#countable and uncountable sets

[floralfractals]

@the-uk-is-jk real numbers is like a soup unfortunately

countability

[notlongtolove]

to be an accountant of the heart

because it’s utterly, bone-deep terrifying. to look into the eyes of the person you love most in the world and feel the weight of a possibility that you might love them more than they love you.

pairing: spencer reid x fem!reader (second person, no y/n)

genre: angst-ish, fight and makeup

content: established relationship fight and makeup woof woof rookie bau reader feels insecure about how much she loves spencer, worries she's too clingy, spencer reid best bf ever

word count: 5k

note: this was haunting me in my drafts for the longest time... please be nice my heart can't take it (psa guys don't ever tell ur partners that they love you more than you love them bc 5 years down the road they'll cope by writing deranged spencer reid fics like this)

a line: You’ve always been this way—more flame than moth, more lightning than thunder. It’s one of the things he loves most about you.

and then it is hundreds of hours later, and you are still hunched over your flowcharts and abacus, trying to decide if you have gotten enough. This is the loneliest job in the world: to be an accountant of the heart. - tony hoagland

The English language draws a neat line between many and much. It divides the countable from the uncountable.

The word many is meant for things you can count. How many cups of coffee have you had? How many days will you be gone for? 

The word much belongs to what cannot be counted, what cannot be numbered. How much longer do we have in bed? How much did you miss me? How much do you love me? 

How much?

It’s an innately impossible question. Love, after all, is supposed to be infinite, unbound, unquantifiable. Any attempt to measure it—to reduce something so sacred to a number, a unit—is to taint it. And why would you want to do that? Why would anyone? There shouldn't be any need to measure something so inherently immeasurable. 

Deep down, you know there's no actual way to count love. You suppose this instinct to measure has always been there, to wonder if the love you received can be tallied like time. It’s buried deep, old as the child you once were. 

Still, the question begs itself. How much? How much more? How much less? If comparison is the thief of joy it’s only because it leaves you with the revelations nobody asked for, the truths nobody ever wants to see. 

Put love on a scale, wait and see—Will it balance or won’t it? 

“Glaring at the clock isn’t going to make time pass any faster,” Elle teases from two desks away, her eyes locked on the report she’s skimming.

You don’t bother hiding your sigh as you glance up from where your chin rests heavily in your palm, elbow propped against the desk. The pencil in your other hand twirls idly, betraying your impatience. “He said they landed an hour ago,” you grumble. Only the faintest trace of a pout slips through.

“Working hard or hardly working, ladies?” 

Your head perks up at that. Trust Derek Morgan to know how to make an entrance, arriving right on cue, grin wide and swagger intact. 

JJ, seated beside you and noticeably more amused by your restlessness than concerned, spins her chair around as she asks, “How was the convention boys?”

“It was great—more than great actually,” Spencer says, appearing from behind Morgan. He’s lugging a bag that seems twice as heavy as when you’d helped him pack it five days ago. “All the speakers were incredible. I got to talk with Lonnie Athens himself. He gave me a signed copy of his latest book.” His grin widens tenfold. “It’s not even out in stores yet.”

You’re halfway out of your seat, ready to pounce on Spencer the moment he sets his bag down. But instead, he offers a halfhug and a light squeeze to your shoulder. It’s understated, but it’s Spencer. Public displays of affection aren’t his thing, and you know better than to expect more. Still, five days without him makes you ache for just a little more.

“It was alright,” Morgan interjects with a casual shrug as he takes a seat at the edge of your table, narrowly missing your nth mug of coffee. “Great sandwiches though.”

“Yeah, you sure seemed interested in the sandwiches,” Spencer says dryly, the kind of tone that suggests sandwiches were not the main attraction.

Morgan smirks, unbothered. “New York, man,” he says with a grin. “New York.”

You turn your attention back to Spencer. “How’d you sleep?” you ask, your question aimed entirely at him.

“Surprisingly well, actually,” Spencer replies, “Despite the snoring.”

Morgan’s response is immediate—a light thwack to the back of Spencer’s head. “How’d he sleep? More like, how’d I sleep. Lover girl over here had him on the phone half the night.”

“I wasn’t that bad,” you shoot back, narrowing your eyes at him. But then your gaze drifts to Spencer, searching for confirmation. “Was I?”

Spencer hesitates, his lips pressing into a faintly sheepish line. “I did wake up late for one of the panels,” he admits, scratching the back of his neck.

“Oh, you think you had it bad? I’ve never seen someone go through so much coffee in a week,” JJ says, nodding in your direction, “She wiped out the entire stock.”

“Almost bashed her over the head with a cup of coffee myself when I had to settle for the instant stuff,” Elle chimes in. A collective shudder goes through the group. “No offence, Reid,” she adds.

“None taken,” Spencer replies smoothly, just in time to earn another smack on his arm, this time from you.

You’ve endured more than your fair share of teasing—it comes with the territory when you’re part of a team like this. You, bright-eyed and bushy-tailed, three years his junior. Him, more comfortable rambling about the number of kernels on an average cob of corn than talking to any girl, let alone one with a smile like yours that could make his knees buckle. What had been an odd match to some, made perfect sense to others—Though Spencer would argue that Garcia just liked seeing him with any girl who could make him laugh the way you could, especially within three days of meeting him. It’s a feat nobody else has yet to achieve in the year you’ve been on the team. 

“Missed you,” you murmur, just loud enough for him to hear.

Spencer flushes as his lips part, maybe to respond, but Elle cuts in before he gets the chance. “Save it for later, lover girl. Some of us want to hear about those sandwiches.” 

“Oh, they really were better than last year’s,” Spencer begins, now distracted, completely oblivious to Elle’s sarcasm, “Probably because the annual reports showed an increased budget for the global initiatives.”

JJ raises an eyebrow in amused disbelief. “You read the FBI’s annual budget breakdown?”

Spencer looks genuinely surprised by the question. “You don’t?”

Chuckles echo throughout the group and though you smile faintly, it doesn’t quite reach your eyes. You just can’t help it as the tally marks start to stack up in your mind. One for the way his attention is just a little too distant, his excitement seemingly aimed at everyone but you. Another for every time you wait for his gaze and it doesn’t come. He’s too absorbed in recounting a discussion about deterministic causality he’d had with a keynote speaker. 

