Sugar, don't kid yourself, Applejack was first, you're the mistress, she's the real wife.

This is one of the hottest things I've ever seen you produce. Fantastic work.

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Damn, Applejack's got 2 numbers for punching Rainbow Dash in the box. So she's 1 and 3

The fucking description had my laughing, hats off to you sir

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It is amazing to have you back Shakes. Amazing story, just amazing, i love the interaction, the twists & turns of it all. It is just so perfect it fits like hand in glove. And what a sweet story it was, i love everything about this fic.

[The Apple Bloom part was the Best.]

And this is them moment Butter Mac was concived, a Little Brother for Apple Bloom to love & make love with. To take care off, then be married off with. Make love with & making love balls with. Keep the Apple Family tradition of incest strong for yet another generation.

https://derpibooru.org/images/2955014?q=oc%3Abutter+mac
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Your stories are always entertaining.

"So is spaghetti until it gets wet." best. line. ever!

Welcom back Shake!

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Nut!

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>tfw u nut but Applejack keep sucking
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Not really into Applecest but decided to read it at random and was very pleased what I read. Good writing! Well written drama about jealousy and sharing.

And despite it being her own wedding, Sugar Belle suddenly felt very out of place.

That's one line that stuck out. Perfect. Only upon coming to the comment section I realized it was written by Shakes, so it's nice to see you haven't lost your touch. Good stuff.

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it's nice to see you haven't lost your touch.

Well, that was fucking hot. And just how often does Dash get punched in the box anyway?

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This. This is better than anything I've ever written. Enjoyable and, to a certain extent, believable. With their parents dying when they were so young, it feels almost normal that they would have turned to each other for comfort.

If I had to make any compliant at all... it's the timeline. As what was pointed out to me in one of my previously published but now deleted stories, the 9 seasons of MLP take place over a span of 3 years. Apple Bloom would have been 11 or 12 during Big Mac's wedding. I've also never known an 18-year-old to go to a sleepover. A party yes, but not a sleepover.

Otherwise.... prefect.

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I've also never known an 18-year-old to go to a sleepover.

It's called a cover story.

It's actually one of those grown-up slumber parties that Sweetie Belle hosts with Moscato, inviting Scootaloo and Babs and the other ladies, where Sweetie tries to push her Multi-Level-Marketing products from Bad Dragon®.

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Okay, now that's hot. Keep up the good work

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Keep up the good work.

Next up: Rainbow's box gets knocked!

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Next up: Rainbow's box gets knocked UP

Part of me wants an epilogue of Rainbow getting punched in the box.

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Even better.

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[Adult story embed hidden]

Pubic fluff on four legged ponies? Really? I’d prefer sone nice little teats.

Oh, and there was one spot where you used tis instead of his.

Overall, though, pretty great story. Enjoyed the comedic aspects. Especially drunk Rainbow. Time for an epilogue where she gets punched in the box. Repeatedly. By Big Mac’s dick. While protesting the whole time that she’s a lesbian and doesn’t even like dick. But secretly she’s loving every second of it. (With AJ and Sugar Belle supervising, of course.)

"Dammit Rainbow!" Mac cursed. "I'mma punch that mare in the box!"

I NEED to see this more

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Fixed!

Thank you!

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WHAT DOES KNOCKING TACOS MEAN?

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That's when you leave a negative Yelp review at a Mexican restaurant, right?

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... Well, it certainly means that now

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Don't tell Trixie that...

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.............. I dont think I want to know anymore

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I appreciate the correction, I'll be sure to apply it in the future.

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Someday I'll explain ordinal collapsing functions and you'll regret that fact.

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"NO REGERTS"

CHALLENGE ACCEPTED.

Newton believed reasoning about infinite groupings of things was impossible, based on the following argument he understood well as one of the discoverers of analytical calculus:

Imagine the nonzero naturals {1, 2, 3...} and the squares of those naturals {1, 4, 9...}. They can pair up one-to-one, with none left over, suggesting they are the same "size", in a sense. But you can also pair them up only with the corresponding number, leaving the other naturals untouched. If you do, and examine them in the usual order, you find that after taking 1, 100% of nonzero naturals are also perfect squares. When you hit 4, now 50% of them are perfect squares (1 and 4 are, 2 and 3 are not). At 9, 33.3~% are squares, and so on. The limit at infinity approaches 0%, which suggests all nonzero naturals dominate the perfect squares. As these appear inconsistent, the assumption we could compare them meaningfully is absurd.

Newton was a genius, but he was (fortunately, as his contributions to math were more important at the time) mistaken. It turns out that what we think of as natural numbers are actually two different things: cardinal numbers, and ordinal numbers. We confuse the two because these are identical for finite numbers, but when you hit infinity all the rules change.

