#continuum hypothesis

[facts-i-just-made-up]

Any interesting facts about the continuum hypothesis ?

There are no interesting facts about the continuum hypothesis. This fact is, however, interesting, so the only interesting fact about the hypothesis is untrue, thus it is true, and interesting, so it's not.

[omegaphilosophia]

The Philosophy of Set Theory

The philosophy of set theory explores the foundational aspects of set theory, a branch of mathematical logic that deals with the concept of a "set," which is essentially a collection of distinct objects, considered as an object in its own right. Set theory forms the basis for much of modern mathematics and has significant implications for logic, philosophy, and the foundations of mathematics.

Key Concepts in the Philosophy of Set Theory:

Definition of Set Theory:

Basic Concepts: Set theory studies sets, which are collections of objects, called elements or members. These objects can be anything—numbers, symbols, other sets, etc. A set is usually denoted by curly brackets, such as {a, b, c}, where "a," "b," and "c" are elements of the set.

Types of Sets: Sets can be finite, with a limited number of elements, or infinite. They can also be empty (the empty set, denoted by ∅), or they can contain other sets as elements (e.g., {{a}, {b, c}}).

Philosophical Foundations:

Naive vs. Axiomatic Set Theory:

Naive Set Theory: In its original form, set theory was developed naively, where sets were treated intuitively without strict formalization. However, this led to paradoxes, such as Russell's paradox, where the set of all sets that do not contain themselves both must and must not contain itself.

Axiomatic Set Theory: In response to these paradoxes, mathematicians developed axiomatic set theory, notably the Zermelo-Fraenkel set theory (ZF) and Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). These formal systems use a set of axioms to avoid paradoxes and provide a rigorous foundation for set theory.

Set Theory and the Foundations of Mathematics:

Role in Mathematics: Set theory serves as the foundational framework for nearly all of modern mathematics. Concepts like numbers, functions, and spaces are all defined in terms of sets, making set theory the language in which most of mathematics is expressed.

Mathematical Platonism: The philosophy of set theory often intersects with debates in mathematical Platonism, which posits that mathematical objects, including sets, exist independently of human thought. Set theory, from this perspective, uncovers truths about a realm of abstract entities.

Philosophical Issues and Paradoxes:

Russell's Paradox: This paradox highlights the problems of naive set theory by considering the set of all sets that do not contain themselves. If such a set exists, it both must and must not contain itself, leading to a contradiction. This paradox motivated the development of axiomatic systems.

Continuum Hypothesis: One of the most famous problems in set theory is the Continuum Hypothesis, which concerns the possible sizes of infinite sets, particularly whether there is a set size between that of the integers and the real numbers. The hypothesis is independent of the ZFC axioms, meaning it can neither be proven nor disproven within this system.

Axioms of Set Theory:

Zermelo-Fraenkel Axioms (ZF): These axioms form the basis of modern set theory, providing a formal foundation that avoids the paradoxes of naive set theory. The axioms include principles like the Axiom of Extensionality (two sets are equal if they have the same elements) and the Axiom of Regularity (no set is a member of itself).

Axiom of Choice (AC): This controversial axiom asserts that for any set of non-empty sets, there exists a function (a choice function) that selects exactly one element from each set. While widely accepted, it has led to some counterintuitive results, like the Banach-Tarski Paradox, which shows that a sphere can be divided and reassembled into two identical spheres.

Infinity in Set Theory:

Finite vs. Infinite Sets: Set theory formally distinguishes between finite and infinite sets. The concept of infinity in set theory is rich and multifaceted, involving various sizes or "cardinalities" of infinite sets.

Cantor’s Theorem: Georg Cantor, the founder of set theory, demonstrated that not all infinities are equal. For example, the set of real numbers (the continuum) has a greater cardinality than the set of natural numbers, even though both are infinite.

Philosophical Debates:

Set-Theoretic Pluralism: Some philosophers advocate for pluralism in set theory, where multiple, possibly conflicting, set theories are considered valid. This contrasts with the traditional view that there is a single, correct set theory.

