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saintgoths · 8 months ago

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ᴄᴀᴛ ꜰɪɢʜᴛ

DEALER!ELLIE X READER - CAT FIGHT. [COLLEGE/UNIVERSITY AU].

[PART EIGHT TO THE CATGIRL SERIES.]

WORD COUNT - 2,214.

RATING - 18+. [protective Ellie, lesbian sex].

SUMMARY - you and ellie get back together, but an angry doris confronts you.

[follow for more content & feedback would be appreciated thank you <3].

previous chapter - chapter seven.

You smiled, never had you thought you’d have a lover serenading to you. Your hands cupped your face as Ellie hummed, while she easily strummed her fingers against her instrument, you had watched with awe as you listened to the lyrics fall out of her lips, “I watch the moon, let it run my mood, can’t stop thinking of you,” the girl sang, and complimented you had leaned your back against the head board, hands now away from your face and now against your lap as she finished.

“That was amazing,” you said as she put down her guitar against the foot of her bed.

“Thank you,” Ellie acknowledged before she moved herself to kiss you, it was quick and soft, but the smiles between the two of you had mirrored the intense shared passion.

It had been two weeks ever since what had happened, and the two of you were confident to say that it had pushed the two of you closer.

Ellie had now had you in her arms as you comfortably rested on her chest, you stared up at the ceiling. “Did you really mean that song?” You curiously questioned, you looked up at Ellie, hands around her tattooed arm as she stared back at you.

“I can never stop thinking about you,” Ellie truthfully shared and with another kiss, you had gently flipped yourself so you could lay on your stomach. Eager to touch her mouth, you had moved your hands beneath her shirt but the sound of someone knocking on Ellie’s dorm door had stopped the two of you.

Your girlfriend had groaned as she pulled herself off her bed. It had been both Dina and Jesse behind the door, and quickly, you had remembered there was a party the four of you were going to. Jesse said he’d drive, which is why the two of them had gone to pick both of you.

You pulled on your shoes, and now on your feet, both you and Jesse greeted each other, he threw a certain look towards your way before he had commented. “What were you love birds doing?” He questioned; half of his question had sounded serious but you were able to hear the playful tone.

With a knowing look, you arched your eyebrow. “Leave it to your imagination,” you responded as you slipped past him and before he did, Dina slapped his arm.

“Don’t you dare imagine anything,” she comically threatened which had then caused the rest of you to laugh.

❊❊❊

It was loud and packed. Both you and Ellie had clung onto each other as the two of you adjusted to the new atmosphere, there had been led lights, drinking competitions and people passionately kissing each other on one of the couches on display

“I wonder if there’s a room we can be in,” you poorly joked, thus Ellie squeezed your hand to reassure you.

“We should get a drink,” she said and with a quick peck on her lip, you told your girlfriend that you’d be the one to get them.

Once you left her side, you scanned for where most of the drinks had come from and made a bee-line towards it once you had found it. You hadn’t realised Doris had been there, but once you had felt someone’s hard eyes dig into you, you had looked up.

You remained silent as you poured both you and Ellie drinks but as you went to leave, Doris had commented.

“You managed to get Ellie back,” Doris said, as if what had come out of her mouth had been irony.

You turned to look at her. “I’m serious about her.”

“It took you time to become serious,” Doris pointed out, which had been true. “To be serious about people, I mean.”

Flatly, you smiled. “The previous people I were with were just stepping stones to get to Ellie I guess.”

Doris smiled right back at you, and in a matter of seconds, the liquid in her cup had been splashed all over you. Everyone turned to look at what had been happening, and annoyed, you had thrown your drink back at Doris as people encouraged the two of you to fight.

Quick on her feet, Ellie immediately stepped in between you and Doris, her eyes dark as she looked at your ex-partner. “Leave [Y/N] the fuck alone!” Ellie rudely spat, and bold, Doris had taken a step forward, ready to speak, but Ellie had been quicker. “I promise I’d give you the beating I should’ve did back in the toilets.”

With a scoff, Doris looked at both you and Ellie up and down before she left. Your arms crossed, you realised how drenched you were, your excitement for the party had been put out the moment Doris threw her drink on you. “I’m wet but not in a good way,” you muttered as Ellie turned to look at you.

Empathetic, Ellie had sighed as she grabbed your wrist. “Come on, let us look for somewhere to dry you up,” she said and obedient, you had followed your girlfriend through the crowds who returned to party to the loud music.

❊❊❊

The two of you had found a room to sit in, Ellie had been the one to look for a bathroom and a towel, and once she did, she had used it to dry you up. “At least you still look very pretty,” Ellie said and you rolled your eyes to her comedic timing.

“Please you love being told you’re pretty,” she teased before she kissed you. Tenderly, Ellie had pulled away before she tucked a strand of your hair behind your ear. “How do you feel?” She asked as she gave you another peck. “Being confronted, and everything with your dad?”

You burst out laughing. “Ellie, we’re about to have sex and you’re asking me about my dad?” You asked and with red cheeks, Ellie kissed you again.

“I’m so stupid,” she mumbled and you agreed, with a smile you pulled her closer as she slipped off your shirt.

You had gently moaned as you could feel her grope your breasts. You could feel the way she circled her thumb around your breast fore she moved her tongue to suck it, her other hand had slipped off your lower garments and to support yourself, you had rested on your elbows as you had watched her desperately devour you. Her mouth had been hot and wet, and every few times she’s suck on your breast she’d fan her hot breath against your skin.

Ellie had slid down your body, left trails of wet kisses against your naked skin till she met with the opening of your legs. Elie had been terrifically wet and had the obsessive crave to rub herself against you, but she had focused on you, had left kisses between your thigs before her mouth wrapped around your pearl, as she sucked, the texture of her tongue had stroked against your slit, and you had shuddered, sensitive, your hand immediately reached for her hair.

The tip of her tongue repeatedly flicked against your clit before she slid her tongue inside, her swift movements had caused you to shake in surprise, her tongue also being aided by her digits had glided inside of you, her fingers arched as she stroked the soft lump inside of your cunt, eyes sunken into yours as she had watched the way your body had twitched.

“Ellie!” You had whined as you could feel yourself quickly cum, with a smirk, Ellie moved herself back to your lips, she had a smirk on her face, a smirk of arrogance, proud of how easy she had made you climax. The Williams girl helped herself to pull off the rest of her clothing, and immediately, your eyes were trained on her lean body.