Compared to Spencer, who was often so reserved, it was easy to feel like your emotions were too big, too eager. Dragging him, wide-eyed and stammering, up the stairs to Hotch’s office six months ago had been nothing short of a test of strength and sheer determination. You’d been the one to silence him with a gentle kiss to his knuckles, promising him that everything would be okay. You were a live wire compared to him, everyone knew that. Lover girl, they teased, though never cruelly. In the field and out of it—Clingy to a fault, always wearing your heart on your sleeve. 

Lover girl through and through, you wait patiently for Spencer to look your way. 

He doesn’t. 

“Yours or mine?” Spencer asks as you stand side by side on the curb, bags in tow. 

“Think I’ll go to mine,” you reply curtly. You don’t trust yourself to say anything else right now.

“That’s fine. I’ve got an extra day’s worth of clothes with me.”

“You can go home,” you say, cutting him off. It comes off sharper than you intended. Then, softer, as if trying to backtrack, you add, “If you want.”

He looks at you, baffled. “Why would I do that?” 

It’s not a rhetorical question, he genuinely doesn’t understand. Weekends apart have never really been your thing. 

“Because—” You cut yourself off mid-sentence. What could you even say? Because you seem so perfectly fine after 120 hours apart. Because the tally marks said so. Because the scale said so. Instead, you huff an exhale and settle for, “No reason. You look tired. Thought you’d want to go home or something.”

“Again sweetheart. Why would I do that?” he repeats, incredulous. 

You fight off a resigned sigh, though you’re sure he catches it, and pull out your phone. “I’m calling a cab,” you mumble, thumbing at the screen. “Are you coming or not?”

“Yeah, I’ll come with you,” he says, still calm but clearly confused.

“Fine.”

The ride home is quiet, save for the driver’s rambling complaints about freeway traffic at this hour. Normally, you’d be the one to humour any conversations with strangers, chiming in with polite nods and oh, reallys while Spencer watched, bemused by your ability to make small talk with anyone. But today, you’re just not in the mood, leaving poor Spencer to fend for himself.  

Which to his credit, he does—By turning the conversation into a tangent about how traffic patterns correlate with certain hours and commuter behaviour, and delving into a detailed explanation of the queueing theory. He does this till eventually, even the driver goes silent, though whether it’s out of confusion or exhaustion, you’re not quite sure. 

You can feel Spencer’s eyes on you in the silence, flicking toward you every now and then. The concern in his attention does nothing to soothe you. If anything, it only fans the flames of your irritation. When the car finally rolls to a stop outside your building, you hand the driver a $20 bill, wave off the change, and stride toward your door without another word. You’re out before Spencer can even pull his door open.

Inside, you drop your things on the couch resignedly and kick off your shoes without so much as a care. They land in a scattered heap that you don’t bother to fix. Spencer lingers behind you, ever patient.

“What do you want for dinner?” His voice is soft, tentative, as he bends down to pick up your discarded shoes, lining them neatly by the door. “We could order something. Chinese, maybe?”

Spencer knows you well—knows how your mood sours when you’re running on fumes. Particularly on days like this, when your only sustenance has been cups of crappy coffee and a few stale crackers he’d coaxed you into eating earlier just before you left, bribing you with a quick kiss on the cheek—After checking that nobody else was in the break room, of course. 

Sullen as you are, you can recognise the offer for what it is. It’s sweet. A thoughtful acknowledgement of how well he knows you, how much he cares. He’s offering you a lifeline, a quiet invitation to let the storm pass without forcing you to name it, something you’re evidently trying not to do. 

But tonight, it feels almost patronising. It’s a spotlight on the hurt you can’t quite temper, like he’s trying to fix something you’re not yet ready to admit needs fixing.

“I can run down to the—”

“I’m not hungry.” 

You walk straight into your bedroom without another word, leaving him standing there in the doorway. You hear him exhale quietly, not quite a sigh but close. Probably one of resignation. Another tally mark falls on the scale. 

“Sweetheart,” he starts. You know he’s testing the waters, trying to find an opening. But you don’t look at him, don’t give him anything to work with. “Can we talk?” he asks, his fingers brushing yours as he takes a seat at the edge of your bed.

“Talk about what?” You’ve always been good at feigning ignorance, but the way you pull your hand away from his is anything but subtle. Spencer sighs, pinching the bridge of his nose as he closes his eyes briefly. He’s clearly exhausted. This is exhausting. You’re clearly exhausting. You can’t help but wonder why you always do this. 

“Was it Elle? Morgan?” he ventures cautiously. “The teasing?”

“They always tease me,” you say with a shrug, your voice dismissive. “I don’t care.”

It’s a half-truth, and you both know it.

Spencer nods slowly as he tries to piece this together. He knows you’re not usually one to let things fester. You’re never angry for long, and even when you are, you laugh it off, always quick to join in on the joke. He knows better than to profile you—it's an unspoken rule within the team and, more importantly, within your relationship. But Spencer’s anything if not desperate to understand.

He watches you slip into the bathroom with a sigh, shoulders dipping. The light flickers on, but you don’t meet your own gaze in the mirror. You’re not angry. That would be easier. There’s something quieter in your eyes. Defeat, maybe. 

“I missed you,” he offers, stepping into the doorway. His tone is softer now, pleading.

“Did you?” It’s almost sarcastic, but not quite. Irritable but undercut by something raw, as though you don’t really believe he did.

Spencer swallows. “You don’t think I missed you?”

“A little hard to tell between the fawning over Lonnie Athens,” you say, wiping mascara from under your lashes. “Or was it the in-depth analysis of sandwich platters?”

It’s a snap, all sharp edges and fire, and for a second, he forgets the minefield he’s meant to be tiptoeing through. Has to bite back a smile. You’ve always been this way—more flame than moth, more lightning than thunder. It’s one of the things he loves most about you.

“Is that what this is about?” The words slip out before he can stop them, and the second they do, he knows. Rookie mistake. Your spine straightens, your jaw sets, and he wants to take it back, rewind, try again.

“This,” you echo, turning to face him. “What exactly do you mean by this?”

Spencer reminds himself that fire is never snuffed out with ice. You douse a flame gently, carefully. So, he steps forward, quieter now, fingers grazing yours before he takes your hand in his, guiding you toward the bed. He doesn’t pull, doesn’t rush, just leads you toward the bed with the same patience he knows you need when you’re fragile and burning.