Ordinality is how children first learn numbers: by counting. Zero, one, two, three, and so on, or as they are often called in this respect, first, second, third, fourth, and so on. You know you've reached three when you pass zero, one, and two (the three numbers which precede it). So if you follow this pattern, you can count objects.

Cardinality is "how many". This is the more intuitive idea, but learned by children after ordinality. For example, if I know I have five fingers, and there are some apples, I can tell if the number of apples is five if it is possible for me to "pair up" each finger with a single apple, with no two fingers pointing to the same apple, and with no fingers or apples left over. This is what we usually mean by "five". Simply counting to "sixth" (a.k.a. "five") is easier than pairing things up with your fingers, however, and it always works because the only way you get to the sixth number is if you have counted a total of five things. In other words, for finite naturals, every cardinal number corresponds to a unique ordinal number, and vice versa.

What Newton didn't understand was that these concepts are not as identical or stuck together as they seem: counting isn't just a calculus, it represents an idea that is very different, especially when you have infinitely many things. The set of squares is the same "size" as the set of naturals because there is at least one way to pair them up with none left unpaired. It doesn't matter that you can also pair them up and leave some missing, there just has to be some way to do it. If there is, the groups have the same cardinality—even if the way you chose to order them was different.

To explore this further, we need to define what ordinals are formally.

An ordinal is the order-type of a well-order. A well-order is a special kind of linear order. A linear order is a binary relation between some type and itself, with the following properties:

• An ordering: if A ≤ B and B ≤ C, then A ≤ C (transitivity).
• A total relationship: either A ≤ B or B ≤ A.
• Antisymmetric: if A ≤ B and B ≤ A, then A = B.
• Reflexive: A ≤ A. (This one isn't necessary because you can deduce it from totality and antisymmetry.)

A well-order is a linear order that is also well-founded: if you take some of the things (all different) together and look at them, one of them will always come before all the others.

For example, the integers ordered by ≤ are not well-ordered: if I consider, say, the negated prime numbers {-2, -3, -5... } then there is no "smallest" one. But the naturals in their usual ordering are well-ordered: for any subset of naturals, there will always be a "smallest" one. Note that well-ordering only goes in one direction: if I look at all the perfect squares, there's a smallest one (0) but no largest one. However, any sub group of squares will always have a smallest one.

This probably seems uninteresting, but something very special happens when we reach infinity. You see, there are many different ordinals you can construct from an infinite group of natural numbers, provided we move things around a little. Keep in mind that the labels like "5" or "sixth" aren't special, it's just what we use to describe them; what defines an infinite ordinal is how the objects are arranged, so try to imagine an infinite line of identical ping-pong balls with a starting point (which would be "0" if we used a typical label). They're all interchangeable; the only thing that makes them different is where they appear in the order. The infinite string of balls here is the smallest infinite ordinal, which we'll call ω (lowercase omega).

If we take out a ball, maybe the one at the seventh place (which might have been "6"), the ordered list of balls hasn't changed at all. It still has order-type ω. But what happens if we then put the ball we took out past the end of the infinite line?

One way to do this is to imagine we have a new number, X. And our ordering will be all the numbers in their usual order, followed by X. Or, alternatively, pretend the numbers are labeled as they normally are, and we literally remove 6 and "place it at the end". What this means is we're ordering the numbers like this:

Our revised "≤" now says A ≤ B is true, when and only when:

• A = B, or...
• B = 6, or...
• A ≠ B and B ≠ 6 and A ≤ B in the "usual ordering" for ≤ on naturals.

The question is now: is this a well-ordering? Well, based on the definition for "≤", if you have a subset of numbers there's still always a first number. Just apply the rules. Since "6" comes after all the others, it's only the first number if it's the only number you have. It's still a linear order, and there's still no way to build an infinite set for this "≤" where it gets smaller and smaller forever.

We call this new ordinal "ω+1". It's kind of like saying "forever and a day". Naturally, we can keep going. For example, consider the order type of the following "≤": all the odd numbers are followed by all the even numbers. It's still linear, and it's still a well-order, too (think about it: if I give you a bunch of numbers, you can always tell which comes first, even if it's an infinite number of them). We call this fella "ω+ω", or "ω*2". (We can't call it "2*ω", because that's the same as "ω": an infinite line of pairs of things is still an infinite line of things.)

You can even get really crazy with it. What about an infinite number of infinities, one after the other? That's ω*ω, or ω². And if you do that an infinite number of times, you can get to ω^ω. You can even have an infinite tower of ω exponents! This one is called ε₀. And all of these are possible just by rearranging the natural numbers. The length (ordinal) can get longer and longer, while the size (cardinal) remains the same. We call the "size" of the natural numbers ℵ₀ (aleph-null or aleph-zero).