Constructivism vs. Platonism: In the philosophy of mathematics, constructivists argue that mathematical objects, including sets, only exist insofar as they can be explicitly constructed, while Platonists hold that sets exist independently of our knowledge or constructions.

Applications Beyond Mathematics:

Set Theory in Logic: Set theory is foundational not only to mathematics but also to formal logic, where it provides a framework for understanding and manipulating logical structures.

Philosophy of Language: In philosophy of language, set theory underlies the formal semantics of natural languages, helping to model meaning and reference in precise terms.

The philosophy of set theory is a rich field that explores the foundational principles underlying modern mathematics and logic. It engages with deep philosophical questions about the nature of mathematical objects, the concept of infinity, and the limits of formal systems. Through its rigorous structure, set theory not only provides the bedrock for much of mathematics but also offers insights into the nature of abstraction, existence, and truth in the mathematical realm.

[futurebird]

What if the Library of Babel contained books, not just texts.

I love the Library of Babel-- but it's important to remember that it contains all *texts* (up to a certain length) this is not the same as containing all *books* -- books are objects, they can be different shapes and sizes, made of different materials, bound with different bindings-- books have highlighting and margin notes from previous readers, old lending cards with the names of readers and dates.

Books can have illustrations. But we could make a Babel style indexing system for possible print images. Then represent the images as their index number. Now we have illustrations. We could have codes for size, color, cover material, paper weight, illustrations, margin notes, the style of handwriting of the margin notes. It would be a complex system and the final index numbers would need to be much longer. Possibly impractically long. But, a finite thing can be impractically large.

Here is a related question. How many distinct (as in you can distinguish between them yourself) objects can fit in a breadbox?

(Not all at once. I'm asking how many "different" things are small enough to be placed in a breadbox. This is either permutations or philosophy. )

[mathhombre]

Curious what people think. Will be discussing with students soon.

[floralfractals]

yknow that law thats like. if you want the righy answer on the internet you only need to post the wrong answer

[onlyhurtforaminute]

CONTINUUM-HYPOTHESIS:EVOLUTION

[zyafics]

GIRL UNDER THE MOONLIGHT | Rafe Cameron

MASTERLIST (Oneshot)

Pairing – Rafe x Mermaid!Female Reader

Summary — Rafe invites you out to the Druthers for a sunrise event with Sarah and his friends.

Word Count — 2.3K

Content — fluff, protective!Rafe, Sarah being a good sister (and considerate to you!), you being clingy and possessive of Rafe, and suggestive scenes. A continuum of this and this and this!

“She can’t be a mermaid,” Sarah announces unexpectedly.

Rafe stops what he’s doing to turn to his sister, “What?”

“I poured some water on her skin,” she diligently informs, leaning against the doorframe of his office, her arms crossed over her chest with this vindicated look. "Nothing happened. Therefore, she can’t be a mermaid.”

Rafe scoffs at Sarah’s hypothesis. “What did you do? Chuck it at her?”

“Who do you think I am?” She rolls her eyes. “I just dropped some water on her… accidentally. I even brought towels—just in case.”

“A scientist,” Rafe drawls sarcastically, returning to his work.

“Precisely.”

Rafe had nearly forgotten that little quirk about you. It’s been almost a month since your arrival, and while there have been some occasional odd moments, nothing has proved evident about your supposed mermaid abilities. Finally, Rafe tucked it in the back of his head as nothing more than a phrase—a figment of your imagination, your fantasy transcending into the natural world.

Nothing more.

“Why is this relevant?” Rafe asks stodgily, flipping through the account books of Cameron Development, his fingers trailing the edge of the sheets.

“Because now you can bring her to the sunrise trip,” Sarah declares.

It takes Rafe a second to remember what she’s referring to. A summertime tradition where Sarah and Rafe host their friends on the Druthers, taking it out to sea to stay a night and wake up before sunrise.

Sarah had tested whether you were truly a mermaid to make you a candidate for the journey.

Rafe scoffs, “So that’s what that little experiment is for,”

“I had to,” she smiles sweetly, “Didn’t want her to turn into a fish when we’re out at sea. It’ll ruin the fun.”

