Points of Canon: S4x10 - Rainbow Falls

In which we once again observe that Wonderbolts are jerks.

• This is the episode in which Bulk Biceps’ name is settled, as previously he was never named on screen.
• As a side note, there’s an interesting house on a hill overlooking all of Ponyville. That’s not the first time it shows up. I wonder, who lives there?
• “I like your attitude, Bulk Biceps, but it’s gonna take more than muscles and Yeah!s to get us to the Equestria Games! We are the Aerial Relay team, and it’s up to the three of us to make sure that we qualify at the tryouts.” Ponyville is fielding a team for Equestria games at least in the Aerial Relay competition, and possibly others, and Rainbow is on this particular one. I wonder, is there a solo pegasus competition? I don’t remember one. If one existed, surely that’s what Rainbow would be trying for? Thunderlane later states that one is only permitted to compete in a single event, and that he is competing with the air sprinters, who are also a team…
• Actually, have we seen cheerleader outfits prior to this episode? In any case, they are front and center here and cheerleading is a thing. Notably, cheerleading is an American thing, rather uncommon the rest of our world over, and has a known history which is not very long. Notice that with ponies, we primarily see adults cheerleading, rather than schoolchildren.
• “And after that, for when we win gold medals in the Equestria Games!” Notice that rodeo competitions give one ribbons, with blue being the highest rated, and so do most (but not all) of the competitions in Equestria we have seen, but Equestria Games have gold medals.
• “And we will make it to the Games once everypony carbo-loads on my Apple Brown Bettys.” Notice that for humans, pastries like these would not be the recommended way to carbo-load.
• Rainbow Falls has its own flag – a four-color rainbow in an inverted-U shape on white. Where did they lose the rest of the typical pony six-color rainbow is anyone’s guess.
• There’s a griffon team at the tryouts, which later qualifies with everyone else.
• Large chunks of Ponyville pegasi population appear to be there as well, doing stretches. Presumably, they’re the air sprinters and possibly the teams for other aerial events.
• So which region in Equestria has a stylized letter “R” in orange on brown for their flag?
• “Hah, nice to see there’s some real competition here!” For once, Wonderbolts recognize Rainbow, which is kind of a change.
• There are three ponies wearing headphones with a microphone in the vicinity of the Cloudsdale team.
• While Spitfire’s glasses have visible ear loops – which never even reach the ears – Fleetfoot’s glasses for some reason do not.
• Applejack uses her tail to toss the brown bettys into the air. She is not very precise with it, however.
• “Okay, the passing of the baton needs to be seamless.” While Rainbow calls it a baton, it is most definitely not a stick – but rather, something derived from a horseshoe.
• “And it’s not like anypony else has color-coordinated horseshoes or anything!” When Rainbow is saying that, she is pointing at her left hind leg. I wonder, does it mean there’s an unseen horseshoe on it?
• There is a rather large amount of stable rainbow rings in the air as course obstacles and used for other purposes – one is used as the base for the finish line. I wonder how exactly does this work, cause they’re obviously not clouds.
• Notice that Soarin clipped his wing on a ring. See the origins of his Wonderbolt callsign nickname in Newbie Dash.
• Soarin’s wing looks dislocated. We do know pegasi have some rather complicated bones in there…
• Rainbow Falls medical cart has a rainbow on it.
• “You know, you grew up in Cloudsdale.” “So that means you could fly with us.” This would mean that only the current residents and natively born are permitted to compete for their respective region.
• While training, the Wonderbolts use muzzle feed bags for snacking. Just catching stuff thrown into the air wasn’t enough for them, apparently.
• The pit stop scene involves the Wonderbolt support crew using powered polishing brushes with an obvious electric wire, as well as a hair dryer – likewise, with a wire.
• Bulk Biceps is lifting a barbell marked 6200. Even assuming these are pounds and apply to the whole thing, rather than each ball individually, that’s ~2812 kg and a bloody lot, especially for a pegasus.
• “Woe is me! Oooh!” As Rainbow starts her performance, Minuette, Lemon Hearts and Twinkleshine run up. Long time no see, girls! Are you with the Canterlot team?
• “I, uh, tripped on a-a foam hoof…” Where would she even find a foam hoof, and what would one be used for?
• Notice that Rainbow is using a sitting wheelchair for this.
• “We’ll take care of you… or, at least, somepony will. Like, a medic or-or a doctor, or a nurse.” Which clearly implies Fluttershy is not qualified as any of the three.
• The Rainbow Falls hospital is equipped with fading-beam CRT heart monitors. Well, we already knew ponies had those for a while, they turn up on cutie marks on adult doctors…
• “So, the medic pony isn’t sure what’s wrong with you.” I wonder why they use the word “medic” this particular time and almost never otherwise.
• Derpy gets to be Rainbow’s replacement. I wonder, though, why didn’t any other team want her? There’s got to be a marathon competition.