Regardless, you try to resist, to hold yourself upright. You’re fighting the urge to sink into it—His touch, the bed, all of it. 

“Sweetheart,” Spencer murmurs, taking a seat beside you. “I know you’re not angry. You’re sad. And I’d really like to know why. Tell me, please?”

Deep inside, you know you’re just clinging on to the last embers of your frustration. But it’s hard—impossible, really, when you’re a fire with no kindle left to burn, and Spencer is all soft whispers and gentle hands, featherlight and soothing. 

You hesitate, twisting the fabric of the duvet between your fingers. “I just—I—You were being mean.”

Spencer lets out a slow, quiet breath. Relief, almost. Not because he agrees—He knows himself well enough to be sure that ‘mean’ isn’t the right word. But he knows you well enough to understand what it means when you say it.

Mean is what you say when you’ve been hurt and don’t know how else to put it. 

So he follows your lead. Doesn’t fight it.

“M’sorry, sweetheart,” he mumbles stroking your hand with his thumb. His touch is warm as it is gentle. 

Because it’s not about whether he was mean or not. Spencer knows that. Knows you. Knows that kindness has never been a given for you, knows that you wouldn’t recognise patience if it came knocking. And he knows you well enough to know that you think in some twisted way, that you’ve brought this hurt upon yourself, that you deserve it. 

What matters is that you were hurt. And that’s the one thing he never, ever wants to do.

“I didn’t mean to hurt your feelings. Can you tell me how I did?”

“You just kept going on and on about the stupid conference. You didn’t even hug me or—And then you—” 

You don’t continue. You can’t. You feel ridiculous. Stupid, even. Mopey and small over something that shouldn’t matter this much. Over the realisation that he doesn’t need you. And why should he? It’s not Spencer’s fault. Not at all. 

His indifference is what it is and what it was. Indifference. It sits like a weight on your bones—Cold, sharp-edged, piercing. He can go 5 days without you. You can’t. The tally marks accumulate, unbidden.

“And then I…?” Spencer prompts gently, prying your fingers from the duvet and replacing the tension with his thumb, tracing slow, soothing circles into your palm instead.

“You ignored me, and I just—” Your voice wavers, frustration bubbling over. "I just felt so—so ignored!"

Wonderful vocabulary. Of course, your words would fail you now.

“And the teasing—I know, I know, I can be impossible sometimes, but I just—I just really missed you! And I get it okay? I’m clingy and you’re not and god forbid anybody else is but it’s because I love you!” You inhale sharply, your hands slipping from his to curl into fists in your lap. “And you didn’t react at all, you didn’t even care! You made me feel like—I thought that you—” 

You cut yourself off before the flurry of tears take over and drown you out. 

Spencer waits a beat, choosing his next words carefully. 

“You thought… that I don’t love you?” His voice isn’t laced with sarcasm, nor does it carry incredulity. It’s a genuine question, as though he’s retracing the moments between you, trying to understand how you could possibly come to such a conclusion.

“No, it’s not that—” you’re quick to say, desperate to correct him. You know Spencer loves you. Of course, you know that. How could you not? It’s Spencer. He loves you like it’s his life mission to show you just how much he loves you. “I know you love—I know that. I just—” 

You bury your face in your hands, fingers pressing into the hollows beneath your eyes—A feeble attempt at hiding.

Because it’s utterly, bone-deep terrifying. To look into the eyes of the person you love most in the world and feel the weight of a possibility that you might love them more than they love you.

To want to shout: Love me. Please love me, and please feel it with every fibre of your being as I do with mine. The kind of love that makes you want to scream from rooftops, to etch it into the sky, to burn the world down just to prove its enormity. 

Because then the question comes: Which would be worse?

To shout into the vast, open air and hear nothing in response? No echo of the same intensity. Or to stand amidst the smouldering ashes only to look into their eyes and find they don’t recognise you anymore? To see confusion or pity where love used to live.

You blink your watery eyes open, but you can’t bring yourself to look at him. Instead, you settle on the knobs of your knees, tracing their shape with your gaze. 

Anything but Spencer. Not right now. 

You take a sharp breath, steadying yourself before continuing.

“Sometimes, I feel like you don’t need me as much as I need you and that scares me. And I know it’s stupid, even I feel stupid thinking about it. I don’t even want to be codependent or whatever but I—I just can’t help but think that sometimes—” 

Your breath shudders out of you, long and uneven, “I love you more than you love me.”

To say Spencer feels his heart break would be an understatement. It’s not a clean break, not a single, shattering moment—it’s a slow, relentless unraveling. It’s a gut punch, pain and duress packed tight, failure laced in every syllable. His heart shatters, splintering into pieces so sharp they lodge in his throat, in his lungs, in every part of him that has ever loved you. 

Silently, he’s always known the teasing would hit a breaking point. You’ve worn that insecurity for as long as he’s known you—too young, too green, too desperate to prove yourself. He just didn’t think it would carve its way between you the two of you like this. He’s watched you lean into it, let the jokes land, let them chip away at you. Newbie. Rookie. Lover girl. As if laughing along might soften the edges of it all. 

You flop onto your back on the bed, boneless, the confession stealing the last of your fight. There’s a splotch of blue paint on the ceiling from last month, when you both tried to repaint the room and got distracted halfway through. It doesn’t make you smile, not even  a little.

“That’s not true.” The mattress dips under Spencer’s weight as he settles beside you, thumb tracing your hairline. His arm moves, coaxing you to toward him, gentle in the way only he knows how to be with you.

“You’re not impossible, sweetheart, you never are. And I know they tease,” he murmurs, fingers of his other hand grazing over your knuckles, “but I also know for a fact that you don’t fall apart without me when I’m gone. That would be co-dependency. And I know that’s not you. You passed your requalifications with flying colors while I was away,” he says. “Garcia sent me the records. You know you even beat Morgan’s old score?” 

You sniffle, startled. That had been your surprise. You’d wanted to tell him yourself. 

“She told you?” 

He shakes his head. “I asked. I always ask for updates on you when I can’t be there.”

A small “Oh,” is all you can get out. 

