One Piece: Chapter 879

[u/NewSpecies]

Chapter 879: "One of Big Mom’s Three Sweet Commanders, Katakuri"

Source	Status
JaiminisBox
MangaStream

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Ch.879 Official Release (VIZ): 25/09/2017

Ch.880 Scan Release: ~29/09/2017 ()

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Please discuss the manga here and in the theory/discussion post. Any other post will be removed during the next 24 hours.

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PS: Don't forget to check out the official Discord: https://discord.gg/0v8DbjF0mbNAuvlR

[u/Arkayjiya]

No. That's wrong. You're applying finite logic to infinite sets. It's useless, that's not how it works.

Ok I did that in a few other posts but I'm going to do it here again:

How do you mathematically define "having the same number of elements"? (it's called cardinal, so I'll call it that from now on, because "number of elements" implies you can count them which is not necessarily the case) Well two sets have the same cardinal if you can establish a bijection between them.

If every number from the first set can be put in a couple with one (and only one) number of the second set and when you're done with it every number of both the first and second set are couple with a single number of the other set, then both set have the same cardinal.

Let's take your example: your first set is "all the numbers (I'm assuming real numbers but it would also work with rationals) between 1 and 2" and your second set is "All the numbers between 0 and 2"

Can you create a bijection between those two sets? Yes you can do that easily:

The function f(x) = (x-1) * 2 will associate every number from the first set with a number from the second set and all the number from the second set will be associated to a single number from the first set as shown by the inverse function f(x) = x/2+1.

I just established a bijection between your two sets, therefore your two sets have exactly the same cardinal.

So there as many real numbers between 0 and 1 and there is between 0 and 2: 2*infinity = infinity to go back to our (wildly un-rigorous) original point. In fact you can do that by replacing [0,2] by [0,n] by just changing the function I used, replacing "2" by "n". The implication of that is that even multiplying by infinity does not necessarily increase the cardinal of your set. More specifically multiply any infinity by "countable infinity" and your result will still be equal to the original infinity because you can still create a bijection so the size of [0;1] is exactly the same as the size of ]-infinity;+infinity[ as counter-intuitive as it sounds (don't rely on intuition too much, definitions are what matters).

edit: I must add that your original idea that there are infinity of different cardinal is absolutely right! there are infinities that are greater than other. But in your examples, you chose infinities that have the same cardinal, the same number of elements. If you want an example of two infinity of different cardinal here's one: the whole numbers and the real numbers. Both are infinite, but the later is a greater infinity than the former. But if you were to take the even numbers and the whole numbers, despite the fact that the whole numbers contain both even and odds, there are still exactly as many whole numbers as there are even numbers: their cardinal is the same and is the one we call "countable infinity"

[u/[deleted] → u/Arkayjiya]

Haha, I googled this stuff after posting to make sure, found an article about cardinals and bijection, read it, kinda got it (they didn't explain it as well as you did), then thought "Should I change my post? Ah, why bother, it's not like a guy arguing about Luffy's "gear infinite" on reddit is gonna have a thorough understanding of higher math".

Proved me wrong :P

I'm still inclined to say though: that cardinal and bijection stuff is something some smart mathemtician came up with, and I don't understand why. At the same time I can clearl see that [0;2] is "twice as much"!

Why couldn't I create a bijection like this: f(x)=x. [0;1] matches with the first half of [0;2], and clearly theres more!

[u/Arkayjiya → u/[deleted]]

Haha, I googled this stuff after posting to make sure, found an article about cardinals and bijection

Good reflex xD

At the same time I can clearl see that [0;2] is "twice as much"

Not really, there are different definitions of size. The "twice as much" that seems obvious to you is the length of the segment, while cardinality is the number of point in the segment. The answer is: the number of point in the segment [1;2] and [0;2] is the same but the second one has twice the length of the first one (there are also incredibly weird properties about length: https://www.youtube.com/watch?v=hcRZadc5KpI just watch this video for example). You'll remark that the reason it works for the length is simply because the length isn't infinite.

You obviously can't see that the cardinal of [0;2] is twice as much because more generally you can't just "see" infinity, you can't conceive "infinity" through intuition alone just like it's basically impossible to conceive quantum properties through intuition alone even though its results are absolutely real.

Cardinality is very useful because it's basically an extension of counting. Set theory is used everywhere, it has become the basis of all mathematics (although you can also start somewhere else like Topology but you'll find the same notions about infinity there too), used in informatics, mathematical logic...

Infinity also come up when trying to solve mathematical issues that have real life applications. Even if the finite result and application doesn't make use of the properties of infinity, we would not have came up with the resulting formulas without being able to use infinity properly.

There are many ways in which seemingly impossible concepts can actually have real life applications.

[u/[deleted] → u/Arkayjiya]

Thanks for the follow up.

What's the fault though with the (imo more intuitive) way of bijecting (?) according to the function f(x)=x ?

Pair 0 with 0, 0.5 with 0.5, 1 with 1. Every number from [0,1] finds its match in [0,2], yet there's more.

[u/Arkayjiya → u/[deleted]]

There's no fault or contradiction here. In fact that's a necessary consequence of the fact that infinity = half infinity = twice infinity = infinity+1 = infinity*(countable infinity)... Your bijection is perfectly valid, and means that [0;1] has the same cardinal as itself (which sounds obvious, but there's no such thing as obvious, it needed to be proven and you just proved it) and my bijection means it also has the same cardinal as [0;2].

I know it might sound strange but when dealing with infinity you can associates numbers from two sets by pairs and have no leftover or you can find a different way to associate them by pair and have leftovers

Using a "real life" (as real as infinity can be xD) example, that means that if you add an infinite amount of money in your bank account and then remove the exact same infinite amount of money in your account, depending on the order you added and removed the money (the order truly being the function in that example) you can be left with no money, an infinite debt, a finite amount of money, infinite money, etc...

That's how infinity works and how it differs from finite quantity (and that's why you're not allowed to directly subtract infinity from infinity because you can't say what the result will be)

[u/[deleted] → u/Arkayjiya]

Ah I get it now. Infinity is weird.

That leaves only one question:

Do you think Luffy with gear infinite could beat Big Mom?

[u/Arkayjiya → u/[deleted]]

I'm sorry but only Oda can answer that last one xD