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bubbloquacious · 1 year

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thoughts on the axiom of choice

sub question: countable choice or FULL GENERALITY

:3€

There's a quote that I'm sure many of you have heard that goes "the axiom of choice is obviously true, the well-ordering principle obviously false, and who knows about Zorn's lemma". It sums up my feelings about the axiom pretty well! In general, I think it's a bit silly to say that a mathematical object 'exists' if you can't explicitly point to it. I'm hesitant to say, for example, that the real numbers are well-orderable, or that there exists a Hamel basis of the reals over the rationals.

But to be honest, my issue isn't even really with the proper, full axiom of choice (concerning collections of arbitrary cardinalities), but rather the principle of finite choice: the statement that given a finite collection of non-empty sets we can form a set that contains exactly one element from each of the sets in the collection. This is not actually an axiom of ZF set theory; it follows instead from the underlying logical framework. The axiom of choice is taken as 'obviously true' by analogy to the principle of finite choice! The issue is that even finite choice is non-constructive in nature; it 'produces data' from nothing. Let's reduce to a single set. We assume that this set is not equal to the empty set ('non-empty'). How do we now get an element of this set? Well, we know it's non-empty, so just pick one of its elements! We know it has at least one by assumption. In general, though, this does not tell us which element we have. This is exactly the issue with the full axiom of choice (or countable choice lol)! It's non-constructive.

This is why constructive set theory distinguishes between a set that's 'non-empty', which is taken to mean that assuming the set is empty leads to a contradiction, and a set that's 'inhabited', which means that we can construct an element of the set. The axiom of choice (finite or infinite) is valid for collections of inhabited sets, but not for non-empty sets (compare this to the fact that, even in ZF, it is provable that the category of sets has all small products, even though AC is equivalent to the statement that any set-sized collection of sets has a non-empty Cartesian product). I'm not all that attached to the framework of intuitionistic logic that's employed for constructive mathematics (I'm currently in the process of being dialetheism-pilled by @apovivdic), but I think this is a good perspective on the axiom!

This whole thing makes the axiom of choice kind of a muddy concept, I think. It looks different from different perspectives. Of the three famous equivalent statements, I prefer giving Zorn's lemma the ontological primacy of being chosen as an axiom. It's a bit more esoteric to get used to, but I think that conceptually it's got the most power behind it, and is most closely attached to the idea of taking a process that you can continue as far as you want finitely (such as picking linearly independent vectors in a vector space, or partial choice functions, or well-orders of subsets) and then declaring 'do this infinitely' and you'll get the 'end product' of the iteration process.

#math #thank you for the ask owo #though it is really interesting how countable choice is sufficient to prove most of the non-constructive results in analysis #though i do prefer dependent choice for that #she at least attempts to do something different with the whole thing

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b00tyyyshker9000 · 3 months

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OH MY GOODNESS😨

AHHHH! FAMOUS-AHHH ONE OF MY FAVORITE ROTTMNT ARTISTS JUST-*Happy stims* MY FIRST FOLLOW!?! *Kicks legs happily while parents stare at me like I’m insane* THANK YOU!🥹🥹🥹🥹

YOU HAVE NO IDEA HOW HAPPY I AM!!!!

@ghosty-0w0

#rottmnt #rise of the tmnt #rise tmnt #rise of the teenage mutant ninja turtles #ask me things #tmnt 2018 #random #crappost #rottmnt fandom #rottmnt future leo #future leo #thank you #@ghosty-owo #AHHHHHHH

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ikiprian · 2 months

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what's ride or die au in yuour wips pinned post? owo

Ride or Die AU is my baby! I'm working on and off, trying to get the first three chapters down and edited before I post 'em.

It's a canon-divergent AU where some months before the Batmobile parks in Crime Alley, Jason steals the tires off the Drake family car. Mid-theft, a nine-year-old Tim pops his head up from the backseat! What should be an adorable first meeting between unlikely friends turns for the worse-- the Drakes had been in the Alley for a reason. The reason catches up to them.

Now it's Tiny Tim and Baby Jay, two against the world! Or against Batman, CPS, the GCPD, and whatever happened to the Drakes, more accurately. Thrown together, their relationship goes from complete strangers to blood brothers quick as a flash, thus: Ride or Die!

I love to overshare but I'd hate to spoil the whole thing so I'll stop here, haha. But I can post WIP snippets if anyone is interested!