It's worth mentioning here that there is only one way to get to any ordinal. Every ordinal is an extension of all the ordinals that came before it, in a linear fashion. To get to ε₀, you have to pass through ω^ω + ω*7 + 198, for example. So ordinals can be compared in a linear fashion by their lengths. If you chop an ordinal in half, the initial part is an ordinal and the final part is also an ordinal (sometimes the same ordinal, e.g. there's no way to chop ω in half without the final part being ω).

What makes this truly interesting is, counterintuitively, there are different infinite cardinal numbers! You just have to go a really, really long way to get to the next infinite cardinal. Since you can think of an ordinal as the first length that is longer than everything that comes before it, the first infinite ordinal that has no possible way to pair up with the natural numbers (the first "uncountable" ordinal) is the first ordinal that follows every possible ordinal that can pair up with the natural numbers. Eventually you run out of ways to rearrange the naturals to form new orders. The reason we know this is true is because we can prove that certain collections of things are "too big" to count. For example, the rational numbers, which are formed by putting an integer over a nonzero natural, e.g. "–3/12" or "0/444", can be counted. You just have to show progress where every rational eventually pairs up with a natural. You need to skip some that have already been counted, but this is possible to calculate. For example, our function may line up the rationals like: 0/1, 1/1, –1/1, (0/2), 1/2, (2/2), –1/2, (–2/2), (0/3), 1/3, 2/3, (3/3), and so on. If you name a natural, I can figure out what rational pairs up with it, and vice versa, with none left unpaired.

Rationals are "dense" in that between any two, there are an infinite number of rationals. Real numbers are like this too. But the real numbers are too "big" to be countable. There are several ways to prove this, and when Georg Cantor (not the Georg here on Fimfiction) found a proof, this was the birth of set theory, the first discovered foundational theory of all mathematics. I'm not going to bother proving it here, though, because ordinal collapsing functions are defined on countable ordinals. If you accept that there are ordinals which are uncountable, you'll be able to follow along.

Let's get back to ε₀, because ordinal collapsing functions are part of a quest to name as many countable ordinals as possible. This ordinal number is the first solution to the equation ω^α = α (remember, it's an infinite tower of ω exponents). It turns out that any function that takes ordinals to ordinals and always increases (called a "normal function") has "fixed points" where the function spits out the same thing that you input, just like ε₀. You can say ε₁ is the second time it happens, which is the supremum (first thing larger or equal to) of the sequence: 1, ε₀, ε₀^ε₀, and so on.

But aha! Notice that f(x) = ε_α is also a normal function (it takes ordinals to ordinals and always increases)! So we can take the fixed points of that function, and call the function that enumerates ε's fixed points ζ (zeta)... but now we have a problem, because we're introducing new symbols. Eventually we'll run out of names for things, so this doesn't get us as far as we'd like. It turns out we can get really creative about how we do this (one of the strongest is called the Veblen hierarchy), but if we want to define some really, really large (still countable!) ordinals, we need a different approach.

And this is what ordinal collapsing functions are for: an incredibly clever way of giving names to insanely large countable ordinals. The initial idea (which I'll refine in a moment) is that we define our function ψ(α) to be the first number you can't reach starting with {0, 1, and ω} and applying a finite number of sum, product, exponentiation, and ψ(α) itself operations (restricted to ordinals no larger than we can create this way), so the function is still well-defined. So ψ(0) is just ε₀, and ψ(1) is just ε₁, and for finite α, ψ(α) is just ε_α. None of this is helping us, but that's because I haven't introduced the trick.

The trick is we're going to cheat. We're going to borrow some ordinals that are uncountable and use those as labels, which is safe because we can't possibly get to an uncountable ordinal using this method.

To explain how we know this is safe, the first uncountable ordinal is usually called ω₁ or (for brevity) Ω. Since we can't "count" that high, there's no way to define it recursively. One way to realize this is every "name" for a countable ordinal we make has to be finite in length, and you can count every possible description that's finite using shortlex order. It's like dictionary ordering, except you check the length first. So all the strings you can make with {a, b, c} can be enumerated { (empty string), a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc, aaa, ... }, and thus are countable. If the total number of possible descriptions we can come up with is countable, the ordinal that maps to it is also countable. That special ordinal is called the Church-Kleene (clean-ee) ordinal, and it's the first ordinal we can't explicitly get to by defining all the ordinals below it—no matter how hard we try, we run out of labels before we reach Church-Kleene, and remember, that's still countable. So any uncountable ordinal we "borrow" is safe to work with, even though it's technically cheating (in math we'd say it's "impredicative", which is sort of an insult).

If we add Ω to the set, now ψ(α) is the first number you can't reach starting with {0, 1, ω, and Ω}. This doesn't change anything! At least, not initially. Although we can now generate numbers like Ω^Ω, which isn't even countable, remember the definition is "the first number we can't reach" which is still countable. So why does this trick help us at all? To see, we'll need to reach a point where we actually need it because our ψ(α) function will get "stuck" if we don't use the trick.