With every other guy you dated, you’d attempted to play it cool, dialling down your enthusiasm, biting back your texts, and pretending to care less than you did. But every relationship seemed to end the same way: you were “a lot” and they weren’t equipped to handle it. It never quite stuck though, and thank god for that. 

Because then you met Spencer.

Sweet, steady Spencer, who didn’t just tolerate your spark but cherished it. Spencer, who had let you cling to his hand during every takeoff and landing on the jet the first week on the job. He never flinched, never teased—Even when everyone else casted him sympathetic looks, the kind that silently acknowledged how your grip was probably cutting off his circulation. Spencer who has kept every scrawled doodle and note you’ve ever given for him, even the ones scribbled haphazardly on napkins or receipts. He knows carbon prints fade within months so he stores them in a shoebox tucked away in his cupboard—Just so they can last that much longer. 

Spencer didn’t just accept the parts of you others found overwhelming. He singlehandedly brought them back to life. Every bit of your spark that had been dimmed or snuffed out by someone else had found new light in his presence.

Spencer’s fingers tighten around yours, a quiet kind of reassurance that draws you back to the present. 

“Being clingy is not the same as being codependent. I know you know that. There’s a clear psychological difference in brain chemistry.” His lips twitch, the smallest hint of a smile slipping through. “You’re clingy, yes. But I love that about you. I love coming home with you. I love coming home to you. I love how hard you love me, how proudly you love me. I know I haven’t been the best at reciprocating that around the team, and I’m sorry. I hate that I made you feel like I didn’t love you, or miss you.”

He shifts closer, eyes searching yours, open and earnest. “Because I did miss you. So much. I nearly blew a month’s paycheck in the gift shop. Spent half of it stocking up on those jelly crackers you told me about.” He shakes his head, like he can’t believe himself. “Morgan said I was whipped when I paid thirty bucks for a pair of souvenir socks.”

With a raise of your eyebrow you ask tearily, “and exactly how many pairs did you buy?” 

“Got you three pairs.” A sheepish little laugh escapes him as he ducks his head. 

And just like that, you’re smiling too. Albeit a small one, but that’s progress nonetheless. “And I don’t think you quite understand how much I love you when you say you love me more.” He leans in, his voice dropping, teasing. “I don’t know if you know this about me, but I’m very competitive.”

“Oh, so I’ve heard Doctor Reid,” you quip, eyes rolling. Spencer’s lips curve, just slightly. You don’t even notice the way you press closer to him, but Spencer does. He takes the opportunity to go on.

“In a way, you’re right. I don’t need you,” Spencer says. Whiplash doesn’t even begin to describe the way your head snaps toward him. Flame and lighting, no doubt. 

“Sorry, sorry,” he says quickly, his expression already twisting in regret. “I shouldn’t have phrased it like that.”

“I don’t see what other way you could possibly phrase something like that,” you snap pettily, already pushing yourself up to stand. 

“Hey, hey.” His hand reaches out, not quite grabbing yours but close enough to make you pause. “Lie back down, honey. Please.” 

Against your better judgment, you relent, sinking back into the bed. “What I meant to say was, I don’t need you,” he repeats, slower this time, deliberate.

You scoff, a bitter laugh slipping through your lips as you swipe harshly at your damp lashes. “I get it, Spencer. Clearly you don’t.”

“No, I don’t think you do,” he says, his voice unwavering. “Biologically speaking, I wouldn’t cease to exist without you. My heart would continue to beat, my lungs would continue to expand and contract, my brain would maintain its synaptic functions. I would survive.” He pauses then, eyes searching yours, “And can I tell you something?”

You don’t answer, but you don’t pull away either. He takes that as permission to go on. “You don’t need me either.” 

Your lips part, the beginnings of a protest forming, but he cuts you off gently.

“I know you said you do, but your autonomic nervous system would still regulate your breathing, your neurons would still fire, your body would persist.” He swallows, voice dipping lower. “But that’s not the point, is it? Love isn’t about biological necessity. It’s not about survival. It’s about choice.” 

The word “choice” feels almost ironic when it comes from Spencer Reid. You knew that the moment you met him. It was never really a choice, not for you. It was him, or nothing. Desperately, you'd like to think it was the same for him, too.

Your answer comes in the form of his thumb brushing lightly over your cheek. He’s patient, always, even when you aren’t. Kind in a way that sinks deep—Like you deserve it. You’re all sharp edges, brittle and worn, and he’s five days off a lumpy hotel mattress, yet the only thing he cares about is brushing away the tears from your skin. 

“Sweetheart, I don’t love you because I need you. I don’t think that would be love at all. That’s survival. I love you because I choose to,” he continues. “Because you are the strongest person I know. Because you are kind, even when the world hasn’t been kind to you. Because you give so much of yourself without hesitation, without ever expecting anything in return.” 

Spencer smiles, shaking his head. “Because you’re the only person I know who will spend thirty minutes on a call recounting every little thing everyone did in the office that you think I’d like to hear about—before you even think to tell me about your own day.”

“It was funny! Since when has Hotch ever tripped on the stairs?”

It’s unfair really, how easily his laugh breathes life back into you. Your heart stumbles over itself as his hand brushes tenderly along your jaw. 

“I’ve spent every day in awe of you since the moment I met you. And I fall more and more in love with you with each one. Even on the days I’m not with you. Even on the days I’m miles away. Even then.” Spencer presses his lips against the back of your hand as he adds, “Especially then.” 

“Really?”

You can’t help it, the quiet little thing in you that wants to hear it again. 

Your tears have dried, but their traces still shimmer faintly on your skin. Spencer presses a kiss to your forehead, his fingers tucking a stray strand of hair behind your ear. He’d say it again. A hundred times. He’d make that speech a thousand times over, if you needed him to. If it meant you’d never doubt it again.

“Really, my love.”

And just like that, a million tally marks fall at your feet.

A million for the way he presses another kiss to your lips, unrushed. A million more for the way his nose bumps against yours, lingering, breathing you in. Another million for the spark that creeps back into your eyes. 

It’s infinite, unbound, unquantifiable—The way he loves you, the sheer depth of it. You feel foolish for ever having questioned it. You thank your lucky stars—all of them—for Spencer Reid. For the way he’s looking at you like you strung the constellations together yourself. For the way he chooses you, again and again, even when you don’t choose him, when you shut down, when you go quiet. 