“Look at me,” she commanded and immediately you did, your lower lips tucked behind your teeth, as you felt her adjust on top of you, your wet skins had barely touched but there was a ghostly gasp that left your lips, and when you had felt the pressure of her weight lean against yours, your eyes softened.

Your hands against her waist you had supported Ellie’s weight on top of yours as she commenced to rub her clit against yours. She moaned as she lowered her body against yours, her face buried between the crook of your neck as she had moaned, her eyes rolled back as the gentle sparks began to increase.

“Kiss me!” She begged and immediately, her lips were pressed against yours, the slick shared between your heat had augmented, and the lewd sting of her pressing your clit against yours had boosted, you had tucked your tongue into her mouth as she had sucked, her thrust had become quicker and more animalistic, greedy to chase her high, Ellie had moved her lips to the sharpness of your jaw, hungry as she used her tongue to lick down your neck.

Intense, your nails had dug into her back, you were almost there, entranced by the way your clit had moved and circled against hers, your eyes watered. “Fuck!” You cried out just as your body started to twitch. “Ellie!” You screamed as you climaxed, stubborn, Ellie had remained against you, her eyes rolled to the back of her head as her figure slowly cloaked with heat.

Her hand that was tightly clasped with yours had dug into your skin, she was wordless, looking down on you as she bounced on your swollen clit, her orgasm coming seconds after yours. Her nectar, spilled against your opened heat as she had then dropped beside you. Briefly, she had pushed a deep sigh as her wet eyes looked up at the ceiling.

“We should get back,” you breathed out as you had then sat up at the side of your bed, your back faced towards her, aware that the sight of her eyes glided down the figure of your back. You held back a smirk, if it had been you two back at her dorm, you would’ve gone for second rounds.

❊❊❊

It had been another week, you had remained by Ellie, spent almost the entirety of your term in her dorm, your finals were soon, but reluctant to stay away from you, Ellie had encouraged you to crash at hers.

The both of you knew how’d it end, kissing and touching each other. Ellie singing to you as you wrote about your fashion research, truthfully, the sex had taken over the amount of time you had spent with her, but it had been one of your favourite things to do, lips gently locked with hers as your fingers circled her clit, her tongue slipped into yours as she arched her fingers into your depth.

You had gasped into her mouth as you could feel your orgasm reach, around your fingers you had released, and Ellie had been after you. Currently, with your forehead pressed against each other, you had been the first one to roll against your back.

“After all this,” you said out of exhaustion. “Where should we go?”

“London,” Ellie quickly said and excited your eyes brightened.

You had then sat up. “London,” you said with a smile. “I’d love to go there,” you encouraged as you slipped on your panties.

“And then to Paris, to Rome,” Ellie listed, in early anticipation of what the future held for the both of you. “We go to the Eiffel Tower and we make out under it.”

You hit her shoulder as you teased her. “How corny,” you joked and with the dramatic roll of her eyes, Ellie pulled her hands behind her head.

When you put on the rest of your clothes, Ellie decided to put on her briefs, you phone had then rung and with a dismissive look you had muttered. “I’ll reply to it later.”

Unsure, Ellie stood to her feet, her gazed fixed on your phone, it had vibrated as the screen shined brightly white. “I think it is best that you answer now,” she suggested, though with the sound of her voice, it had appeared more commanding.

With a side smile you reached for it, unaware that the number of missed calls had been detrimental.

It was from your mother.

With a frown you had picked up the call, you had quickly pulled a strand of hair behind your ear as the sound you were met with were the frantic cries of your mother. You didn’t need to hear what she had to say because you knew, it had come to you like a moth to a flame, but with your silence and the stiffness of your body, you didn’t know if you had become paralysed or had been in shock to what your mother finally said.

You knew it was always going to happen, but there was always a delusional hope you had within you that believed that your he would survive, but it was just the inner child naivety. A scared daughter clinging onto her dying father. You had already wept before you mother had spoken, but when her woe words eventually came out of her mouth, clarifying the death of your father, your whole world had come to a stop.

feedback would be appreciated! if you want to be apart of the tag list comment <3

plus, it is ending there's only one chapter left.

tag list - @liasxeatt

masterlist

catgirl masterlist

#the last of us #ellie williams #the last of us 2 #tlou #the last of us two #tlou2 #ellie williams x reader #ellie williams fanfiction #ellie williams smut #ellie williams imagines #ellie williams imagine #ellie williams x female reader #ellie williams x y/n #ellie the last of us #ellie tlou #ellie williams tlou #ellie williams the last of us

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serenityfails · 7 months ago

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Meowrrigan

y'know like nya?

(my friends said Morrigan's new double bikini made it look like she had catgirl nipples. thus:)

#dragon age #dragon age: the veilguard #datv #datv spoilers #morrigan #i made this

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catgirl-lucy · 2 months ago

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Cardinal Numbers

Nyaa!~

I have too many drafts :c

Hi everyone! Today, we are going to learn about infinite cardinal numbers! I hope this'll be fun! ^^

I was also planning to explain something about the axiom of choice, but I decided that it's better if that becomes its own blog-post.

So, first: what is a cardinal number?

✦ Motivation ✦

Well, cardinal numbers are numbers that represent the size of a set of things. For example, {owo, uwu} is a set of cardinality 2: it has two elements, owo and uwu.

However, we can't simply count the elements of a set to figure out its cardinality. What if we come across a set like {0, 1, 2, 3, ...}? I mean... you can try counting the elements, but it's gonna take you a veryy long time.

So! What do we do?

Well,,, first, observe that cardinals can't exist in a vacuum. If we say that {owo, uwu} has cardinality 2, what does that actually mean? ‘2’ is just a symbol we use to describe the cardinality of {owo, uwu}, but has no meaning on its own. Where it gets interesting is when we compare the cardinality of {owo, uwu} to {10, 3}: they're the same! Now, that is interesting.

So, we use ‘2’ to describe the cardinality of sets like {owo, uwu} and {10, 3}, and we use this symbol so we can more easily recognize when two sets have the same cardinality, without having to search for statements like “this set has the same size as this other set” exhaustively having to apply transitivity of (cardinal) equality.

But what makes {owo, uwu} and {10, 3} so similar that they can be considered ‘equinumerous’?