“My fun or yours?”

Sarah doesn’t answer, giving him a knowing wink, before departing from his office.

That night, Rafe asks you. He was getting ready for bed, turning off all the lights, before you patter your way into his doorway, shyly inviting yourself into his room. Rafe no longer is surprised by your arrival, and with a wave of a hand, he beckons you forward and you sink in his arms.

You’re always giddily, full of soul, and when Rafe has you in his arms, it amplifies. You detail him about your day—the time spent along the coastline of his estate, traveling barefoot along the empty roads, interacting with land critters. You’re always so fascinated by the mundane, the landscape and sights, but the way you go about it—it’s a soothing sound, full of bursting energy.

He can, and has, fallen asleep to it.

Knowing you’re in a good mood, Rafe decides to pop the question. He tells you about the trip, taking his yacht out, and watching the flaming palette of orange-blue light in the morning sky. He thinks you’ll enjoy it; after all, you’re a self-proclaimed mermaid with a fascination for all human derivatives.

But, for the first time, you say no.

“Why not?” Rafe asks as you lay on his chest, shaking your head at the invitation. Your nails are tracing the fabric of his shirt, drawing doodles in similar manners you would do at the bottom of the ocean floor.

“I don’t want to,” your voice is quiet and tiny as if you don’t like the idea of saying no to him.

“It’s just for one night,” Rafe assures. Perhaps you’ve gotten used to the stability of the Tannyhill estate.

You persist, declining the offer.

“It’ll be fun,” Rafe reasons, but there’s a bitterness in the way he’s pushing the topic. Truthfully, if you don’t attend, Rafe doesn’t have much incentive to join either. Yes, it’s been a long-standing tradition, but he wants to experience it with you. Ever since you entered into his life, he’s been feeling that way.

Yet, he knows he has to go. Sarah doesn’t know how to drive the Druthers, and she’s been looking forward to this all summer. Despite their bickering, he doesn’t want to let her down.

You shake your head quietly, slouching your shoulders inwards, making yourself small. It’s as if your body is physically recoiling at denying Rafe.

He doesn’t know what’s going on. You never do this. You’ve always been pliant, and obedient, agreeing to every little concoction he conspires. It’s one of the many things he adores about you; yet, for the first time, you’re being wayward.

“Are you afraid of the water?” Rafe asks gently, stroking the curve of your spine with his finger, in a way that makes you relax your muscles. He accidentally hooks it underneath the shirt—his shirt—drawing it up to expose your skin; soft, tender, and perfect.

Sarah had been right. Normally, you don’t like wearing clothes. Only when Rafe asks you to whenever you go out together, but preferably, you choose to remain as close to naked as possible. It’s too hot, you told him. You’ve gone years without clothes, and the actual barrier produces heat. The only exception, however, is if you get to wear his.

Again, you don’t answer. Your fingers coils around the loose fabric of his shirt, bundling it into a fist, as if you’re frightened by the suggestion. Rafe sees it—feels it—emulating from your body, and he stops for a second and relinquishes his touch.

“We’re just going to be on the boat. You don’t have to go into the water if you don’t want to,” Rafe reassures, hoping his words soothe something over you. He knows he’s been persistent, but he truly doesn’t want to leave you alone—not even for one night. “I’ll protect you.”

Normally, under that advisement, it would palliate all concerns; and would coaxe you into an affirmative yes. But you say nothing, and finally, with a tick of agitation pulsing through him, Rafe gently grabs your chin and lifts your tender gaze to his.

“Don’t you trust me?”

Your teeth sink into your bottom lip; plumped, fresh, coasted with this perpetual wetness that makes Rafe burn with desire. And you nod your head, listening, but not actively responding.

His thumb traces your lower lip, pulling down the plumpness and forcing it to split apart. Your eyes meet Rafe with a tenderness, almost hunger, while your breathing slightly stills.

You still don’t answer him.

And this time, Rafe decides to let it go.

“If you don’t want to go, you don’t have to.”