Because love to Spencer isn’t desperation, isn’t need—it’s choice. The deliberate, unwavering act of reaching out, of staying, and of saying over and over: I choose you. 

Not because he has to, but because he wants to. To be the one to put you back together again when you’re all embers and ash, to cradle you back onto earth when stare past him into the ceiling, to remind you that there’s still warmth in you left to hold.

To breathe the spark back into your eyes—It’s a choice he made the very moment he met you. It’s a spark Spencer swears he’d spend his whole life keeping alight.

⋆✴︎˚｡⋆ hi if you're here! thank you so much for reading! likes, comments or reblogs are very much appreciated!

ᯓ★ song recs if you feel like it: daylight by taylor swift intrapersonal by turnover

[cams-math-rants]

Finite and Infinite Sets

Recently, in my Discrete Mathematics course, we have been discussing the title of this blog post. It took me some time to understand these concepts, but it is finally clicking in my mind and it’s all I can think about at the moment.

First, I’d like to discuss some definitions.

When I mention the notion of a set, this refers to a collection of objects/numbers (we call each number in a set an “element”) in which the order doesn’t matter (unlike with sequences). An example of a set is ℝ, the set of all real numbers. Another example that will become important later is ℕ, the set of natural numbers (positive integers starting at 1).

A bijection refers to when every element in one set maps onto a unique element in the other set. When we map a set onto another, we write X → Y. This is shown in the image below, because it helps to see this visually.

When two sets are equinumerous (written X ≈ Y for two sets X and Y), this simply means there is a bijection between these sets.

By definition, in order for a bijection to be possible, the two sets have to contain the same amount of elements, or have the same cardinality (written |X| = |Y|, which in English means “the cardinality of X equals the cardinality of Y”). This means the two sets have the same size.

Now, we can talk about finite and infinite sets. The notion of a finite set is rather intuitive: there are n elements in a set X where |X| = n. An infinite set also makes sense conceptually, where the elements never reach a boundary. However, there are different kinds of finite and infinite sets that makes things interesting.

We can say a set is denumerable, which means we get a bijection when mapping ℕ to said set. An example of this is the set of integers, ℤ. When we have ℕ → ℤ. We can show this by representing this as a piecewise function and separating the integers into even and odd, and we find that there is a one-to-one, unique correspondence between the two sets (perhaps I will go through this proof in another post but there are examples online).

A denumerable set can be infinite or finite. In the example above, ℤ is infinite and denumerable. When a set is countable, that means a set is either finite or denumerable.

What is interesting is when we have an infinite and uncountable set, such as ℝ. In this case, there is a beautiful way to prove such a thing using Cantor’s Diagonal Argument. I won’t discuss this in full here, but here is the link to the Wikipedia article: https://en.wikipedia.org/wiki/Cantor's_diagonal_argument?wprov=sfti1

Thanks to anyone who has read my post. My name is Cameron and I’m a math major that loves to discuss what I’m learning in my classes. Keep in mind that I am not an expert, so I am open to anyone scrutinizing my explanations if you believe something is represented wrongly.

This is the song I’m currently listening to on repeat:

[thydungeongal]

So okay there's two different types of infinities. Countable and uncountable infinities. And in the context of the possibility space in analogue roleplaying, it is infinite, but it's a countable infinite. Like, of course not everything is possible in D&D or GURPS or whatever, because there are some things where the game will simply say "oh actually that shit can't be done," and even in freeform roleplay there are usually some limits imposed upon what can happen by the fiction and social contract.

But the range of things that can take place within the medium is still meaningfully infinite.

For an example: we know that there are infinitely many prime numbers. That doesn't mean that the set of prime numbers contains every number. Of course not. But just because the set of prime numbers does not contain the number 8, for example, doesn't mean that it isn't an infinite set. It's simply a countably infinite set.

[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

[xyymath]

Exploring the Infinity Paradox: What Does Infinity Really Mean?

Infinity is a concept that challenges our intuitions. It isn’t just a number; it’s an idea that can stretch across mathematical fields, appearing both simple and mind-boggling.

1. Zeno’s Paradox: Halfway There… Forever

Zeno’s paradoxes, particularly the famous Achilles and the Tortoise, deal with infinity in the context of motion and time. In the paradox, Achilles runs ten times faster than the tortoise, but the tortoise gets a head start. Zeno argues that Achilles can never overtake the tortoise because, by the time he reaches the point where the tortoise was, the tortoise will have moved further ahead.

While the reasoning seems absurd, it introduces the idea of infinite divisions of time or space. In reality, calculus helps us solve this paradox—by summing the infinite series of steps, we conclude that Achilles will indeed surpass the tortoise in a finite amount of time.

2. Cantor’s Set Theory: The Sizes of Infinity

The concept of infinity is explored profoundly in Cantor's set theory, which shows that not all infinities are equal. Cantor introduced the idea of countable and uncountable infinities. A countable infinity is an infinity where you can list all elements, like the set of natural numbers {1,2,3,4,… }\{1, 2, 3, 4, \dots\}. This type of infinity is represented by the cardinal number ℵ0\aleph_0 (aleph-null).

However, uncountable infinity is much more interesting. Consider the set of real numbers between 0 and 1. No matter how much you try, you can’t list them all in a sequence because for any number you list, there’s always another real number that isn’t on the list. This is a larger infinity—the continuum. Cantor showed that the real numbers form an uncountable set, and the cardinality of this infinity is represented by 2ℵ02^{\aleph_0}, often referred to as the cardinality of the continuum.

3. Hilbert’s Hotel: A Hotel with an Infinite Number of Rooms

Hilbert's Hotel is a famous thought experiment that demonstrates the paradoxes of infinity in a concrete way. Imagine a hotel with infinitely many rooms, all occupied. The hotel manager is faced with a new guest arriving, yet there is no available room. But because the hotel has infinitely many rooms, the manager asks each guest to move from room nn to room n+1n+1. This frees up room 1 for the new guest. The paradox shows that even with all rooms occupied, infinity can accommodate more guests.

The real kicker? If an infinite number of new guests arrive, the manager can still accommodate them by shifting guests in a similar pattern. Infinity has no "end," and this seemingly impossible scenario exposes the counterintuitive nature of infinite sets.