#kipanswer #anon #ride or die au #thanks for the ask friend!!#owo #are you. like is the owo at the end your anon tag or are you just owo-ing? i'll tag owo anyway you can be owo if you want #tim drake #jason todd

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pink-november · 4 months

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Not incredibly high effort, but I was thinking about him today

HOLY SHIT STUBBORN???????????

Asterion you are giving me brainrot... literally had to pause and stare at this blinking because you inflicted me with Stubborn feels 🥺

#slay the princess #kisses you kisses you mwah mwah #thank you for this my sweet dove im gonna devour it :)))#always so happy when you send me asks owo #novem asks

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tricitymonsters · 3 months

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*Scrubs catnip onto my own skin like some expensive perfume.

mori after smearing his face all over you for 15 solid minutes

#direct route to him being like OWO WHATS THIS #thank you for asking #mori

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sketchingstars03 · 5 months

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Special christmas delivery from keuwi!

AGAHHSHSHSGAHHSHSBS OH MY GOODNESSSSSS

THANK YOU SO MUCH SHE’S SO CUTEE!!!!

Merry Christmas and Happy Holidays to you too Keuwi!!!!!!! <3 <3 <3

#mutual ask #christmas #merry christmas #splatterutmv #splattertale #fanart!?owo?!#I love this so much!!!!#Thank you again!!#cute lil opal in the corner <3

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theelvishfiddler · 8 months

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Hello!!! I'm here!(again, haha) I'm wishing you all the happiness!! You deserve to be so happy and to feel your best!! Thank you for being alive and so very kind!! I hope you have a wonderful day/evening/night!!! Please remember to drink water and eat!! You bring me joy and give me inspiration!! You make me happy by being you!!

(I am sorry for the spam!! I get hyper and just want to love people /p!!! Thank you for bearing with me and being wonderful!!)

whoops sorry didn't see the first one! This is very sweet, thank you so much! I hope the same for you (o•ᴗ•o)ﾉ♡

#asks #pls take care!#thank you for the good vibes #and the reminder to hydrate #i forgot #water ( owo)b #i am very tired

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pixthepixel · 4 months

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What kind of buffs do you get for the year of the dragon?

I PROBABLY GET VERY BUFF AND WILL THROW CARS AT BUILDINGS!

I also get an EXP boost!

#answer #Been a while since I got an ask!#Thank you!#You made my year owo #New year #Hehehehe

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ragnarokhound · 6 months

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“Tell me to stay and I will be here for as long as you’ll have me.” for the h/c prompts? 💞

Thank you for the prompts!!! I was going to respond to both of them today but uh. This one got a little out of hand lmao

I hope you enjoy!!

9. “Tell me to stay and I will be here for as long as you’ll have me.”

Jaytim, warnings for: Thoughts of self-harm, canon-typical violence, panic attacks/triggers w/heavy emphasis on breath, Jason being a defensive, self-deprecating asshole <3

---

Jason doesn’t like to think of himself as a couple cards short of a full deck, but some nights? Are harder than others.

It’s so stupid. It’s always so stupid. Half the time it’s not even a trigger— and Jason knows his triggers, intimately. Has them filed away in a corner of his mind so that when the glass descends and his hands stop feeling like they belong to him, he can think to himself, oh, it was the beep of that guy’s wristwatch that set him off. That’s why he needs to be outside right now. That’s why he wants to punch brick until his knuckles are red and white and mangled.

Take tonight for instance.

He was doing his rounds. In between cases and expecting a slow night, when he found some guy getting worked over by a couple of mobsters. This would be easy, he thought. I eat chumps like these for breakfast, he thought. They aren’t even packing.

But a few meaty thwacks in the exact wrong rhythm, followed up by the poor sucker screaming in the exact wrong tone, and he’s left playing fifty-two-card pickup with only jokers to substitute.

Ha. Ha. Ha.

Bloody shins aren’t as satisfying as busted braincaps no matter what Tim and the other bats say, but relationships are about compromise. He puts the guys in cheap suits on the ground and calls them an ambulance.

Their hapless victim gets a talking to about when it’s smart to borrow from loan sharks (never), directions to Leslie’s clinic (yeah, no, you’re gonna wanna take 15th—), what to say when the Red Hood personally intervenes on your behalf (thank you, strange man, bye), and Jason makes it to a safehouse without once giving in to the burn of the goading green fire in his blood.

See? Compromise.

Now he’s on the couch with blood on his bare knuckles, and the shards of glass in the bathroom are all that remain of his mirror. He sits and tries to breathe, but his brain is full of static; shuffling and shuffling and shuffling the deck but there’s no one to tell him when to stop. He’s not really sure how much time has passed when he hears the quiet whir of his security system shutting down.

