Hm… what is yours?”

 You are curious. What sort of fate-bending, death-defying power could this seasoned warrior have?

“Mine?”

“Your ability. You must have one, being the head of such an elite corps.”

“My ability…” he pauses. 

You raised a teasing brow. “What? You’re not going to tell me?”

“Just considering the risks of doing so. You have proven yourself to be both smart and deceitful. A deadly combination.”

“Are you saying you don’t trust me?” You place a hand on your chest in mock offence, scoffing in indignation. 

Fukuzawa laughs—that familiar smooth rumble that you have come to find placating. “Would I be wise to?”

“Of course not.” You wave a dismissive hand. “But you should tell me anyway because I am curious and stubborn and will likely find out on my own regardless.”

The general’s gaze is filled with a kind of warmth that is unknown to you, only interrupted by a flicker of a melancholy that twists his expression momentarily." It happens so fast you almost mistake it for a trick of the light.

“You remind me so much of her…” He mumbles under his breath so softly you pass it off as a whisper of the wind. “Very well. I will tell you.”

The sun has all but disappeared from the horizon, the shimmering moon slipping in its place. The dark, glittering night falls onto Fukuzawa’s features beautifully, making  him seem a little more weathered and a little less mundane as he explains his decidedly non-mundane powers. 

“It allows me to control my soldiers’ own abilities. I am able to manipulate their capabilities, help navigate their potential, and expand the boundaries of what they can do. That is my ability,” he explains. 

You mull over Fukuzawa’s words, a bit surprised at the nature of it all. The powerfully built military veteran looks at you like he knows what you are thinking—knows that you are confused on why someone with his battle prowess has such a passive skill. 

“You forget, Your Highness, that before I am a warrior, I am first and foremost a leader. Without my men, I am nothing, and without me, many of those men would not have survived until now,” he states. He says it like a fact, and perhaps in some ways, it is. It makes more sense the longer you think on it, his ability is almost perfectly suited to his position. You wonder what yours would be if you manifested one. What about Dazai? Would his ability reflect bloodthirst and coldness? Or would it be the opposite of what you know him as?

You make a mental note to come back to that question later, and direct your attention back to the conversation at hand. 

“[Name],” you correct.

Fukuzawa blinks. “Sorry?”

“You called me ‘Your Highness’ just now.”

“I apologize. Force of habit,” he drops his head in a slight bow and the moonlight streaming through the open windows reflects off his gray hair, transforming it into a silver mane. 

Fukuzawa apologizes to you a lot, like a father fumbling for words in front of his newborn, careful not to be anything but kind. If anything, you find it endearing. As well as a little… disappointing. 

“General.”

Fukuzawa’s smile drops at your change in tone. The worry in his eyes is clear. “Is something wrong?”

You give him a small smile, a tad tense. “No. Not really. Though, I would like to ask you something. Would you humour me?”

“Of course. I will answer anything within reason,” he reassures. 

You rest your cheek against your palm, curiosity and wariness burning bright. 

“Why are you so kind to me? I know how this country views the Empire—views me. I am not blind to the scornful glances nor hidden insults thrown around. I am numb to them. But you… Kunikida… that peculiar doctor as well, you are all much too cordial with a sworn enemy. Is it pity? Some misplaced sense of duty? Or perhaps it is all fake and you are all laughing behind my back as we speak.”

Silence spreads through the empty library, the only noises are the crackling of the fireplace and the gentle swishes of the willow branch behind you. The only thing you hear is your pulse thrumming against your skull.

If Fukuzawa is taken aback by your bluntness, he does not show it. Despite only knowing you for this short period of time, he is probably already used to your brusque manner of speech. He folds his hands in front of him and leans backward, taking some time to come up with a suitable answer. You can practically see the gears turning in that head of his. 

A few moments pass before he finally speaks in a serious, yet gentle, voice.

“Do you think yourself undeserving of our respect?”

You shake your head and answer: “Not at all. I am only surprised you would willingly impart it to me.”

“I cannot speak on Sir Kunikida or Dr. Yosano’s behalf—although, I imagine they share the same thoughts—but I am kind to you because it is common sense. I am kind to you because I am honoured to serve under your reign,” Fukuzawa assures. His expression softens. “I am truly sorry about the harassment you have had to endure. I will do my best to keep them in check, but if it happens again, do not be afraid to use your status. You are their ruler. Do not let them forget it.”

A lump forms in your throat and you force yourself to swallow it down. The support eases your heart, but the anxiety does not fully disappear, nor does the cold tingle of resentment in your chest. They probably never will. For now, you will accept his words, but with caution, as you are still very much in enemy territory. You will need to lead with your mind to survive, not your heart.  

And Fukuzawa? The gentle general is merely a stepping stone, not a friend. 