4. The Paradox of Infinite Sets: The Size of the Continuum

Cantor’s discovery wasn’t just theoretical; it had deep implications for how we think about the continuum of real numbers. The real number line is uncountably infinite—there are more real numbers between 0 and 1 than there are natural numbers, even though both sets are infinite. This raises the issue of infinite sets with different cardinalities. While ℵ0\aleph_0 is the size of the set of natural numbers, the set of real numbers between 0 and 1 has a larger size, denoted 2ℵ02^{\aleph_0}. This revelation left mathematicians questioning the true nature of infinity, as it seems there’s no upper bound on the size of sets.

5. The Infinite Hotel of Cantor: A Deeper Dive

Cantor’s paradoxes are further developed by thinking of sets of different "sizes" of infinity. Consider Hilbert's Hotel again, but now with infinitely many floors, and each floor is also infinite, forming a 2D array. This two-dimensional infinity shows that even within the same "size" of infinity, there are many different levels of infinity. As Gödel and Cohen later proved, infinity is not fully understood: questions like the Continuum Hypothesis—whether there is a set whose size is strictly between the size of the natural numbers and the real numbers—remain unresolved.

6. The Unknowable Future: Limits of Infinity

Infinity isn’t just something that mathematicians wrestle with—it’s tied to the very limits of knowledge and computation. Turing’s halting problem and the limits of computability further illustrate the practical consequences of infinity. No matter how powerful a computer becomes, it will never be able to calculate all the possible outcomes of an infinitely large problem.

Infinity is also central to modern physics—from the infinite expanse of space to the singularity at the center of black holes. As much as we strive for understanding, the paradoxes of infinity challenge even our fundamental laws of the universe.

[max1461]

Most real numbers are not arithmetically definable. This means, more or less, that there is no statement in the first-order language of arithmetic which is capable of uniquely identifying such a number. There are a countable number of arithmetically definable reals and an uncountable number of reals overall. A number is analytical if it is definable by a formula in the second-order language of arithmetic. Again, almost all reals are not analytical, because there are only a countable number of second-order arithmetic formulas. Likewise, most real numbers are not definable in the language of ZFC.

Humans and human minds are finite (or finitistic) things; presumably most real numbers are not "humanly definable", or perhaps even "physically definable". It may be the case that most real numbers cannot be individually picked out, named, or specifically described in any way given the constraints of the physical world. This does not mean that nothing can be said about them: we can still confidently conclude that an undefinable number greater than 6 is also greater than 3. Roughly, they can only be spoken about in generalities, with statements that apply to infinitely many of them at once. It is impossibly to even conceptualize any one of them specifically.

I am puzzling over two things right now:

Are there truths which are true of individual undefinable reals? These truths could never be stated or even thought, and almost by definition they could not have any bearing on the real world, but are they "there"? It seems like there should be unique truths about undefinable reals; for any undefinable real r, surely x=r (free in x) is uniquely true for r. But maybe this is a cheat, maybe there is no well-defined predicate "x=r" for undefinable real r. If you do believe there is such a predicate, I am tempted to ask: what does it mean? Of course by definition no answer can even in principle be formulated.

Do undefinable reals even have independent existence? I mean, in set theory they arguably don't: for undefinable real r and s, the statement "r ≠ s" does not correspond to any valid sentence in the language of ZFC. We know "from the outside" that they are distinct, but... do we? We can say tautologically "distinct undefinable reals are distinct", but surely general truths should in some way just be families of specific truths. Like "all dogs are smaller than the moon" is true because each dog individually is smaller than the moon. But we cannot individually say that any two undefinable reals are different from each other, or in fact individually say anything about them.

All this assumes that the universe, or at least human experience, is in some sense "finitistic" and therefore that most reals are in fact undefinable to us.

[algebraic-dumbass]

✨✨✨hey, first glitter guy here:

suppose you were to store all the pieces of glitter in a warehouse with infinite storage, so that the warehouse was full and there was no space left. if you later found another piece ✨ under the couch (as happens with glitter), could this glitter be fit into the warehouse?

✨✨do we consider the set of all pieces of glitter to be countably or uncountably infinite?✨✨

In mean, in both cases, you can fit the piece of glitter into the warehouse though, any infinite cardinal + 1 is the same infinite cardinal

since it is possible for me to isolate only one individual piece of glitter i will say glitter is countably infinite. though too much glitter in a continuous space tends to get dense in it much like Q is in R

[sigmaleph]

Noun sleep (countable and uncountable, plural sleeps)

[...]

(uncountable) Rheum, crusty or gummy discharge found in the corner of the eyes after waking, whether real or a figurative objectification of sleep (in the sense of reduced consciousness). 1886, Peter Christen Asbjørnsen, translated by H.L. Brækstad, Folk and Fairy Tales, page 233: When she had rubbed the sleep out of her eyes and wept till she was tired, she set out on her way and walked for many, many a day, till she at last came to a big mountain. 1980, “Daydream Believer”‎ performed by Anne Murray: But it rings And we rise, Wipe the sleep out of our eyes […]

That is what rubbing the sleep out of one's eyes means?

[sentient-sunset]

Currently reading up on gender identities.

Too many people have assumed that I am non-binary recently for me not to want to check if they're on to something.

So far, I have not found anything that seems like it will fit better than cisgender, however, I have seen the word subset enough times that I'm now wondering mathematical properties of gender, set theory stuff mostly. Is gender a group? A Ring? A Field? Is it infinite? Countably infinite or uncountably infinite? Would I be having these thoughts if I hadn't just finished a Group Theory assignment?

[khaosophist]

We've had some updates...

X= (A,B,C)

X1= (A,∅,B,∅,C,∅)

Change in entropy, A, becomes 'impossible'

X2 =(∅,∅,B,∅,C,∅)

If this is true, then we need a way that ∅ entropy points collapse! Else All dead universes have 'N' ∅ entropy points, which means we have an amount of ∅ points which, counted without discrimination, mean that after KP, with ∅ points, end not with one possibility, '∅', but 'N'∅ possibilities, which would make the death of a universe entropically indistinguishable from KP. The only way to distinguish KP from K∅ is their cardinality. If this is true

Which means entropically, without a ∅ collapse rule, KP = K∅.