• This is the dialog that anyone writing anything about Wonderbolts needs to print and hang on a wall:

  Soarin: You sure have nice friends. Nopony’s been by to visit me. Eh, too busy practicing, I guess.
  Rainbow Dash: Well, hopefully your wing will be better soon.
  Soarin: Oh, it’s fine. I-I’m just keeping it warm in case my team wants me back.
  Rainbow Dash: What do you mean, ‘wants you back’? Spitfire and Fleetfoot told me that you were still too injured to fly!
  Soarin: And they told me that they were worried I wouldn’t be one hundred percent by the tryouts, so they were going with somepony else!
  Rainbow Dash: That somepony else was me! Until, uh, well, until I got hurt, that is.

  Notice that they didn’t even tell him who would be replacing him.

• And that’s our second Chest Of Harmony Key epiphany: Rainbow Dash gives up her ambition for friendship, which motivates Spitfire to do… something. We don’t exactly see Spitfire being significantly more friendly afterwards, but the pin she gives Rainbow in the end as her souvenir clearly counts as the key.
• “Even with Rainbow Dash’s replacement, I don’t think we’re gonna qualify. No offense.” Say it, Twilight. Say her name. I dare you.
• “Maybe my Equestria Games uniforms will still be in style next year.” I wonder, what sort of occasion would Rarity want to use them for next year? Could it be the next Equestria Games? If not, what else?
• Spitfire is drinking from what has to be a PET bottle. PET is a plastic patented in 1941 in our world, and PET bottles did not become widespread at least until the 80s.
• Notice that Soarin is present, sans bandages, during Rainbow’s reveal, which would require him to be in on that when she left the hospital.
• “You may be a winning team, but you’re still not the kind of team I wanna be a part of.” Remember those words, Rainbow. You’ll be eating them in less than two seasons.
• Notice that the referee overseeing the tryout competition is a unicorn standing on a rainbow. And operating a huge mechanical stopwatch. While the actual lines spoken say that Ponyville must be one of the first four teams to complete the relay to qualify, the presented cutoff seems to be a certain numeric time instead – the stopwatch is marked up with a section of the circle painted red. Was there a rule change, or was Rainbow just wrong?
• Qualifying already involves an award – some kind of medal with rainbow and golden wings. Like most pony medals seen in modern day, this one is worn around the neck.
• Derpy gets on the group shot with the Ponyville relay team and the Mane 6, which is probably her consolation prize.
• “Oh, I can’t deny it. I love to win! But if I ever gotta choose between winning and being loyal to my friends, I’m always gonna choose my friends. ’Cause as much as I love winning, I love them waaaaay more.” Both the text and the photo Rainbow adds to the Friendship Journal are present in the printed Journal verbatim. But the actual phrase Rainbow is narrating is on the opposite side of the page from where she is writing, and completes, rather than begins her entry. The obvious scene break would permit the actual writing to occur later than the rest of the episode, but Rainbow is still wearing the award she got for the qualification, so it can’t be very long.

Not a particularly big haul this time.

Comments

[Trick Question]

derpicdn.net/img/2016/5/7/1148513/full.png

(I don't really agree, but the image is still hilar.)

[Mitch H]

So which region in Equestria has a stylized letter “R” in orange on brown for their flag?

Rock Valley! Actually, it sounds like there's at least one secondary canon source which refers to Pinkie's hometown as 'Rockville'.