Well,,, observe that the cardinality of a set doesn't change when we replace some of the elements with other elements: the cardinality of a set is completely independent from what the members of that set actually are. If we have a set like {catgirl lucy, my dearest friend, Semi}, and I replace ‘my dearest friend’ with ‘my worst enemy’ (so we get {catgirl lucy, my worst enemy, Semi}), the size of the set doesn't change!

I hope these two observations are enough to motivate the definition of a cardinal number:

～ Cardinal Equality ～

[Definition] Two sets, A and B, have the same cardinality, denoted |A| = |B|, iff there exists a bijection f: A → B.

There are three kinds of functions that will be important to us in this blog-post:

An injection is a function f: A → B from some set A to some set B so that, for all x,y ∈ A, if f(x) = f(y), then x = y.

A surjection is a function f: A → B from some set A to some set B so that, for all y ∈ B, there is some x ∈ A so that f(x) = y.

A bijection is a function f: A → B from some set A to some set B that is both an injection and a surjection.

A bijection thus pairs every element of A with a unique element of B, and vice versa.

So now, we need to show that this definition of cardinal numbers actually makes sense. I.e., that it is an equivalence relation:

An equivalence relation is a relation ~ such that:

~ is reflexive: x ~ x for all x

~ is symmetric: if x ~ y then y ~ x

~ is transitive: if x ~ y and y ~ z, then x ~ z

...

What?

OK, fine... I'll give you the answer: reflexivity of cardinal equality is witnessed by the identity function on a set, symmetry is from the functions inverse and transitivity comes from composition. I do want you to start thinking about how to solve these problems in the future!

Given an equivalence relation ~ on some class C, we can define a function F with domain C so that, for all x,y ∈ C, F(x) = F(y) iff x ~ y. If ~ fails one of the conditions of an equivalence relation, such a function does not exist.

This time, I will leave you to ponder why~

So! We can choose some function |·| that maps sets X to objects |X| such that |X| = |Y| iff X and Y have the same cardinality, the choice of mapping doesn't really matter.

The cardinality of ℕ, the set of natural numbers, is denoted ℵ₀ (‘aleph-nought’), this is also the cardinality of ℤ, the set of integers, and ℚ, the set of rational numbers. ℵ₀ is an infinite cardinal (not 0, 1, 2, 3, etc), but it isn't the only infinite cardinal!

|P(ℕ)| = 𝔠, the continuum, is another infinite cardinal. It is the cardinality of the set of subsets of ℕ, i.e. the set of sets of natural numbers.

Why are these cardinals different? Well, given some function f: ℕ → P(ℕ), try to find some subset of ℕ that is not in the range of f~ Thus, show that f cannot be surjective!~

Here is a hint for when you get stuck: if there is some x for which A and B disagree on whether they contain it (x is in A but not in B, or x is in B but not in A), then A and B are different.

Now, that we have gone over the basics of cardinal equality, let's look at something more exciting~ cardinal comparison:

⊰ Cardinal Comparison ⊱

[Definition] For two sets A and B, |A| ≤ |B| iff there exists an injection f: A → B.

≤ is a partial order on cardinal numbers:

≤ is reflexive: x ≤ x for every cardinal x

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

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

Reflexivity and transitivity are easy exercises for the reader~ (the proof is literally just the same as refl and trans for cardinal equality :p). Antisymmetry, on the other hand, is a lot more difficult!

That cardinals are antisymmetric is known as the Shröder-Bernstein theorem. The proof may be hard to follow, so good luck! :D

Let A and B be sets and suppose there are injections f: A → B and g: Β → A. We want to define a bijection h: A → B.

We can see that f and g form chains a₁ ↦{f} b₁ ↦{g} a₂ ↦{f} b₂ ↦{g} ... alternating between elements of A and elements of B. Since g is an injection, for every a ∈ A, we either have that it comes from some b (i.e. g(b) = a), or from nothing. The same holds for f. So every element in A and B belongs to a unique chain ... ↦ * ↦ * ↦ ... defined by f and g. Some of these chains have a sudden start (i.e. some a ∈ A or some b ∈ B that doesn't have an inverse under g or f), and some of these chains are infinite.

For a ∈ A, we can define h(a) based on what kind of chain a belongs to. If the chain a belongs to starts in A, we can set h(a) = f(a). If the chain a belongs to starts with some element in B, we can set h(a) = g⁻¹(a) (i.e. we set h(a) to an element b in B such that g(b) = a). If a belongs to a chain that goes infinitely far left, we do whatever. I'm just going to choose h(a) = f(a).

I'll leave it to you to verify that h is indeed a bijection.

We can also define strict comparison:

[Definition] For cardinals x and y, x < y iff x ≤ y and x ≠ y.

Since ℵ₀ ≠ 𝔠, as shown in the previous chapter, and n ↦ {n} forms an injection from ℕ to P(ℕ), we have ℵ₀ < 𝔠. In fact, for every cardinal x, there is some cardinal y so that x < y.

Here is some basic terminology for partial orders:

[Definition] Let ≤ be a partial order on some class P and let A ⊂ P be a subclass of P. A maximum element of A is some x ∈ A so that, for all y ∈ A, x ≤ y → x = y. A minimal element of A is some x ∈ A so that, for all y ∈ A, y ≤ x → x = y. The greatest element of A, if it exists, is some x ∈ A so that, for all y ∈ A, y ≤ x. The least element of A, if it exists, is some x ∈ A so that, for all y ∈ A, x ≤ y. An upper bound of A is some x ∈ P so that, for all y ∈ A, y ≤ x. A lower bound of A is some x ∈ P so that, for all y ∈ A, x ≤ y. The infimum of A, denoted inf(A), if it exists, is the greatest lower bound of A. The supremum of A, denoted sup(A), if it exists, is the least upper bound of A.

Given x,y ∈ P, the meet of x and y, often denoted x ∧ y, is the infimum of {x,y}. The join of x and y, often denoted x ∨ y, is the supremum of {x,y}. If every pair of elements in a poset (partial ordered set) has a meet and a join, then that poset is called a lattice. x and y are comparable iff x ≤ y or y ≤ x, x ⊥ y is used to denote that x and y are incomparable. Examples of lattices are: the powerset lattice (P(X),⊂) of any set X, where x ∧ y = x ∩ y and x ∨ y = x ∪ y, linear orders like ℚ, where x ∧ y = min(x,y) and x ∨ y = max(x,y), etc.