There’s a gun in his hand with the safety off, pointed at Tim before the window’s even open.

Tim’s voice is loud in the quiet room, casual and light like the liar he is, but to Jason it still sounds muffled. Boxed up. Buried.

He takes another breath.

“Bad time?”

“Fuck off,” Jason says, just as faux-casual, because that’s what they do.

Only it wheezes out of him instead, thready and weak. The gun shakes in his hand. Tim pauses on the window ledge.

Jason can practically hear the gears whirring as he reassesses the scene: Jason’s bloody knuckles and the distant sound of water still running in the bathroom sink. Haphazard parts of his uniform shucked off almost at random, the helmet nowhere in sight, but his top missing and the holster straps hanging off his waist.

Air whistles in and out of his tight chest in shallow, streaking streams that get closer and closer together the longer he looks at Red Robin’s expressionless white eye sockets. He knows what Tim sees.

The silence weighs on him. It pisses him off.

The only thing worse than being a scattered mess is having a witness. The only thing worse than having a witness is if that witness is Tim. They haven’t been fucking that long, but it’s been long enough that Tim probably came over tonight with more than one item on his to-do list.

“Not tonight, princess,” Jason bites out when Tim still hasn’t said anything. “Daddy’s tired.”

Tim steps inside.

The gunshot rings sharp and stinging in the air before he even realizes he’s pulled the trigger. The smoking hole on the floor by Tim’s foot is the only warning Jason feels capable of giving right now.

“Did I—did I fucking stutter?” He tries to say, but black spots bloom at the scattered edges of his vision, and he wavers.

One second Tim is frozen with his palms out, showing that they’re empty, and the next he’s easing the gun out of Jason’s hand and letting Jason twist a fist in the front of his costume. He helps Jason sit back down, and Jason doesn’t want to look at his face, to see the pity there.

He gasps, muscles tensing, and under his fingers he catalogs the new mesh Tim is trying out with his suit. It’s probably something with a stupid, long name and the word “polymer” thrown in for added pretension. It sounds like something Tim would do.

“...something happen tonight?” Tim is asking quietly.

It’s not Robin-steady, because he knows Jason well enough to know how bad an idea that would be, but it’s not the levity they fake with each other either. Whatever it is, it’s making Jason itch.

Then he catches Tim taking his pulse and peering into one of his eyes with intense scrutiny, and what he’s said catches up to him.

Jason laughs. It’s an ugly sound.

“Oh, quit it. I wasn’t gassed,” he snaps. “There’s no toxin, there’s nothing wrong with me.”

“That’s not what I—” Tim tries to backtrack but it’s way too late for that weak shit.

“Except for the everything that’s fucking wrong with me,” he snarls. He didn’t know he was going to say that until he'd already said it, bubbling and burning out of his mouth like melting plastic, acrid and noxious— but he feels afloat, drifting; like he’s watching himself from the other room. So it doesn’t matter.

“It’s not going to be fixed, there isn’t any antidote for you to cook up in your cute little lab for this—” he whirls a sharp-fingered hand around his temple, “fucking garbage.”

“Jason,” Tim says, fingers tight and prying at Jason’s grip.

“There’s no cure for being beaten to death, no little step-by-step handbook for kids who don’t fucking stay dead, so you can take your shitty costume and your knock-off batarangs and fake-ass fucking concern—”

“Jason, take a breath for me.”

“You—”

Take a breath? Oh, he’ll take a fucking breath, alright—

He’ll take a breath and then he’ll—

He’ll take a breath and—

He’ll take a—

He takes a fucking breath.

Tim’s face is bare.

His eyes are very blue. It takes Jason approximately three years to realize that the heavy weight on his shoulders is Tim’s cape, unclipped and pooling in their laps.

It takes another five to realize that the heavy weight in his lap is Tim himself, straddling his thighs and breathing steadily under their joined hands. Tim’s heart beats an alternate rhythm, almost three beats for every breath in, almost four for every breath out, and Jason closes his eyes.

“Fuck.”

“Hey,” Tim says.

Jason doesn’t say anything more.

What else is there to say when your kind-of-colleague, kind-of-booty-call, complete-pain-in-the-ass comes over when you’re in the middle of some kind-of-mental break? Fucking ‘sorry’ ?

“Sorry,” he says. “Not exactly fit for company right now.”

Tim smirks. “I noticed.”