“I… am grateful. Tha—”

“General Fukuzawa!” In a very familiar fashion, the doors to the library burst open to reveal a man, effectively cutting you off. 

Kunikida stands beneath the frame, face alarmingly red and breaths coming out in short, laboured puffs. Out of the corner of your eye, you catch Fukuzawa grimacing. 

“What. Are. You. Doing. Here?.” The minister spits out each word with barely contained anger—more accusation than actual question. 

“Chief Minister.” Fukuzawa bows and slowly inches himself towards the door, closer and closer to the fuming blonde. “I see you are… upset.”

Kunikida’s eye twitches. “Upset? Upset?!” His voice hits an impressive octave and you briefly wonder if he’s ever considered a career in opera. He certainly has the knack for it. 

“I—” 

“The outdoor arena is on fire.”

The general clears his throat. 

“Right. I did tell them not to try out those new techniques without me around, though His Majesty’s soldiers were never ones to adhere to the rules.”

“A black hole opened up in the ceiling and swallowed three stable boys. They were… fully nude when they fell out an hour later.”

Fukuzawa blinks. 

“That’s… new.”

“You have five seconds,” Kunikida says flatly. 

“Well. Duty calls. I shall have to put out some fires… er… literally.” Fukuzawa makes his way to the open doors and is about to leave when he adds: “Have a wonderful  night, [name].”

“Good luck,” you laugh. 

He gives you a small wave before disappearing down the hall. 

You turn your attention to Kunikida who is now slightly less red, though still glowing a nice shade of pink. 

“Good evening, Chief Minister. To what do I owe the pleasure?” You ask. 

“I am here to bring you to dinner service. Perhaps you have forgotten? You seem to be engrossed in our literary offerings,” he answers plainly. 

Kunikida stays standing, but has walked further into the room, hands clasped behind him as he studies the books you chose with furrowed eyebrows. 

“I enjoy reading. Is that such a crime?”

“I am only surprised you were able to find this place. After His Majesty banned entry, most just ignore it as they pass by.”

You cock your head to the right. “I was curious about that. Why? It is a beautiful library—a sunlit treasure trove of knowledge. I would imagine most people would be clawing at the doors for just a glance, yet it is as barren and untravelled as the deserts in the West,” you muse.

 Your curiosity is only a mild interest until Kunikida’s gaze sharply turns away from yours, blatantly avoiding your poking and prodding. His averted eyes cause what little inquisitiveness you had just felt to balloon into a wave of eager investigation. 

“Kunikida.”

He adjusts his glasses and nervously glances at his timepiece. “We are going to be late if—”

“Kunikida.”

He sighs, relenting. 

“If nobody uses this place, why is it so well kept? There are no dirt patches or cobwebs, but the dust between pages suggests that no one has opened them for many years. ”

“If I were to make an educated guess…” Kunikida stops for a moment to think. “I would wager that His Majesty misses what it used to be, and is only trying to preserve the last of that magic. Though the memories here are much too vivid and much too painful for him to come back to.”

What it used to be… 

A flicker of something… a fleeting feeling… No. A memory. At the very back of your mind—

“But I do not think he will continue to do so.”

It vanishes, and you fall back to reality, grasping at nothing and nowhere. 

You shake yourself out of your daze, a bit peeved at the interruption, but curious all the same. 

“Do what? Preserve this place? You believe he will let it just… crumble to ruins?”

Kunikida takes a seat and folds his gloved hands together. The lines on his forehead appear as he tenses, preparing his next words with careful precision. He works his jaw, tension releasing and forming with each movement, as if he is warring internally, fighting to either let the words out or keep it in. 

You hope he chooses the former. The more information, the better. 

His expression settles and a stern look replaces his calm visage. Whatever he has to say must be serious.

You catch yourself tapping the side of your thigh anxiously under the table and clamp your fingers down on your leg… hard. Your father did always say that a royal must be poised and perfect, and he made it extremely clear that such emotions were to be erased and forgotten. 

And if they weren’t… 

A chill runs down your spine at the memories.

“I am well aware that you are, and pardon my candor, untrustworthy.”

You almost snort. Not the first time you’ve heard that and it certainly won’t be the last.

Kunikida continues. “But I believe it is only right to tell you as His Majesty’s spouse. King Dazai is… he is…” Kunikida pauses as he fumbles for the right word. 

A clock ticks. Kunikida settles on a phrase. 

“Unwell. A disease of the mind and heart that has stolen his will. He is here only to serve a purpose and that purpose is not to live out the rest of his life. He exists, but for years now he has not been… here. Almost as if one wrong move and the line His Majesty balances upon disappears and takes him with it.”

Time slows. The air thickens. Are you breathing?

“Slowly but surely, he is fading away,” Kunikida pauses and swallows as he tries to work out his next words. 

“Some days I believe he is better. Most days I do not allow myself to indulge in such a lie.”

—

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