This should make you happy, for him to drop it, but the coated disappointment in his tone causes your stomach to twist. You don’t like upsetting him, don’t like the idea that you’re not meeting the standard and his needs.

“But you’ll still go?” You ask softly, gently, like an ocean breeze.

“I have to. Sarah doesn’t know how to drive,”

Your brows pinch, furrowing together. “Will there be other females there?”

“Yeah,” Rafe nods. “Some of Sarah’s girl friends.”

You purse your lips, eyes squinting. You don’t like that. You’re possessive about your mate. You understand Sarah’s his sister, and that company is natural, but with other women? Unrelated to him? It’s wrong.

You can’t stand it.

“Okay,” you murmur softly, conceding in a way that Rafe likes. “I’ll come.”

The next morning, everyone’s at the docks of the Tannyhill estate, loading onto the yacht. Sarah brought a variety of fruits and snacks, while Kelce and Topper helped her and her friends abroad. They climb up the slippery steps and enter into the cockpit, settling with their things.

You stay close to Rafe, timid among the new crowd.

Out at sea, everything is smooth sailing. Today’s a beautiful day, with steady waves, and it’s meant to last the entire week. Rafe parks the Druthers off the coast, where you can’t see Kildare anymore, save for a small coastal cove that’s within view. The boat gently bobs against the rolling tides, and the sounds of Sarah and her friends are screeching with enthusiasm as they take a swim around the yacht.

You watch from above the deck, your focus on the distance, staring at the island cove.

When Rafe slips out of the cockpit, his hand slides over your waist, snapping you out of your concentration. You lift your gaze to meet his, and the furrowed crease between your brows disappears, shoulders relaxing upon his touch.

Rafe offers you a rare, gentle smile. “You wanna swim?”

You shake your head, “Not with them.”

He likes the fact that you don’t entertain his friends, that you want him and only him. “You were waiting for me?”

You nod, leaning your head against his shoulders. “You looked busy,”

“You could’ve told me,” Rafe declares, “Better yet, you could’ve joined me.”

You huff softly, amused, as Rafe pulls you closer to his side. Again, he smells the scent of the sea—but it’s fragrance, exuding from you. His eyes drift to the direction you were looking at, “What's that?”

“Nothing,” you hum, but there’s a pang of longing. You tip your chin skyward to find his gaze once more. “Can we go inside now?”

A couple of hours later, Sarah’s right. Again. The whole crew is having dinner on the main deck, and someone accidentally spills a cup of water on your arm, but nothing happened. Rafe was ready to see something—a twinkle, a glow, or a glimmer—but it was absolute zilch. One of her friends who did it apologizes, and you chuckle softly, wiping it away with a towel, not a care in the world.

He truly doesn’t understand this mermaid business. He really doesn’t.

Maybe you’re someone who loves the sea so much, you claim it as part of your identity. You want to be closer to the ocean, to the marines, to the corals and the sea creatures that the title is merely an expression of self, rather than a true folktale.

You can’t be a mermaid, Rafe reasons, you don’t even have a tail.

Later, everyone shuffles off to their individual cabins. Rafe claimed the biggest one—because of course he did. When you step out of the small bathroom, in nothing but a large shirt of his, Rafe swallows thickly. Because most times, when you come into his room, it’s night, punctured with darkness saved for a glow of moonlight through his curtains.

Now, the cabin lights remain perpetually on, at low brightness, and it allows Rafe to see everything. He’s reminded of the tidbit from Sarah—how you hate panties—and his eyes drop to your thighs, where the shirt casually brushes mid-level, almost revealing more. His heart beats heavily, and you slowly climb onto the bed, wrapping yourself around him.

You fall asleep on his chest, as you normally do, and the weight is like a natural blanket to him. Something he knows, expects, and remembers. It tames all the raging emotions inside of him, silences all the busy thoughts, and hones in completely and only you.

During the duration of the night, while the yacht slowly rocks against the stronger currents, his hand falls on your back protectively.

Until it doesn’t.