“Oh, fuck off.”

“You gonna shoot at me again?”

Jason scowls.

“I fucking might.”

Tim only rolls his eyes. Jason finds it oddly settling. He finds all of it oddly settling; bickering with Tim is normal. Easy. It reminds him of what he’s like now. The full fifty-two.

His fingers are still trembling, but Tim is stroking absent-mindedly over the skin on the back of his hand. He might not be aware he’s still doing it. Tim exudes an air of total comfort, like the only place he wants to be right now is literally babysitting Jason; but Jason knows better. He sighs.

“Listen,” he says, staring at a point past Tim’s ear. “I could have picked a better way to say it, but I’m— I’m really not up for anything tonight—”

Tim stiffens.

“Jason, what the fuck are you talking about?”

“Sex, obviously.”

He watches in real-time as Tim processes that. It’s honestly kind of fascinating, in a terrible, gut-stabbing kind of way. Tim takes a deep breath.

“Dude. I get that you are in an emotionally vulnerable place right now—”

“Hey—”

“—but I’m not that much of an asshole,” Tim says flatly. “You really think I’m gonna bail the second I see that you’re having a bad night? Or that I’m only gonna stick around if you put out after you’re done having a panic attack? Very sexy. Fuck you very much,” he finishes, hissing and affronted.

Jason stares at him.

“You can’t tell me this is how you wanted to be spending your night,” he says, but his hands creak convulsively around Tim’s, as if to hold him in place. The traitors.

“You don’t know how I want to spend my night,” Tim retorts.

“Oh, come on. I’m not gonna ask you to babysit me and check the closet for monsters before I go to sleep so I don’t have wittle nightmares—” Jason scoffs.

“So don’t ask,” Tim says, eyes burning; Jason’s breath catches. “Tell me to stay. I’ll be here.”

“You’ll be here,” he says skeptically. But Tim’s hands squeeze around his, so hard and fast it hurts. The pain is grounding.

“As long as you’ll have me.”

He wants to scoff again. It’s the kind of saccharine after-school-special corny bullshit that he always hated coming from Bruce.

But his grip is just as tight on Tim, their bones rubbing together through the barrier of their skin. The thought of letting go makes his stomach swoop, like the moments just after stepping off a plane with no chute. That window between stepping and being caught; when his heart still needs convincing that someone else will save him.

(It’s the kind of earnest, rock-solid sincerity he always craved from Bruce; the kind he always ate up, hook, line, and sinker.)

He swallows his pride. His dry throat clicks.

Fine. Stay. If you’re so fucking eager, is what he tries to say.

But all that comes out is, “Stay.”

Tim leans forward, and drops his forehead onto Jason’s bare shoulder. Jason’s shaky breath stirs the fine black hairs beneath his nose, the mild scent of his shampoo wafting gently into Jason’s nose. After an uncertain beat, he lets his cheek fall. Then the rest of him. They stack against each other, two cards forming the foundations of a house.

When Jason shakes apart, Tim holds steady.

#thank you tauria!! <33 you picked out some juicy ones OwO #jaytim #my writing #ladytauria #asked and answered #prompt game #*pats jason on the back* sorry bud I had to do it to em #i wrote the last paragraph first and the One metaphor at the end prompted me to write another thousand words OOPS #i'll collect these asks and post them to Ao3 for easy reading when I'm done with them lol

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justaregularken · 3 months

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Tulip and orchid???

Favorite number/Lucky number?- 27. Always has been.

Favorite fruit?- That’s actually kinda tough.. I’d say mangoes or strawberries.

#ken things #shitpost #ken asks #anon ask #owo #finally someone asked me about that goddamn flower post thank you

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procyo9 · 1 year

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I didn't know about Crystal and Elegy!!! Could you tell more about them? How they met? Anything else you wanna share?

OMG yeah sure!! I'm happy to talk about them anytime <3

We (me and @quaddmgd) decided we want them to be officially together recently and yeah!! They lesbian together lmao

Though I'm not sure when we'll be able to take pics of them together (npv is a complicated process :') but I'm sure we can do it!! Heather is a smart girl after all uwu)

So… It might be a bit boring but they met at the Afterlife bar! Although Elegy was crushing on C a lot earlier than that. One day both her and Crystal needed help with something, and Elegy offered her netrunning abilities in exchange for Crystal's in-field experience; I'm actually in the middle of writing a fic about that!! But I'm not sure when I'll be able to finish it (and I haven't written a real fic in years so it's a bit difficult :P)