So, we had the idea that every material point is either placed, positioned or in orbit, and all that. This means there can be rules to the order of points, rather than just their cardinality; Transfinite rules; we also have Trans-ordinal rules.

Since, we have a log of A, we can tell that the FIRST 'N', A, became ∅ in the next iteration.

X1 =(A,∅,B,∅,C,∅)

X2 = (∅,∅,B,∅,C,∅)

The order did not change, but the cardinality did.

Again, entropically, X1=X2,

So we establish a rule, that any ∅ which then follows another ∅ collapse.

X3= (∅,B,∅,C,∅)

Why? Well...Hrrrm...if not, then every universe is entropically the same. They are infinitely impossible on their death S = (∅,∅,∅...). I'd say it's more neat that nothing collapses when next to nothing...The materiality is the difference...but...either entropy is a force...or no measure of entropy can differ from KP once a universe starts to die...making KP insertion uncertain, we could Inject ourselves in any point in time...including the one where every possibility is dead...we want to avoid that...

Anyways, this would also mean that we can distinguish an aborted universe [∅] from a dead one [∅,∅,∅...], also from an immaterial one [], a birthed one [A], and a K one [A,B,C...N], KP no longer exists since it is equivalent to K∅.

We can use Trans-ordinal rules to demonstrate an asymmetry of entropy.

Y= (∅,A,∅ B,∅,C,∅)

Z= (A,∅,B,∅,C,∅,D)

Tree thing:

(A,∅,B,∅,C,∅,D)

(A,∅,B,∅,C)

(B,∅,C)

(∅,C)

(C).

Vs

(∅)

(∅,A)

(∅,A,∅)

(∅,A,∅,B,∅)

(∅,A,∅,B,∅,C,∅)...

But, this also means there can be symmetry from the order of the set.

(∅,A)

(∅,B)

OR

(B,∅)

(A,∅)

...

Problems aside...the recognition of ∅ points can give a kind of mosaic effect, where KP is now [∅,A,∅,B,∅,C,∅...] Rather than [A,B,C,D...] Doubling our countable/uncountable base...and adding another layer of security. Then, the death of a universe is seen in the collapsing of possibilities into ∅, and with the Trans-ordinal rule, Make them collapse into a singleton, and showing the decay of entropy, until it Reaches [∅] However...this means entropically, a dead universe is also an aborted universe...(???)

Buuut.

What about meta-entropy?

We have multiple ∅ universes next to each other...no?

S= [A(∅),B(∅)...N(∅)]

Although the entropic measure of ∅ is 1 for a K-bit, we have the rule of collapse...would these dead universes on a K-curve collapse as well? If so, then all K-curves share the same end...differing only in the permutations that lead to it. In other words, the best place to build a sanctuary is where both meta, and classical entropy dies, that way, if anyone does an 'oopsie' the worst that can happen is that the bartender at the end of all universes ribs you. Thanks Douglas Adams.

Meh, why not third thing?

G=[A,∅,()]

So we have points which are neither possible nor impossible...but is still a possibility...entropically we would think this;

G ≈ [1(1),1(0),0]

The maximal value of {1(1)} is the KP of N^KP...and the maximal value of {1(0)} is N^{1(0)}...the ∅ singleton of both meta-entropy, and Closed-entropy. While 0...is...well...NULL, an open K-gate...by definition, any value in there is NOT entropic, else it would be either 1(0), or 1(1)...This is how we fetch our identifiers through Non-Linear K-Gates.

Materiality is countability?...No, it's more that anything material clearly has a countable limit of KP...

*FLIPS TABLE*" THE RESTAURANT PRINCIPLE!"

"The...restaurant principle?"

" Basically, the material arrangements and permutations of every universe from N→KP, to KP→∅ are all a transistor for computing. From the fact that in possibility A the fork in that restaurant is blue, and that in possibility B, it is green, then it represents a computational gate that is Unique to the N-th degree. Like a mixed quantum packet logged by entropy...maybe?

Anyways...

WEEEEEEE

[the-least-undefinable-ordinal]

What is the Aronszajn line? Can you describe it to someone who knows calculus and a wee little bit of set theory?

I was planning to make a post about ordinals to ease into talking about the Aronszajn line, but I think I can manage a quick explanation.

So, in a previous post (hope the link works) I talked about Suslin lines, which are dense complete total orders with no endpoints, aren't separable and follow the ccc (countable chain condition), note these can't be order isomorphic to the real line, I also [think I] mentioned the existence of these lines is independent from ZFC.

The Aronszajn lines are esentially weakened forms of Suslin lines, they are dense complete total orders with no endpoints, aren't separable and have no uncountable well orderings embedded into them (that last condition is the one that could be best explained with ordinals, coming soon!); it can be proved that all Suslin lines (if they exist) are Aronszajn lines, the cool thing about Aronszajn lines is you can actually construct them within ZFC!

The minuatiae of their construction are part of my thesis so I don't have all the kinks ironed out yet, I'll try to give a more thorough explanation after I post a brief introduction to ordinals (they're just too important to set theory not to talk about them!).

Feel free to ask any follow up questions!

(Can't guarantee I'll be able to answer them though)

[omegaphilosophia]

The Philosophy of Infinity

The concept of infinity has fascinated philosophers, mathematicians, and theologians for centuries. In philosophy, infinity refers to something that is unbounded or limitless. It challenges our understanding of the world, pushing the boundaries of human cognition and logical reasoning. Here's an exploration of the philosophy of infinity:

1. Mathematical Infinity

In mathematics, infinity is used in various contexts, such as the idea of an endless sequence of numbers, the concept of limits, and the notion of infinite sets. The work of Georg Cantor in the 19th century revolutionized the understanding of infinity by showing that there are different sizes of infinity, known as cardinalities. For instance, the set of all natural numbers is countably infinite, while the set of all real numbers is uncountably infinite, meaning there are more real numbers than natural numbers.

2. Metaphysical Infinity

Metaphysically, infinity often refers to the nature of the universe, God, or other ultimate realities. Philosophers like Aristotle and Aquinas discussed infinity in relation to the nature of the divine. Aristotle distinguished between potential infinity (a process that could continue indefinitely) and actual infinity (a completed state of infinity). Thomas Aquinas later used the concept of infinity to argue for the existence of an infinite being, God.