Something doesn’t feel right; missing. His eyes slowly blink awake, drowsiness coating his features, while his eyes adjust to the low cabin lights. His hands weaved through thin air.

You’re gone.

With the door of the cabin wide open.

Consciousness strikes Rafe, and he jumps out of bed, rushing out of the cabin, and following the hallway lights to the deck. Slowly, with the rocking of the tides, Rafe climbs up the stairs, to the main deck, and finds you in the stark darkness.

Standing on the ledge.

You’re at the gap where the railing ends, allowing an opening to jump to the swim platform. You’re standing dangerously close to it, his shirt flapping against the wind, a loose hand wrapped on the safety handle.

Rafe calls your name, but you don’t turn around. He suspects you’re sleepwalking, entranced in a dream, that led you up here. Ocean calling you home, it’s evidence for his theory.

But you’re not a mermaid and you can’t survive that leap.

Cautiously, Rafe approaches you, slowly, tenderly, calling your name. He’s afraid of waking you, afraid of startling you from your dream and causing you to release and fall. With each step closer, he hears the thumping of his own heartbeat and the prize within reachable fingertips.

He’s almost there.

He’s so close.

Until you jump.

Rafe screams as he reaches the ledge, his eyes adjusting to the dark currents of the sea. Nothing is visible, not a stream of light underneath, except for the glowing reflection of the full moon bathing the dark waters.

He’s calling your name, again and again, trying to see if you’ll surface to the sound of his voice.

But nothing happens.

Rafe’s already taking off his shoes, taking off his shirt. He’s gearing up to jump in after, especially if you don’t surface within the next minute.

He’s praying. A godless man as himself, who doesn’t believe in a higher power, is begging for you to come up unscathed.

But he still sees nothing.

Until something cuts the waves, a sharp prodding sculpture that slices through the harsh currents.

A tail?

He isn’t sure if his eyes are deceiving him, especially with the drowsiness of his sleep, but he sees another cut in the ocean, this time paired with an iridescent color of a fin, scaly and glimmering.

He calls out your name once more, a little timid, a little frightened.

And you raise to the surface.

Attached to a long, kaleidoscopic tail, with skin full of scales, climbing up your shoulders and throat, you’re flipping through the water; your smile bright, eager, and real.

Rafe breathes out a sigh of disbelief.

“Holy shit,”

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[thursdayg1rl]

what the hell do you mean 'inaccessible cardinal' touch some grass fr. maths silliest subject ever

[geometea]

also ngl some of y’all take your headcanons way too seriously (continuum hypothesis fans im looking at u)

like im not saying you’re wrong but canon isn’t exactly complete and it’s just as consistent that actually it’s not

and yes i know that argument is kinda forced and relies on fuzzy logic BUT that’s not exactly very constructive of some of y’all so take ur L lol

The maths fandom is wild. “Real” and “imaginary” numbers? I think you mean canon and non-canon. You guys seriously go “this is my number oc his name is i and he is the square root of -1” when in numbers canon lore it’s actually impossible to square root a negative but sure whatever. “Complex numbers”? I think you mean a character x oc ship. “f(x) = 3x - 5”? That is self-insert fanfiction.

[algebraic-dumbass]

"i have 99 problems" lame. "i have an amount of problems strictly in between aleph0 and the cardinality of the continuum" exciting. takes a stance on the continuum hypothesis

[transmutationisms]

The popularity of drugs and the eagerness of conservative politicians to cut state hospital budgets explains the quick pace of deinstitutionalization in California.86 From 1970 to 1974, California reduced the number of state hospitals within its system from ten to seven and converted or limited services in the remaining hospitals to specialized types of care. This physical shrinkage translated into an eighty-three percent reduction in the state hospital population, from 36,000 patients in 1955 to less than 6,000 patients in 1977.87 When taking into account the number of patients who passed either through the doors of a state hospital or county ward for treatment, California also reduced the number of patients it institutionalized for any given amount of time by fifty-three percent (558,922 to 193,436) from 1957-1977.88 Under the LPS law, this new system reflected the belief that mental illness existed on a continuum between “normal” and “violent” states by only temporally holding the mentally ill when they were considered dangerous and releasing them into society when their florid states subsided.