So let's see what happens if we continue naively. ψ(α) can continue into infinite ordinals, but the first fixed point of the ε function is where it will get stuck: ψ(ε) would be the ε'th epsilon number, or the first fixed point of the epsilon function, which I previously called ζ₀ (zeta-null). But our function gets stuck here, because we can't construct ζ₀ from our ψ collapsing function. So ψ(ε+1) is also ζ₀, and so is ψ(ε^ε), and even when we get to ζ itself: ψ(ζ₀) = ζ₀. ζ₀ is the first fixed point of our collapsing function, which seems to have taken a dump on our plans.

We can't move past it, at least not naively, because: ψ(ζ₀ + 1) = ζ₀, too! Remember, we can't construct ζ₀, so that's as far as we can go. But that's where using Ω can do something special. See, our function ψ doesn't have to be defined for every value, and we can talk about ψ(Ω) even though we haven't named all the ordinals below it yet (hell, even though we can't name all the ordinals before it, ever). This is why it's impredicative: we're using an ordinal that we haven't defined under this process to define a smaller ordinal (but we still can prove it exists, so it's only semi-cheating).

ψ(Ω) is the first ordinal we can't construct from {0, 1, ω, and Ω} and ψ(α), where α < something we can construct. This is still ζ₀, but something amazing happens when we hit ψ(Ω+1). Remember, we've shown that ψ(Ω) = ζ₀. Since Ω is in the initial set of numbers, ζ₀ can now be constructed because Ω < Ω + 1. This means the value defined by ψ(Ω) is now constructable (in a single step: just apply Ω to ψ). So ψ(Ω+1) = ζ₀ + 1! Okay, that's not impressive yet because we've made this ginormous function that required an uncountable ordinal just to name ζ₀ + 1, but it illustrates the trick.

So our function "unsticks" and can continue giving names to ordinals. The point where it finally gets super-stuck is when we get to ψ(Ω^ΩΩ...) with an infinite tower of Ω's. That ordinal is called the Bachmann-Howard ordinal. We can go past it by adding another ordinal that's too large, say, the initial ordinal of the third infinite cardinal, ω₂ (i.e. go up one more level), and we can keep taking larger ordinals to define smaller ones... but eventually we can't name any more large ordinals, so we can do something totally nuts and define the ordinal α we use to name ω_α by doing the same thing to it: make α large by using an ordinal collapsing function! Now ψ(α) is super sus, or "doubly impredicative". And we're still nowhere near the Church-Kleene ordinal, which again, is a countable ordinal (and thus smaller than almost all countable ordinals).

Final thought...

WHY??!?!!!!?

Well, it's fun to make numbers go up. But why do we care about large countable ordinals? One reason is that they can be used to define the strength of a proof system, usually one that's much weaker than set theory. For example, Peano arithmetic is a very simple model for arithmetic (it's a few axioms that define the natural numbers with addition and multiplication) in which you can't prove most things in math, but you can prove a lot of basic stuff. But in Peano arithmetic, you can't prove that there is a way to well-order natural numbers like ε₀. In fact, it's the first ordinal that Peano arithmetic can't prove exists, which means ε₀ measures how powerful Peano arithmetic is (which is not very powerful, as you might have guessed). For example, to prove the "hydra game" (see Wikipedia) always terminates, you need to show that each state of the game can be mapped to an ordinal under ε₀. This immediately tells us that Peano arithmetic isn't powerful enough to prove the hydra game always terminates.

Of course, there are more applications, most of them even geekier. But there you have it, a textbook-novel on your comment page because you dared me to.

I'm starting to like this RD-causing-incest-verse.

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This is a thing of beauty, and I understood a lot more of it than I expected to going in. It definitely helped that I have watched several videos on different orders and types of infinity, but you also have a very clear writing style. Well done!

I nearly died when Mac tried to assert a 1/3 claim to his own dick and was denied.

Hilarious.

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Sorry, it's easy to miss it.

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thumbs.gfycat.com/NewWhoppingArchaeocete-size_restricted.gif

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It's also worth mentioning that she's an ex-cultist who probably doesn't understand what normal is.

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I never considered that angle, but yeah, that makes sense too.

Applejack said, leaning forward nuzzle his scrotum again and give attention to each of his kiwi-sized testicles

I need to come up with a term for this kind of error...

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a missing Article (to).

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Eeyup! Looks to be the case

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Thanks. At the time of my first comment i had not noticed that there was one more chapter with two more of the (hilarious) callbacks.
I think i just brought down that "you have the smartest readers" myth.

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But now you learned.
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You have become a smart reader.