Without assuming the axiom of choice, two cardinals needn't have a meet or join. Now you know random useless stuff about posets!!

Oh, and: ℵ₀ is a minimum infinite cardinal. Proof is left as an exercise~

❈ Cardinal Arithmetic ❈

Cardinal numbers are numbers, so it'd make sense if we could do arithmetic on them. And we can!

[Definition] For sets A and B, |A| + |B| is the cardinality of the set A ⊔ B, i.e. the disjoint union of A and B. Members of A ⊔ B are (0,a) and (1,b) for a ∈ A and b ∈ B.

[Definition] For sets A and B, |A| · |B| is the cardinality of the Cartesian product A × B of A and B. Members of A × B are ordered pairs (a,b) for a ∈ A and b ∈ B.

Arithmetic on finite cardinals works as you'd expect: 6 + 3 = 9 and 12 · 2 = 24. Addition and multiplication on infinite cardinals, however, is actually quite boring, as x+y and xy are simply max(x,y). Well, that is, if you assume AC. Without AC, cardinal arithmetic can actually get very interesting (and weird af).

I am not that familiar with cardinal arithmetic without the axiom of choice, so you'll have to do more research on that on your own if you're interested!

Cardinal addition and multiplication are commutative, x+y = y+x, xy = yx, associative, (x+y)+z = x+(y+z), (xy)z = x(yz), and multiplication distributes over addition, x(y+z) = xy+xz.

Personally, I think cardinal exponentiation is more interesting than addition or multiplication:

[Definition] For sets A and B, |A|^|B| is the cardinality of the set of functions from B to A.

For cardinals x and y, if x ≥ 2, then we can prove that x^y > y. 𝔠, the cardinality of the continuum, is equal to 2^ℵ₀ and to ℵ₀^ℵ₀. Also, it's the cardinality of the set of real numbers ℝ. Here's a vid I found that explains why.

I think I might be getting too eepyy to write. I'll write more tomorrow.

[zzz]

Good morning!

Cardinal exponentiation has all the properties you'd think it has: x^y · x^z = x^(y+z) and (x^y)^z = x^(yz). Also, 0⁰ = 1.

There are also infinite sums and infinite products:

[Definition] Σ_(i ∈ I) A_i is the set of tuples (i,a) for i ∈ I and a ∈ A_i. Σ_(i ∈ I) |A_i| = |Σ_(i ∈ I) A_i| is the cardinality of this set.

[Definition] Π_(i ∈ I) A_i is the set of functions f with I as domain and f(i) ∈ A_i for all i. Π_(i ∈ I) |A_i| = |Π_(i ∈ I) A_i| is the cardinality of this set.

Σ_(i ∈ {1,...,k}) x_i = x₁ + ... + xₖ and Π_(i ∈ {1,...,k}) x_i = x₁ · ... · xₖ, so this is a valid extension of sums and products. The sum of x many y's, i.e. Σ_(i ∈ I) x for |I| = y, is the same as the product of x and y. The product of x many y's, Π_(i ∈ y) x, is equal to x^y. If X is a family of sets, I also sometimes write ΣX and ΠX for Σ_(A ∈ X) A and Π_(A ∈ X) A.

Here is a fun fact that idk where to place in this blog post: a Dedekind infinite cardinal is a cardinal x = |A| for which there exists an injection f: A → A that is not surjective. In other words, x+1 > x. Without the axiom of choice, infinite Dedekind finite cardinal numbers (aka mediate/Dedekind cardinals) can exist, and these cardinals are incomparable to ℵ₀.

Here is a fun question: is the product of any number of non-zero cardinals always non-zero?

Well?~

◈ Axiom of Choice ◈

The axiom of choice (AC) states that the product of any number of non-zero cardinals is always non-zero. I.e. for any family X of non-empty sets, ΠX is non-empty (a member of ΠX is called a choice function for X).

Eh... I didn't really plan what to write in this chapter besides what the axiom of choice is. Oh, well!

...

Wait-

Wait, WHAT THE F---

So.. apparently, you cannot take infinite products of cardinals without assuming the axiom of choice. ℵ₀ is the cardinality of ℕ, but ℕ is not the only set with cardinality ℵ₀. If A has cardinality ℵ₀, then in how many ways does it have cardinality ℵ₀? Well, if a bijection f: A → ℕ exists, then 𝔠 many of those bijections exist. If we take ℵ₀ different sets A₀, A₁, A₂, ... all with cardinality ℵ₀, we have, for each k, multiple bijections f: Aₖ → ℕ. We would expect |Π_(k ∈ ℕ) Aₖ| = ℵ₀^ℵ₀ = 𝔠, but how do we choose a bijection f: Aₖ → ℕ for each natural number k? If we only had a finite amount of Aₖ, this wouldn't be a problem. However, we have an infinite amount of Aₖ... so we need to make an infinite amount of choices. So, we can define the set X = {{f | f: Aₖ → ℕ is a bijection} | k ∈ ℕ}, and a choice function for X gives us a bijection for each Aₖ! But... we need AC to prove such a choice function exists.

...

HOW AM I SUPPOSED TO LEARN AD WHEN THIS F----D UP SH-- HAPPENS?

..sorry for screaming at you.

Luckily, infinite cardinal addition still works as normal without AC... probably.

Maybe.

Nope, it doesn't.

【 Aleph Cardinals 】

As you can see, we have run out of symbols to put on the sides of the chapter titles.

Anyways! So, a well order is an order (A,≤) where:

≤ is a partial order on A.

≤ is total/linear: for all x and y in A, x and y are comparable by ≤.

≤ is well-founded: all non-empty S ⊂ A have a minimum element.

Well, ig those random facts about posets weren't completely useless.

I might make a blog post about ordinals later, which'll go more in-depth on well-orders and linear orders.

[Definition] An aleph cardinal is an infinite cardinal number |A| for which there exists a well-order (A,≤).

An example of an aleph cardinal is ℵ₀: ℵ₀ = |ℕ| and the usual order on ℕ is a well-order. The n-th aleph cardinal is written as ℵₙ, starting at 0th of course :3

The statement ‘ℵ₁ = 𝔠’ is known as the continuum hypothesis (CH), which is both unprovable and undisprovable. I might make a blog post about forcing in the future! ^^

Although cardinals (without AC) don't need to be linearly ordered, aleph cardinals are linearly ordered and even well-ordered. A proof of this is left as an exercise~ Because, well, of course it is :P

The well-ordering principle states that every set A has a well-order (A,≤). Equivalently, every infinite cardinal is an aleph cardinal.