3. Epistemological Considerations

Infinity also raises epistemological questions about the limits of human knowledge. Can we truly comprehend infinity, or is it beyond the grasp of finite minds? Some philosophers argue that infinity is a useful theoretical construct but cannot be fully understood or visualized. Others suggest that our understanding of infinity is shaped by our cognitive and perceptual limitations.

4. Ethical Implications

The concept of infinity has ethical implications, particularly in the context of moral philosophy. For instance, the idea of infinite value or worth can influence discussions about human dignity and the sanctity of life. If each human life is considered to have infinite value, it could have profound implications for ethical decision-making and the prioritization of resources.

5. Infinity in Cosmology

In cosmology, infinity pertains to the nature of the universe. Is the universe finite or infinite in extent and duration? The concept of the infinite universe has been a topic of debate among philosophers and scientists alike. Some theories in modern physics, such as the multiverse hypothesis, suggest that there could be an infinite number of universes.

6. Infinity in Literature and Art

Infinity also appears in literature and art, often symbolizing the sublime, the eternal, or the boundless. Artists and writers use infinity to evoke a sense of wonder and to explore themes of eternity, the divine, and the limits of human experience.

The philosophy of infinity is a rich and complex field that intersects with many areas of human thought, from mathematics and metaphysics to ethics and cosmology. Infinity challenges us to think beyond the finite and to grapple with concepts that stretch the limits of our understanding. Whether in the precise language of mathematics or the profound reflections of metaphysics, infinity continues to inspire and perplex, pushing us to explore the boundaries of knowledge and existence.

[greetings-inferiors]

I absolutely adore that the set of rationals are countable infinite but the set of irrationals are uncountable infinite, so there are many more irrationals than rationals, but between any two irrationals there are infinitely many rationals. It’s one of those facts that are so absurd that you laugh because you don’t know any other way to process it

[pyrosomatic-metamorphosis]

SHAPE HELLO IT IS ME!!!! CALLI CUBITOZINC!!!!!!! So erm. gonna be honest dude i fought with myself in my head so hard on what i wanted to send for this especially cause i didn't want it to be silly or stupid but you know what. Who give a shit. and i have been there when it comes to having to fight for writing motivation... boy have i ever. SO!! feel free to run with this prompt if you get any ideas from it but absolutely no pressure :]

"Jaiden spent uncountable moments these days thinking about what kinds of things she could've done in the past."

HI CALLI HII BELOVED MUTUAL CALLIIII . i have never written a jaidn before but for you.... i will try.

Jaiden spent uncountable moments these days thinking about what kinds of things she could've done in the past.

Or- maybe they weren't uncountable? They were probably countable, if she'd cared enough to count, but it hadn't occurred to her to count them when she started, and so she'd lost count of them from the beginning, so. Uncountable.

...There's something almost sad about that. Losing track of things since the beginning, no hope of catching up to the present. No chance to right the now because of the wrongs of the past. It makes her heart ache in her chest to think about, but she can't- she can't stop thinking about it. She keeps poking at it, like a bruise, but a sideways bruise. Look at these flowers, they look just like the ones in Bobby Fields. Drink this tea, it's just like the tea Cucurucho gives her. Look at Roier, smiling at Cellbit, and then quietly leave them to be happy. It's...

One thing that's different, from the then to the now, is how many houses she has. One for each piece of her heart, all of them missing something. Does the party house count as one of hers, too? What about the room she stayed in while helping the Cucuruchos? She wants to say yes. It hurts to think yes. If she said yes, if she went there to call for him, would he still leave her waiting?

...She knows the answer. Part of her wishes that she didn't. More of her knows that it's better that she does. It's like playing pretend, and you need to know when you're playing pretend. She knows how often she plays pretend, and how often she pretends that she isn't, and she remembers the time that she didn't have to and she wants that back so badly that it hurts.

Like a bruise. Poke, poke, poke. She knows she's being manipulated, and still she leans into it. Is it helping? Who is she helping? If Cucurucho is hurting, can she really help him? Can she free him? Is it worth it? Is it about worth, or is it about waste? Is it about anything?

Jaiden groans and throws her arm over her face. Maybe it's about laying in the warm pink sand on Hot Girl Beach while Mouse and Foolish chatter behind her. Maybe it's about leaving her empty houses and keeping her hair dyed Miku-blue and doing what she can, because she can. Maybe it's about tailing along with Foolish whenever he gets more silly Cucurucho tasks. Maybe it's about- whatever else is fun. Whatever else keeps her occupied, and out of bed, and waving goodnight to the sun.

There, is that another one of her uncountable moments? No, probably not, she hadn't done much thinking about the past. She could do that. She could think about Bobby and his raccoons, and Bobby and flowers, and Bobby and the very first painting he made for her.

Like a bruise. Poke, poke, poke. She's almost afraid of finding out what will happen if she lets it heal.

...That's enough angst for the day. The sun will be setting soon, and she wants to say goodnight to Bobby with a genuine smile. She pulls her arm off her face and sits up to turn to her friends, join in on their fun.

Her houses are empty, but she knows how to pretend that she has a home.

[resonant-singularity]

a number so large it cannot have a name i dont think is a sensible concept. bc if it is an integer (which it would have to be for its distance from 0 to be relevant to how complex it is), then given that the set of all integers is countable, it has a name (its distance from 0, and sign). that name may be larger than the universe itself. but it is a finite name and thus it is valid.

if it is not an integer and is instead a transcendental number, its unknowability gets a lot better (and is now untethered to its size), but it can still be defined by its unknowability. see chaitins constant as an example. the uncomputable numbers are the closest thing we have to this concept, but they too can be defined. you can lay out the criteria a number would have to fulfill in order to be in this set.

though... i guess if you had an uncountably infinite set of numbers, and you picked a number out of that set at random. you could not describe the number you picked to anyone. this is probably the closest to unknowable that we can get. size has nothing to do with it though, its just gotta be transcendental, uncomputable, and uninteresting

as such, such a number cannot be uniquely identified, and therefore, not named.

so i guess such a number does exist, but not because it is large

this post is meant to be a reblog to a recent post by @cryptotheism but it turns out you cant copy and paste an entire post on mobile into a reblog for some reason so yippee.

i also said a few things on the subject earlier but i think they were wrong