We called the ship "Nothing's Gonna Hurt You Baby" because that's one of our favourite songs and the lyrics are so soft and match the vibe just right <3 Both Elegy and Crystal went through a lot in life and both need someone that's their little safespace they can turn to anytime and just be happy :3

We're still developing them so their relationship is a constant work in progress, but right now they're #1 thing on my mind and I keep putting new screenshot ideas in my notes app; it might still take a little while for them to appear together in the same photo but!! We also plan on doing some screenshots separately <3 I even have some drafted already (but I'm too shy to post them.. buut maybe someday this week) :D

Thank you for asking really!! I love yapping about blorbos from my games <3 I hope this is at least somewhat interesting haha

#.ask #this is so nice fjhdskjfhalhfjk i love talking about this stuff!! aaaaa thanks yet again owo #otp: nothing's gonna hurt you baby

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leaphia · 9 months

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uh uh uh OC questions!! hehehe yesss >:] For Sookie 30. Who do they most regret meeting? For Aaron 21. Why do they get up in the morning? For Noah 27. What causes them to feel dread? For Tristan 28. Would they prefer a lie over an unpleasant truth? And for Astrid 34. How hard is it for them to shake a sense of guilt? And for you of course (and any or all of them) H) What trait of theirs do you admire most? :>

Waaaaah thank you so much for all the questions! ;v; <3

This will be quite long omg

Sookie - Who do they most regret meeting?

Aside from like half her classmates back in high school, who were bullying her for being "the weird comic book geek" I'd say… Aaron. He's not only the type of villain who embodies everything she hates (arrogance, possessiveness, narcissism, etc.) after their first encounter he wouldn't leave her alone anymore. He's planning any kind of evil schemes to lure her into a trap, to catch her attention and to eventually "make her his", as he developed a crush on her. And Sookie gets incredibly mad and annoyed by him, as the only one she really loves is Noah…

Aaron - Why do they get up in the morning?

There was a time when the only thing that would get him out of bed were his inventions and robots, he could tinker on. Now he has other reasons like his cat Giselle, who he wants to care for. (He loves her very much <3) And (as weirdly as this sounds looking at the previous question) another reason would be Shelter/Sookie and their encounters. It's the only thing that really excites him and lets him forget about his "boring" life. (As already mentioned, he has a crush on her and since he never felt like this to anyone before he wants to... "explore" these feelings.)

Noah - What causes them to feel dread?

There isn't much that Noah is scared of… except Spiders. If he has to fight a bunch of giant spiders I guess he would die of fear immediately xD But what would cause him to feel real dread is if Sookie or any other of his loved ones (his mother, his best friend) would be in danger, something that could threaten their life and he wouldn't be able to save them.

Tristan - Would they prefer a lie over an unpleasant truth?

No. He hates if somebody lies to him. And he rather wants to know the cruel reality than being under an illusion. (That doesn't mean that he hasn't lied himself before, he actually did lie to others if it was necessary xD)

Astrid - How hard is it for them to shake a sense of guilt?

It's really really hard for her. If it's something grave she feels guilty for, she would lie awake at night, thinking about it and how she could make up for it. (the first night after she started working together with Tristan - a criminal in her eyes back then - she didn't sleep a wink)

What trait of theirs do you admire most?

Sookie - her kindness and will to help. She tries to see good in everyone. :>

Noah - his sarcasm xD And I think his tolerance and acceptance towards any kind of people? He puts others needs first instead of his own, for example he also accepts Sookie's boundaries and loves her for who she is <3

Aaron - His confidence. Even though some family issues and other bad sh*t that happened to him, he knows that he's of worth.

Tristan - Everything. Uhhh I think his patience. He can stay quite calm in stressful situations most of the time. ovo (Which is funny because contrary to his fire-magic-powers, he's not impatient or aggressive xD)

Astrid - Her determination. When she sets her mind on doing something, there's nothing that can stop her.

[OC Questions from here :>]

#ask game #hjdgrs thank you again for your interest! ;w;#Me while typing this: suddenly forgets any information about my OCs ever... why am I like this x'D #oc: sookie cantrell #oc: noah hale #oc: aaron sinclair #oc: tristan #oc: astrid #I feel like I overshared a bit. But what if someone needs to know this??#(sneaking a drawing of Shelter&Aaron in there which I never shared on here owo")#I would have loved to doodle something for every answer but that would be a lot xD #also look at me gushing about Sookie and Noah I'm sorry but I already love these beans so much

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