It turns out: AC and the well-ordering principle are equivalent! Although AC, when written in its original form, seems more obviously true (to me at least), I think the well-ordering principle is a lot more useful.

I might go more in depth on why this is in my post about ordinal numbers. (why AC and the well ordering principle are equivalent)

❖ Cofinality ❖

There are two definitions of cofinality: one for cardinals, and one for ordinals. Since this blog-post is about cardinals, I'll give the one for cardinals:

[Definition] The cofinality of a cardinal x = |A| is the minimum cardinality of a partition X of A into sets of cardinality <x.

I often use cf(x) to denote the cofinality of x and cof(x) to denote the class of cardinals with cofinality x, though people also often use cof(x) to denote the cofinality of x. A partition of a set A is a set X such that all members of X are subsets of A, none of the members of X intersect and the union of all members of X is A.

We have cf(0) = 0, cf(1) = 1, cf(2) = 2 and for every finite n > 2, cf(n) = 2. 3 = |{a,b,c}| can be partitioned into {{a},{a,b}}, the partition {{a,b,c}} doesn't work as this has a set {a,b,c} of cardinality 3, which is not < 3.

[Definition] A cardinal κ is regular iff cf(κ) = κ.

Equivalently, every partition of κ either has cardinality κ or an element of cardinality κ.

We have cf(cf(x)) = cf(x) for every cardinal number x, so cf(x) is always regular.

0, 1 and 2 are examples of finite regular cardinals, though often, 2 is not regarded as regular. ℵ₀ also is a regular cardinal.

A cardinal that is not regular is called a singular cardinal. An example of a singular cardinal is ℵ_ω, which is the sum of ℵₙ for finite n. cf(ℵ_ω) = ℵ₀ as {ℵ₀,ℵ₁,ℵ₂,...} is a partition of ℵ_ω = Σ_(n ∈ ω) ℵₙ into ℵ₀ many cardinals each of which is <ℵ_ω.

Here is the ordinal definition of cofinality, which is (a lot) more often used if you have the axiom of choice:

[Definition] The cofinality of an order (A,≤) is the minimum cardinality of a cofinal subset S ⊂ A. S is cofinal if ∀x ∈ A ∃y ∈ S, x ≤ y.

Assuming the axiom of choice, the cofinality of a cardinal κ is often defined as the cofinality of the minimum ordinal α such that |α| = κ. An ordinal is the order-type of a well-order, and one ordinal is less than another if there exists an order preserving injection from one to the other. cf(κ) with the above definition agrees with the one I gave earlier for infinite κ, but if κ > 0 is finite, then cf(κ) is 1 instead of 2. I might talk more about this in my blog-post about ordinal numbers. For now, we'll just use the partition definition of cofinality.

If x is infinite Dedekind finite, then cf(x) is 2, which I think is kinda funni. We can also have cf(𝔠) = 2 if we omit choice :3 Though with choice, cf(𝔠) is uncountable (i.e. >ℵ₀).

I don't know how to end this blog, so here is some random terminology:

A cardinal x is finite iff κ is 0, 1, 2, 3, etc.

A cardinal x is countable iff it's ≤ℵ₀. It is uncountable if it's not ≤ℵ₀.

An alpeh cardinal is an infinite well-orderable cardinal.

A strong limit cardinal is a cardinal x for which, for all y < x, we have 2^y < x.

ℶ_0 = ℵ_0, ℶ_(n+1) = 2^(ℶ_n) and ℶ_λ = sup{ℶ_α | α < λ} for limit ordinal λ. A cardinal of the form ℶ_n is called a beth cardinal and ℶ₁ = 𝔠.

Assuming AC, x⁺ is the next cardinal after x, called the successor cardinal. AC is needed to ensure there exists a cardinal directly after κ.

The continuum hypothesis (CH) states that ℵ₁ = 𝔠. The generalized continuum hypothesis (GCH) states that 2^x = x⁺ for all infinite x. Both are unprovable and undisprovable from the usual axioms of ZFC.

Assuming AC, a weak limit cardinal (or simply limit cardinal) is a cardinal x for which, for all y < x, y⁺ < x. Assuming GCH, weak and strong limit cardinals are the same.

A strongly inaccessible cardinal (or simply, inaccessible cardinal) is a regular strong limit cardinal. Assuming AC, a weakly inaccessible cardinal is a regular weak limit cardinal. The existence of an inaccessible cardinal implies the consistency of ZFC, and thus, by Gödel's incompleteness theorems, ZFC cannot prove the existence of an inaccessible cardinal.

That's all I had to say about cardinals (for now), bye!~

#set theory #maths #mathblr #cardinal numbers

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endcant · 4 months ago

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Crossposted from my Tumblr Community: The Trans South

The Trans South Monthly Bulletin Board

Our pinned post for the month of November, 2024.

comment anything you want to add, and i'll add what fits in this month's post. i'll be making a new post each month, so you may need to resubmit any links that are applicable for multiple months. check the comments for anything i missed!

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the bright side:

the fact that you're alive is good news to me!

"Good News: Some States Passed Protections for Abortion Access, Marriage Equality, and More" | Here is an article with some upshots and progress from the 2024 election.

"Lipstick Lounge Owners Break Ground on Sports Bar Venture" | "The Lipstick Lounge is one of only 32 open and operating lesbian bars in the United States registered with the Lesbian Bar Project."

"Florida's LGBTQ+ community defies queer repression with advocacy & preservation" | "I will not paint a desolate picture of the state. And, there still is hope."

"The Candy Shop: Columbia SC Rallies to Honor Landmark Black Gay Bar" | "Researchers and preservationists at Historic Columbia have joined the struggle to correct the way the club has thus far been "written over" in the historical record."

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TTS crowdfunds:

help fellow transgender people in your community today!

Catgirl-smash needs help with bills and moving!

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fun links:

things that gave me dopamine and kept me going

Read "When We Call a Place Home" by Chinelo Onwualu | "A utopian tale of historical memory, the dangers of forgetting…and vampires."

WEBFISHING | "WEBFISHING is a multiplayer chatroom-focused fishing game! Relax and fish (on the web!)"