West sought to advance racially liberal psychiatry by conjoining psychotherapeutic concerns around child development with the fields of biological psychiatry and psychopharmacology. In an era of growing alarm about the availability of public funds, he believed the future of community mental health required money be diverted away from “non-medical” anti-poverty programs and towards resource- and capital- intensive lab-based psychiatric research and medical education. Although he began the mid-1960s using interpretive frames of culture, race, and class to explain violence, he increasingly became interested in unlocking an underlying biologic basis for violence that rendered such frames less meaningful. His attempts to find a universal color- and class- blind basis for violence, however, ended up strengthening policing of poor people of color by reinforcing their spatial segregation from society in prisons, state hospitals, and segregated neighborhoods with new ideas about the social transmission of violence.

Comments West made after the 1965 riots/uprisings demonstrate he initially agreed with most race-based community mental health proponents that normal black personalities existed among black populations where strict cultural adherence to normative ideas about gender and sexuality was exhibited. On a panel titled “Prevention of Racial Violence in the Urban Ghetto,” for example, West suggested psychiatrists study the Nation of Islam’s program of black nationalism and its strict moral codes for working class black communities because, unlike the psychiatric community and the black middle class, they seemed to be the “only ones who have made significant progress with the prostitutes, alcoholics, addicts and recidivistic criminals who prey on their fellows in the ghettos.”89 His comments suggest he sympathized with Cannon’s belief that separate black cultural spheres had therapeutic effects.

By 1969, however, West was convinced violent behavior had nothing to do with race and/or class, but had everything to do with childhood development. He crafted new arguments about the etiology of violence after closely observing the social relationships of the Tarahumara Indians of Northern Mexico.90 Despite contact with dominant Mexican society, West believed the Tarahumara’s reverence for children and their collective protection of childhood accounted for the perceived absence of violence and its strong commitment to peaceful conflict resolution. In spite of their minority and impoverished status in dominant Mexican culture, he hypothesized the Tarahumara’s limited exposure to violence and dysfunction during childhood created biologic benefits that prevented violent behavior in them as adults.

His hypothesis captured the imagination of UCLA’s search committee and was prioritized as a guiding research agenda for UCLA after he was appointed as Chair of Psychiatry and Director of the NPI in 1969. In a series of joint psychiatric studies conducted by the University of Southern California and UCLA, West and his research team analyzed the case histories of violent offenders “from all walks of life” who, as a group, could not be “distinguished by belonging to any particular groups.”91 His findings suggested that a significant percentage of the study’s violent offender subjects tended to share the same history of being “victims of violence in childhood themselves.”92 His observations produced a new color- and class-blind psychiatric theory he eventually termed epidemiology of violence theory, a concept which proposed dysfunction witnessed as a child created a psychobiologic basis for violent behavior as an adult. In other words, he argued that childhood trauma altered brain functioning enough to make violent behavior a predisposed inevitability in adulthood.

West was convinced his theory had enormous potential for renewing community mental health’s moral discourses of prevention and hygiene and for generating a whole host of research investigations for psychopharmacological-based rehabilitation programs for a group he termed “violent people,” individuals biologically disposed to violence because of childhood exposure to dysfunction. For race-based mental health professionals, his research sustained focus on the importance of childhood development and the place of positive role models in producing well-adjusted personalities. For mental health researchers accustomed to research in biologic psychiatry, particularly in the fields of electroshock therapy, electroencephalography, psychosurgery, neuroscience, and psychopharmacology, West’s hypothesis offered a useful bridge to the popularity of social science and psychotherapeutic study that defined psychiatry since the 1920s.

Nic John Ramos, Pathologizing the Crisis: Psychiatry, Policing, and Racial Liberalism in the Long Community Mental Health Movement. Journal of the History of Medicine and Allied Sciences, Volume 74, Issue 1, January 2019, 57–84

[futurebird]

Urge to make dadaist fake mano-sphere content rising…

That aleph 1/2 grindset. For the men who don't conform to the continuum hypothesis.