[FanOfMostEverything]

There is a rather large amount of stable rainbow rings in the air as course obstacles and used for other purposes – one is used as the base for the finish line. I wonder how exactly does this work, cause they’re obviously not clouds.

Well, we've seen liquid rainbow several times at this point, so solid rainbow doesn't seem too farfetched. What gets me is how they're neutrally buoyant in air.

“I, uh, tripped on a-a foam hoof…” Where would she even find a foam hoof, and what would one be used for?

Presumably as cheering aids for those who find foam fingers unsettling.

“We’ll take care of you… or, at least, somepony will. Like, a medic or-or a doctor, or a nurse.” Which clearly implies Fluttershy is not qualified as any of the three.

Not for ponies, anyway.

Derpy gets to be Rainbow’s replacement. I wonder, though, why didn’t any other team want her? There’s got to be a marathon competition.

I'd suspect she was the designated stand-in, but going by Fluttershy's "We even found a replacement," I think she was planning on just watching before volunteering to fill in for Dash.

Yeah, the Wonderbolts don't exactly have the friendliest work environment. Remember how Wind Rider will insist that subterfuge is just part of the game.

With regards to the stopwatch, that mark could indicate the current fourth place team's time. Granted, that would require adjustable paint, but it's not entirely impossible.

[Trick Question]

Unrelated.

{ x | x ∉ x } is not a set because Unrestricted Comprehension is false. { x ∈ S | x ∉ x }, for a set (or class) S, is, in fact, a set (or class) by Separation. This is true even if you omit Foundation and Extensionality: note that proper classes cannot be contained in other proper classes, so there is no way for a proper class to contain itself.

[Meta Four]

• “I, uh, tripped on a-a foam hoof…” Where would she even find a foam hoof, and what would one be used for?

Probably a variant on those foam fingers we’ve seen before.

[Oliver]

4715044

Seriously?

[Trick Question]

4715054
Yes, seriously. That's the solution to the paradox.

Russell's Paradox isn't a paradox, it's a proof that you can't perform set operations on the class of all sets. The class of all sets isn't a set in the same sense as the sets it contains, in other words.

[Oliver]

4715081

Consider that I have explicitly used the formulation of the paradox that would be familiar, the shortest one I could fit into the oneliner, anyway. Would I have used this formulation without being already aware that this paradox has a proper solution?

[Trick Question]

4715085
I've known people who would, yep.

When you phrase it as a question, it invites a solution. No offense was intended by my response, and I apologize if you felt I was being condescending or rude.

[Oliver]

4715091

None taken. To be honest, the way you phrased it felt more like you didn’t understand the reference.

[Trick Question]

4715094
I probably wouldn't have referred to Unrestricted Comprehension if I didn't understand the source, but it's fine.

My favorite ordinal is omega Church-Kleene.

[Oliver]

4715096

Being a sociologist, I am surely permitted to let this one float over my head. :)

[Trick Question]

4715112
Being an avid practitioner of useless math, I shall not allow it.

Ordinals form the backbone of sets in set theory:

{} = 0 (empty set)
{{}} = { 0 } = 1
{{},{{}}} = { 0 , 1 } = 2
{ 0, 1, 2 } = 3
{ 0, 1, 2, 3 ... } = ω (all of them)

Each ordinal is equal to the set of ordinals less than it, which is convenient because we can define < to simply be ∈.

Ordinals are a kind of number that represent all the ways in which things can be well-ordered. A well-ordering is a discrete order that goes in one direction. Formally, a well-ordering is an order where for any set of well-ordered things, there is always a least element. The integers aren't well-ordered, because (for one) the set of negative integers has no least element. The rationals between -1 and +1 aren't well-ordered: consider the set of rationals greater than zero but less than or equal to one (there's no smallest positive rational number).

But the natural numbers are ordinals. You might even think they're the only ordinals, but we can go further than this. However, first we need to recognize that the natural numbers are also cardinals, which is a different kind of number. Realizing cardinal and ordinal numbers aren't the same thing is what led to the birth of set theory!