Memory Games | I've been using these memory games to try to heal my brain a bit from long covid brain fog.

Mahogany's List of Black Banned Books | Read some books they don't want you to read!

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useful links:

interesting resources and organizations you should know about

Trans Reads | "Trans Reads is an ambitious project created by and for transgender people to openly access writing related to our communities."

Southerners On New Ground | "SONG is a home for LGBTQ liberation across all lines of race, class, abilities, age, culture, gender, and sexuality in the South".

Winners from The LGBTQ+ Victory Fund | Check out this list of winning LGBTQ+ candidates for various political offices from around the U.S. who were supported by the Victory Fund.

Lambda Legal's Help Desk | "a resource for the community we serve in providing general legal information and resources relating to discrimination based on sexual orientation, gender identity and gender expression, and HIV status."

Every State has a DD Council! | Did you know that it's federal law that every state must have a council whose job it is to hold their state accountable on behalf of developmentally disabled people?

Help the Palestinian People with a Click | "Your free click generates donations from our sponsors. Remember, you may click once a day, every day. Donations raised will go to UNRWA to assist the Palestinian refugees."

If you're an adult trans person in the U.S. South, join my Tumblr Community: The Trans South.

If you're a teen trans person in the U.S. South and you create a youth community for trans southerners, let me know and I'll link to it in The Trans South.

#mine #pins #tts

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vividfae · 24 days ago

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About My Document...

Hey. It's been a minute. I know that I have said over & over again that I'm not going to keep posting about the situation. I plan to hold true to that. But understand that it has been a very long & difficult road & I've been spending a lot of time reflecting & trying my best to heal. Part of that means seeking closure & regaining my autonomy.

Over a year ago now I wrote & published a document going over my suicide attempt from the preceding year & what I felt the culpability my group of friends had in the circumstances that led to it. Though my relationships with Zena & Poppy had ended, I initially stuck by that document, partially because I value principles & partially because I hadn't really begun to process anything.

Given new information, a lot of reprocessing, & a lot of recontextualizing events, I can no longer in good conscience completely stand by that document or the video that I published going over it.

A Brief History

Before I explain my new stances, I need to summarize what happened.

Italics represent the story as I told it in my doc, which is representative of the way I was made to think about things after the fact. Regular text is just me trying to tell the story of what happened with as little editorializing as possible.

In May 2023 I attempted suicide. There was a wealth of reasons at the time that pushed me to this place but central among them were my fears of abandonment & anxieties in my relationships due to an at the time untreated BPD spiral. I ended up in the hospital for a little under a week. At the time, I was friends with a group of people who had been splitting time between two discord servers, my personal friend server The Catgirl Cafe, & Zena & Poppy's Foundation server. There were a lot of things that contributed to my mental state. My relationships with Zena & Poppy were rocky (Zena had even broken up with me a couple weeks before my attempt), there was an incident several months earlier with one of my former friends that I'd classify as a form of sexual assault that I never really addressed or told anyone about, & both servers were developing really toxic atmosphere in their own ways.

Due to the group of friends that were in the Cafe egging on a lot of negative thinking towards my relationships with Zena & Poppy & not really taking much care to actually provide much in the way of support, my mental state began to collapse as I felt there was no hope because I couldn't see a way to fix things & my friends were pushing me to think more & more negatively about the situation.

When I made my attempt, I made some concerning posts on twitter before shutting my phone down & driving from Michigan where I had been staying to a place in Indiana where I made my attempt. The Cafe group found my family & contacted them, I was found & prevented from succeeding, & went to the hospital later that night.

My friends mishandled my attempt because instead of contacting the people who were closest in proximity to me & had resources to get me help, Poppy & Zena, they spent time searching for my family without any definitive hope of success, leading to a large chance of me not being found. Instead, they only lucked into finding my family & thus being responsible for me being found.

Amendments, Clarifications

Being out of the Foundation & away from Z&P has given me a lot of ability to process everything that I went through. I'm not going to go line by line, but I need to make some amendments.

For starters, I've privated my video. Idk if my original doc is still up, but as soon as I find it I'm going to be removing it as well, as I no longer fully stand by it. It took a lot of time reprocessing & deprogramming, but I have been able to better understand the situation, it's complexities, & most importantly, how I was being manipulated.

One thing I mentioned was how my former group of friends handled the attempt & finding me. This group consisted of people I knew as Paco, Aztec, Ghost, & Rayne (Ghost is the only member of that group I am still on good terms with).

I do not currently & do not remember if I ever did have Rayne or Aztec's messages with Zena & Poppy from this time. I only saw Poppy's DM's with Paco & the group DMs between Ghost, Zena, & Poppy, as well as the posts in the Cafe. So I need to stress that I am working on those DMs/posts, the claims made by Rayne as to her actions, & the claims made by Poppy & Zena as to the other's actions.

I do not know for sure how much of an attempt was made to reach them. I don't know if it matters honestly. I know Paco's message to Poppy from the night before still sticks with me as being woefully inadequate for the severity of the situation. I know that Poppy & Zena ignoring messages from everyone including myself made the situation much worse than it should have been. I know that everyone had information that should have led them to act quicker. But I also know that I made mistakes & refused to take care of myself, make boundaries, or listen to those around me.

Ultimately, based solely on the way Zena & Poppy were acting - their general distancing from & disinterest in me, their consistent lack of responsiveness to messages from anyone (IIRC Rayne, a person they called a close friend, invited to their house, & offered to host as a streamer on their site, had been ignored by both of them for months up to this point), & their defensiveness whenever anyone tried to work out these issues with them - make it perfectly clear that no one had any reason to trust their ability to act quickly or to take the situation seriously.

Especially because they didn't.

They treated me like I was just attention seeking & faking it. They made me feel guilty for putting them through the inconvenience of dealing with me. They neglected our relationships & refused to hear any conversations about it until my mental state was so fucked I felt I had no other way out. And because my friends weren't experienced enough to know that it's not always a good idea to just assert to an abuse victim that they're being abused, lest they sink deeper into the situation as a defense mechanism, the attempts at getting me to realize what was happening only served to worsen the spiraling. I don't say that to blame them for trying, it's just my read of what happened based on my processing.

What I Stand By

This isn't a full retraction. There are things in that document that are true or are accurate representations of my feelings on things.