[4denthusiast]

I've just noticed an odd combination of beliefs I hold about how maths works. I'm not exactly convinced that every statement in arithmetic (i.e. statements that can be written in terms of 0, the successor function, =, +, ×, and, not, and quantification over natural numbers) is either true or false. (My previously more Platonist views were shaken by some university courses on topics like model theory and Gödel's incompleteness theorems. In order to understand that stuff you have to entertain the possibility that certain seemingly obvious things aren't true, and I then sort of never stopped for some of them.) At the same time, I accept the law of the excluded middle (that P∨¬P is always true). I generally wouldn't describe myself as an intuitinist, even if I am interested in the applications of some intuitionistic logics in computer science.

I think the way I resolve this apparent contradiction is that the reason I don't feel like all arithmetic statements are true or false is that I'm not sure the natural numbers, as a set, are uniquely defined. Any definition of what is and what isn't in ℕ tends to involve some degree of circularity. "It's 0, and S0, and SS0, and so on.", but "so on" for how many steps? A natural number of steps. Hopefully you see the issue. So then, an arithmetic statement may be true of one model of the naturals, but not another. Within any one model, P∨¬P is true (or, more to the point, (∀x. Px)∨¬(∀x. Px) is true), so if it's true in every model it's true, but we can't ever pin down quite which model we're talking about, so the individual statement (∀x. Px) can remain indeterminate.

All this sort of implicitly relies on a separation of the language and meta-language, even though I didn't set out to have a separate meta-language in the first place. I'm not quite sure whether what I'm thinking here even makes sense. Perhaps what I mean is that the meta-language does have logical connectives (and, or, not), so you can form a claim like "(∀x. Px) is true, or ¬(∀x. Px) is true.", but it doesn't have quantification over the naturals, at least not always, because in the meta-language there isn't a unique ℕ, and you can't specify which one you mean because there's no way to totally pin it down. At least I think. But then the semantics of A∨B is meant to be that A∨B is true iff A is true or B is true. I guess we can still recover this by saying that any statement in the language that includes any quantifiers is implicitly with reference to a particular model of ℕ, and a statement is true iff it's true for all models, but then that requires that the meta-language can quantify over models of ℕ, which should be way less possible than quantifying over individual naturals. I don't know how to resolve this, if it even can be resolved. I'm kind of confused.

The true ℕ, if it exists, ought to be the smallest one of course. The trouble is you can't define "smallest" properly without discussing the whole class, which is a less basic concept than the numbers themselves. Also, not every ordered set or class has a smallest element. I think probably if you allow yourself sufficient expressiveness you can prove that in this case there is a smallest (take the intersection or something), but again I don't think you can prove that without making assumptions at least as strong as the conclusion.

The same thing happens with set theory, but there it all feels clearer. In contrast to the naturals where I'm not sure, I feel somewhat more confident that there isn't a single true set-theoretic universe V. There ought to be sets that can't be named (there are only countably many names after all), which makes the universe much trickier to pin down than the naturals. I know there are countable models of ZFC, but they don't feel like they're the real model, and ZFC is itself kind of vague. It leaves a lot of room for rather natural variation in what sets are allowed (e.g. the continuum hypothesis), while non-standard naturals seem much more exotic. If you assume that there's some particular ℕ that's the real ℕ, in the meta-language, this gives you much more solid foundation to use when talking about potential uncertainty in V. You can do induction, talk about constructible sets and stuff. It seems quite likely that the continuum hypothesis doesn't have a definite truth value, even though CH∨¬CH does, but it feels like quite an ordinary sort of indefiniteness, like "He has brown hair." when it isn't clear who "he" refers to. "Is there a cardinal between ω and 2^ω?" What version of the class of cardinals are you talking about?