Cardinal numbers represent the amount of something, while ordinals represent the steps you take to get there. For example, children learn ordinal numbers before cardinal numbers, because they learn to count: zero, one, two, three, four. Four is the fifth number, and the ordinal name "fifth" is another name for the cardinal name "four".

Next, children learn cardinal numbers. Four is four because if you have four fingers and four marbles, you can match one finger to each marble without leaving any fingers or marbles left over.

It turns out that infinity is different. There are lots of ways you can well-order the same infinite "number" of things, but infinities also come in different sizes. There are "more" real numbers than natural numbers, because there is no way to match reals to naturals one-to-one without leaving any reals unpaired. They have different cardinalities.

But what about ordinalities? Let's just look at the number of ways we can order the natural numbers. We can keep going from omega! We'll use the following rule:

Successor ordinal: x+1 = {x union {x}}
Limit ordinal: x = {union of all the stuff less than x}

Note that limit ordinals don't have a predecessor.

So far I've given you two limit ordinals: 0 and ω. We can continue to ω + 1 by adding {ω} to the set ω:

ω + 1 = { 0, 1, 2, ... ω }

This is like "forever and a day". It's still a well-ordering: given any nonempty set of these ordinals, one of them has to come first, and you can always tell which one it is. You can imagine it like an ordering of the naturals where you remove all the labels—like you have an infinite number of unlabeled ping-pong balls in a row, starting from one by itself—and then you start a second row by removing one of them (which doesn't change how the first row is ordered at all) and start a new row where that new row comes "after" anything else.

For a concrete example, imagine ordering the naturals in their usual ordering, except 45 comes after every other number. You can still tell which numbers come first, but even if we remove the labels like '45' this is a distinctly different ordering. It's LONGER, but it isn't MORE. The length is bigger but the size is the same.

We can keep counting ω + 1, ω + 2, ω + 3... ω + ω (or ω*2) is the third limit ordinal, but it's still a 'countable' infinity (still an ordering of the naturals—you can pair each natural with one so it can be 'counted'): just imagine all the even numbers appearing before all the odd numbers. You can go to ω², ω^ω, and even ω^ω^ω^ω^... where there are ω many ^ω's! This last number is called epsilon-naught or ε₀, and it's the first number where x = ω^x. But that's still countable and you can keep going much further.

So how many ways CAN you arrange the natural numbers before the ordinals become uncountable? Well, with ordinals, there's always a first. The first uncountable ordinal must therefore be the set of all countable ordinals (and we call it ω₁). So the number of ways you can arrange the natural numbers is uncountable, even though each arrangement is a countable arrangement. This is similar to how there are infinitely many finite numbers, but each number is finite.

Now we finally come to the Church-Kleene ordinal. ε₀ may seem a little crazy, but it's still small potatoes in the world of large countable ordinals. You can invent crazier and crazier ways of explicitly defining larger and larger ordinals, all of which are still countable. But think about the act of using symbols to explicitly define how to add ordinals to make larger ordinals: there are finitely many symbols, and your definitions all have to be finite in length. That's a countably infinite number of possible definitions.

This means we have to run out of definitions before we hit the uncountable realm, which means there is an ordinal which is still countable that is the first ordinal that can't be explicitly defined by using direct methods like repeated addition, multiplication, and other crazier functions we invent to make stuff big. It's still much smaller than ω₁, but it behaves a lot like ω₁ in computational theory because it isn't computable. It's the first nonrecursive ordinal, and we call it the Church-Kleene ordinal or ω₁^CK.

derpicdn.net/img/2012/10/15/123636/full.png

[howard035]

I always felt this episode was a little unfair to Rainbow, since her other two teammates seemed like they spent more time flirting with each other than practicing.

Brown Betties work as a pony carbo load because of their much higher sugar requirements.

I forgot that the Wonderbolts didn't even visit one of their own team members in the hospital. Yeah Spitfire had a friendship epiphany at the end of this episode, but Discord had a friendship epiphany and later tried to throw a pony into a sock-puppet dimension. Perhaps the epiphany is a moment of clarity rather than a permanent change in outlook.

4715042 Wind Rider definitely feels more like the one who got caught than someone substantially different from the other Wonderbolts.

[hawthornbunny]

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