The SA Allegation

I know most will never believe me, but I didn't make up my story regarding one of the members of the Cafe. That said... I do regret including it. I went back & forth on it for a while. A friend of mine suggested that it could be therapuetic to include it but cautioned me as well. Poppy pushed me to include it in calls we'd had, despite my insistence that it not be used as a way to attack anyone.

I wasn't trying to avoid talking about it (& to be clear I am glad that I'm no longer keeping it bottled up) but I never wanted it to be a thing used against anyone. Had I known that one side was going to weaponize the drama sphere to relentlessly bully me for speaking about it & use frankly concerning rhetoric to try & dismiss me & that the other was going to weaponize it to paint someone who did something really inappropriate due to messy boundaries, lacking/miscommunication, & an admittedly weird social atmosphere as a rapist... I'd have just stayed quiet or at most kept it between me & an IRL friend I trust. While I do believe that it contributed to my overall mental state, it was Poppy & Zena who ultimately placed the suggestion with me that it was one of the top contributing factors & basically responsible for my attempt.

I don't say this to ask for sympathy. I don't need or want people to think about this in any particular way. I don't want anything from anyone other than I want people to leave that person alone. They are not a rapist, or a sexual predator, or anything like that. They did a thing that I would categorically classify as an instance of sexual assault, & I will stand by that definition, but goddamn it I never wanted anyone to go after them for it or go any further than the acknowledgement. I just want people to believe me, nothing more. I don't want anything to be done about it. I just want to stop being treated like I faked it & I want to stop being made fun of for it.

I'm so sorry that I ever allowed it to go unchecked that the rapist branding was ever being applied to them for that. Please leave them alone.

The Weird Social Atmosphere

We really did have a strange social atmosphere where a ton of things were normalized that never should have been. The sexually charged chats & the meangirling that led to cliquey behavior were all fairly common in both servers & I reflect on it as being inappropriate & not something I wish to recreate.

My Treatment

I was treated like utter dogshit during all of this. From before the document's publication all the way until now, I feel that my treatment was unjust for the circumstances I was put in. I did say & do some horrible things. And I will apologize to the people I hurt should they ask me to. But I was used by everyone in this. As a shield by Zena & Poppy & as a punching bag by everyone else.

It isn't right. I didn't deserve any of that. For all the harm that I did, jesus christ I didn't deserve to be treated that way.

TL;DR

I have privated my video & will be removing my document regarding my attempt because, while I stand by the following elements of it:

- The social atmosphere was kind of toxic & weird

- I never made up the event I classify as SA (though I never wanted it to be used as a bludgeon or used to brand the other party as a rapist like Poppy did)

- I was treated like garbage unjustly from the anti Z&P crowd

...the document is built upon notions that were fed to me by Zena & Poppy because I didn't have all of the information & wasn't in a headspace to really question their framing of things, & because the attempts my former social group made to pull me out weren't effective to where I was in the process of it all, I was pushed further towards them out of fear & insecurity. It is for these reasons that I am retracting the document as a unit while clarifying & standing by the true statements or accurate reflections of my feelings it contained.

#saige #saige alexis #saigealexis #saigealexis10 #saigealexisonyt #vividfae #vividfaegarden #zenaandpoppyonyt #zenaandpoppyonyoutube #poppy and zena #zena and poppy #zena #poppyandzena #poppy #trans girl therapist

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thisbelongsto-nohbodys · 1 year ago

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Sasha: -Thinking- Hear me out on this weird crazy theory.

Anne: -Looks up from her laptop- Alright I’ll bite what’s this theory you have.

Sasha: …. If you’re the next god to take D3’s place does that mean by change when you take over you also could potentially turn yourself into a cat form?

Marcy: Oh yeah that would be interesting to know.

Anne: ….. -Whispers- That would be so awesome.

~Later That Night~

D3: …. You want to know if you can be a shapeshifter like I am when you take my place?

Anne: …. Yes.

D3: …. Yes.

Anne: Sweeeeet.

I thought the ask was about Anne being a catgirl and thus this.

#ask #amphibia #sashannarcy #sketch

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dorothylarouge · 1 month ago

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Ayesha Khalil Ghassan Al-Ghazzawi

Catgirl or Miss Kitty or Bastet or Huriah

She’s had a hard time picking a codename, all right?

Birthday: April 25

Hometown: Chicago, IL

Ayesha Al-Ghazzawi always knew she was different from other kids. She just thought, when she realized she was trans, that this was the extent of it. How little she knew.

Ayesha was born Dallas, Texas, the child of a Palestinian academic and a Kuwaiti oil heiress. The family moved to Chicago when Ayesha was six years old, and it was around this time that Ayesha began to identify herself as a girl. Thus the activation of Ayesha’s supergene was delayed until she began hormone therapy. Once it did activate, Ayesha was astonished to find that she could now transform into a catlike form, complete with sharp teeth and claws. At first unable to control the transformation, Ayesha enrolled in the Kirby Beck Shuster Academy for Extraordinary Young People, although she is still unsure if a superheroic career is something she wants to commit to. This lack of commitment is reflected in her flakiness over her own codename. However, Ayesha is reliable in a pinch and her teammates in the Young Brigadiers have come to trust and depend on her.

Ayesha has grown especially close to her teammate Rachel Rahabi, alias Roadrunner, although the two have agreed to keep their relationship on the DL until the homecoming dance. This has created something of an awkward situation, as her other teammate Paco nurses a very obvious crush on Ayesha, which she isn’t sure how to dissuade. Teenagers are complicated, man.

Ayesha has the ability to transform into a catlike, furry form. This form possesses many catlike powers, including enhanced senses, agility, and reflexes, as well as sharp teeth and claws she can use for combat.

#hero school

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corvidcrowned · 3 months ago

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Carmilla having a black cat form is so good not only bc of the interesting metaphor with cats being associated with femininity specifically witches and thus being persecuted and feared for nothing but the crime of human superstition and tradition and how that relates to Carmilla’s lesbianism but also bc ooogh…… big fuckin kitty…….. weird scary catgirl vampire………… cause like Carmilla is supposed to look very human aside from her fangs but what if she’s really subtly inhuman in ways you’d only notice after hanging out with her for a while like Laura does yk….. Carmilla stepping into the sunny outdoors and her pupils narrow into slits. Carmilla second eyelid that Laura only just noticed by watching her blink. Carmilla functioning kinda like Skyrim vampires where going too long without feeding causes her to look steadily less human. ghghgh idk. I like when a girl is a creature

#Laura gets giant cat cuddles… Carmilla purring like a generator…..#Carmilla #im normal guys

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catfox1310 · 7 months ago

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Humanity's Hope AU: The Magical Girls

So I haven't really expanded upon this AU in a while, so I figured I should.