[lilith-hazel-mathematics]

Cardinal arithmetic is easy

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

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

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

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

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

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

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

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

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

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

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

[floralfractals]

*continuum hypothesis is the idea that theres no set size in between countable infinity (e.g. the amount of integers that exist) and uncountable infinity (e.g. the amount of real numbers that exist)

[simply-disconnected]

To add to all of the above, there is another way to construct countable many different cardinality, and it's with power sets! For any given set A, the power set which I will denote P(A) is the set of all subsets of A. For example, P({1,2})={{},{1},{2},{1,2}}. With our friend cantor's diagonalization argument, it can be shown the cardinality of P(A) is strictly greater than that of A, even if A was infinite. That means that we can construct infinitely many cardinalities simply by taking power sets of power sets. If we denote the cardinality of A to be |A|, then |A|<|P(A)|<|P(P(A))|<|P(P(P(A)))|<... and so we have countable many infinite infinities.

You can even complicate things! What if you take the union of all of A,P(A),P(P(A))... ? You get a set that's even larger than all the previous ones! let's denote it A_2. What then of P(A_2)? and P(P(A_2))? We can take the infinite union of all of those and get an even larger set, which we will call A_3. We can then recursively define A_4, A_5 etc. And if we take the infinite union of all the A_n? We get a new larger set! If we denote it B, and then repeat this process taking the infinite union of B, P(B), P(P(B))... we get B_2 and similarly B_3 and so on. We can in fact repeat this process infinitely many times, getting C, then D, and continuing even after we run out of letters! And THEN we can take the union of this infinite process, and denote that Alpha, and take P(Alpha) and P(P(Alpha)) and so on and so forth, and the more you look at this, the more similar it starts to seem to cardinals.

Tell me everything about infinity.

Oh, a very loaded question! All right. Let's start with the sizes of infinity!

Roughly speaking, there are two sizes of infinity; or, in proper terminology, "cardinalities." (There is some debate, as I recall, over whether there are more sizes of infinity. But we know there are two.) The first cardinality is the same size as the integers, which are the positive and negative whole numbers; essentially tick marks going in a line forever. 1, 2, 3, and so on; and in the opposite direction, -1, -2, -3, and so on.

The second cardinality is the same size as the real numbers, which are all the numbers that most people use on a daily basis; think, instead of tick marks, a line, and every place on that line is a number. No matter how close two places on that line are, there's always another number in between them. So you have 2.5 and 2.6, but between them you have 2.55 and 2.5932, and infinitely many more.

The concept of infinity, of course, gets a bit weird once you move into more than one dimension; it's easy enough to point in the same direction as a line and say, "that goes to infinity," but once you have multiple dimensions, is it meaningful to talk about a negative or positive infinity? Oh, and adding and subtracting get weird once you start adding infinitely many things together. Addition loses commutativity (e.g. 5+3 = 3+5), which still blows my mind, even though I've seen the proof for it. It's the kind of thing that keeps me up at night.

Generally mathematicians get around the multiple-dimensions problem by using the modulus of a number, which is the distance from the number to zero, measured using our dear old friend the Pythagorean Theorem: so, if your point is at 4 in one dimension and at 3 in another, you use the Pythagorean theorem to get 5, and then you consider that point to be 5 away from zero. (This would be easier to explain if I had a chalkboard!) Then, infinity is sort of a circle that surrounds the whole plane; or, if you think of your 2d plane like a flat circle, if you folded it up into a ball, infinity would be all the points at the very top of the ball, and zero would be the point at the bottom. (Obviously this gets weirder if we have three dimensions, but you get the idea.) Okay, so that's a quick introduction to infinity from a mathematical perspective. I think there were also some physics questions re: the expanding universe and spacetime? I'm happy to write a bit about that, too, but I think it belongs in a separate post! So if you have questions about that, please let me know and I'll try to share what I do know! Disclaimer: while I DO know more math than the average person, I have essentially a bachelor's-degree level of knowledge in math. I think everything I've typed out is correct, but I may very well have missed some details! Dear readers, please feel free to correct me if you have greater knowledge than I.

Edit: also, I should have mentioned that the pythagorean theorem only works in a euclidean space. But I feel like going into non-Euclidean stuff is a bit past the scope of (this) tumblr post.