This is, of course, a sequel to the last post, I highly recommend people to read it as it is a basic introduction to the AU.

This post meanwhile will introduce you to all seven magical girls that were chosen by Tikal to defeat Dr Ivo Robotnik. All of whom happen to be various human girls (Well, mostly human) from the Sonic franchise.

Without further ado, here they all are:

#1: Maria Robotnik

Our main protagonist in this AU. 50 years ago, she used to be a termally-ill girl living on the Space Colony ARK before Tikal chose her to become a magical girl, now she's cured, in a different time period, and is determined to protect humanity from her evil cousin.

The Knuckles Clan knows her better as "The Angel From The Stars" due to her depiction on an aincent murial on Angel Island. Funny enough, Maria's grandfather, Gerald Robotnik, never noticed the similarites on said murial when he visited the island for research purposes.

Her companion is, of course, Shadow The Hedgehog, his creation was assisted by Tikal in this universe, though his design is mostly the same as his canon counterpart. (This means that the Black Arms don't exist in this AU)

#2: Hope Kintobor

Maria's first cousin once removed who comes from a future where Eggman took over the world, after realizing the truth behind what had really happened to Earth, she became scared. Luckly, Tikal came and gave her a chance to set things right. Thus, she was sent back in time to stop Ivo.

She encountered Maria shortly after she was sent back in time, and is heavily featured in this AU's version of Sonic 2, where she teamed up with Maria when their new friend, Miles "Tails" Prower, was capured by Eggman.

Her companion is Amy Rose, who first encountered Maria in this AU's version of Sonic CD. Since Hope needed a companion, she decided to become one.

#3: Sara

The daughter of the President of the Land of the Sky, while this feisty catgirl may not seem like the ideal choice for a magical girl. Tikal knows that deep down, she's secretly a kind person. And she truly shows it to her fellow magical girls. Other than that, she's also quite powerful in her magical girl form.

Her companion is Knuckles The Edichna, Knuckles doesn't need to guard the Master Emerald since Eggman already stole it in this AU. Much like his OVA counterpart, he's a friendly individual, especially to Maria since he knows about the Angel Island murial.

#4 and #5: Helen and Frances

A parapalegic girl and an adventurous girl who are classmates at Station Square Elementary School. And apart of a friend group consisting of them, Chris Thorndyke, and Danny.

Maria, who enrolled into Station Square Elementary in order to avoid suspicion and to gain knowlege about Earth in the present day, ended up being befriended by the two girls.

At first, Maria had no idea that they were magical girls just like her. But when Eggman attacked the school, Helen and Frances revealed their magical girl selves. In fact, the reason why they befrended her was because they knew Maria was a magical girl. (Tikal told them that)

One notible thing about Helen's magical girl form is that she is able to stand and walk perfectly fine, though it reverts once she turns back to normal.

Their companions are the blue hedgehog and twin-tailed fox duo themselves, Sonic and Tails. It's worth mentioning that both of them idolize Maria and wish to be just as cool as she is.

#6: Princess Elise The Third

The current ruler of Soleanna. Tikal gave her a Blast Processor wand when she was 7. However, while testing out her powers with her father, The Duke of Soleanna, an accident had occured, killing her father and causing her immense greif. Because of this, she would become afraid of her own powers, and never use them again.

However, all of this would change when she was 17, Maria would visit Soleanna and encounter the princess. Maria knew that something was troubling her and wanted to help her. After Elise talked to Maria about her fear, Maria wanted to assure the princess that she doesn't need to be afraid and that she can use her powers for good.

It just so happens that Eggman was planning to attack Soleanna because Elise had a chaos emerald in her possession. Now it was time to prove Maria was right.

Elise's companion is Silver, who took the princess under his wing after the accident ten years ago. He also knows about Eggman's quest for world domination, since he came from the same future that Hope was from, they never met during that time though.

#7: Sage Robotnik

A humanoid AI girl, originally created by Dr Eggman in an attempt to create a "Maria" of his own, she suddenly ended up going beyond her programming, and began to wonder what her pourpose and identitiy even were.

While wandering outside, she would encounter the magical girl group, she knew that they were the enemies of her father, but was suprised at how friendly they were to her. Everyone would then become shocked when Tikal chose her as the seventh magical girl, and Eggman had no idea about this.

This caused her to become even more conflicted about whose side she should be on. But she would make her final decision once Eggman caused destruction in Station Square.

Seeing how horrible her father really was, she finally had the objective she was searching for: Defeat Eggman and protect the world.

In this AU, Eggman created Sage a lot earlier than in canon, around the time this AU's events of Sonic Adventure took place. (And yes, Eggman is the one who destroys Station Square instead of Chaos)

Her companions are Mighty and Ray, for the same reason as Amy. This also makes her the only magical girl with two companions.

Bonus Magical Girl: Eggette

A mysterious girl who somewhat resembles a female Ivo, she's a Dark Magical Girl who is Maria's rival, and later Sage's rival as well. It's unknown what her goals are or if she's connected to Eggman at all, but Tikal knows that she didn't give her a wand.

Eggette pretty much has a similar role to Sonic 3 Knuckles in this AU.

Her companion is Fang the Sniper/Nack The Weasel, but sometimes Fang/Nack battles the girls on his own too.

Possible Shadow counterpart to Maria?

Since this AU will also adapt Sonic Adventure 2. It made me wonder if Maria should have a Shadow counterpart of her own to replace Shadow's role in the original game. Also acting as a Dark Magical Girl like Eggette.

I don't think it's entirely nessicary, since Sage is the closest thing Maria has to a 'Shadow' counterpart. And Eggette can fill the Sonic/Shadow "boss battles" in the OG as well. But It could still be done if people are interested in that idea.

And that's all I have for this. If anyone has any questions about the AU, or whatever else about it, let me know! I'd love to hear em.

#Sonic The Hedgehog #maria robotnik #hope kintobor #sonic ova #Sonic X #princess elise #sage the ai #sth au #Sonic The Hedgehog AU #Humanity's Hope AU

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