Mathematical Girls - Gödel's Incompleteness Theorems

Spoiler for next chapter:

Spoiler

let x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1
Therefore, 0.999... = 1

maid: Math is fun, teach us more!

 

yeah, enjoy the math while it is still fun. once u graduate from highschool, u also graduate from fun math

False. High School Math is little more than entering numbers into a calculator. Boolean Logic, Infinity Theorems, that sort of stuff is fun. Sadly, most people already have their potential passion for Math ruined by that age, so your frustration is understandable. But give it a try some time without any need to be "good" at it. You'll be surprised at how fun and interesting Math can be. Might I suggest the Youtube Channel "Numberphile"?

"Mathematics itself can become a target of mathematical study" Exactly, and i know at least one philosophy ph.d. who was talking nonsense like "only philosophy can say anything about itself".
This series is going in interesting direction but i hope that "romance" won't be too irritating.

That's convincing and true, but the point is that we haven't really defined what 0.1717... is. Once we formally define what a decimal expansion like that represents plenty of arguments are available to show it equals 17/99.

I came into this completely expecting the author to know sweet fuck all about math, like how authors of manga based in science often know sweet fuck all about science, but was greatly surprised to discover that this author does indeed appear to know his stuff.

Spoiler for next chapter:

Well of course, as a connoisseur of Japanese math textbooks I was well aware . . . um, no, actually, I didn't know.

Thanks for the info, but why exactly would you expect people to realize that?

Well, i just checked author profile on mangaupdates and assumed that more people would check who the hell could make manga about mathematics.

This is the sequel to Suugaku girl, right?? So I'm confused about one thing, at the last chapter of the previous series, I thought Miruka confessed- the kiss and her saying for them 'to not remain partitioned'- and they became a couple or something? Soo, did they take a step back in the sequel or just ignore that part.. or what??

You do realize that the author of this manga is also an author of math textbooks?
 

Well of course, as a connoisseur of Japanese math textbooks I was well aware . . . um, no, actually, I didn't know.

Thanks for the info, but why exactly would you expect people to realize that?

The day I see discrete mathematics in a Manga...

This manga is really interesting, the author seems to know what he is doing. It made a person want to study mathematics, which speaks a lot about a manga.

Although the author does have an unfair advantage i.e. mathematics already very interesting for those who can enjoy this.

You do realize that the author of this manga is also an author of math textbooks?
 

This manga is really interesting, the author seems to know what he is doing. It made a person want to study mathematics, which speaks a lot about a manga.

Although the author does have an unfair advantage i.e. mathematics already very interesting for those who can enjoy this.

I read all three chapters, didn't retain a damn bit of it, read for the characters and the interactions. It is WAY too late in the night to try to learn math. Will try again tomorrow maybe.

Oddly enough I'm enjoying this manga tho? Odd that. 

OK, am I the only one amused by the fact that while two of the characters were going on about Piano's Axioms, the other one was playing a Peano?

MATH = Mental Abuse to Humans

thats about all i can understand right now. without the 5th i can imagine a little bit what it would look like but i think it kind of have to create a new of starting point for every step forward u take forward and then apply the first 4 axioms (0,1,2->1,2,3->2,3,4). I hope im close.

@Volandum- thank u for the explanation. As I am not a math student I am only capable of scratching the surface of that explanation. i can understand the basic gist of it ....i think. basicly its saying that every natural number have properties that is transitive from the designated starting point. so it basicly takes the first four axioms and say it must apply to the whole set since the first four only describes the relation of a certain number X to its X' within the set.

thats about all i can understand right now. without the 5th i can imagine a little bit what it would look like but i think it kind of have to create a new of starting point for every step forward u take forward and then apply the first 4 axioms (0,1,2->1,2,3->2,3,4). I hope im close.

Racky: good points and question. But remember 0 and 1 are just symbols, so when he says 1 I'll just read 0.

lastKANASHimi: The fifth Peano axiom is the induction axiom, the important one. (And also dangerous, because it requires second-order logic. Remember that identifying classes (which are equivalent to propositions) in first-order leads to Russell.) The other four just give you a set with successors which doesn't double back on itself. 

The full-power induction axiom says that any proposition true of the designated initial element and which is inherited by successors is true of every natural number. This tells you, among other things, that what you think of as the natural numbers (*, *', *'', ...) [here * is 1] is what you get. This isn't possible in first-order logic, so you end up with models of N that look like (0, 1, 2, ...) followed by as many copies of (... ξ-1, ξ, ξ+1, ...) as you like.

Why do we call it the induction axiom? Because what it says is that induction over the naturals is valid. So, are you familiar with standard induction over the naturals?

It was one of most terribly writen Peano axioms i've ever seen. Why not use standard notation with succesor function? Also if they stick with original version (0 is not natural number) they should keep it or use "modern" version. What kind of math teacher is he?

can someone explain to me the fith axiom? that one is the weirdest out of all of them

I wish they kept the old artist.

Chapter 1 has been re-released with the missing end-of-chapter page translated, thanks to Saiyaku and radioactivekitty.

 

I don't know the specifics of the argumentation of Godel's and Tarski's theorems so I hope this series will shed some light on the most heartbreaking theorem of Math. That's the closest to drama it will ever get.

The following is for those that want to learn in a bit more detail from a mathematics perspective.

As Volandum says, strictly speaking those theorems depend on the "encoding of sentences" you choose (you assign a number to each sentence), this may look quite arbitrary at first, so I would recommend knowing first about decidability, here is a link with further reading and references http://ncatlab.org/nlab/show/partial+recursive+function.

Afterwards you may learn how to actually encode finite combinatorics in the natural numbers and prove the theorems using self-referencial-like paradoxes. I like chapter 7 of poizat "Model theory" book.

Roughly Tarski undefinability comes from something like " this statement isn't true", Gödel first incompleteness from "this statement is unprovable" and gödel second incompleteness from assuming "peano axioms are consistent" is provable and repeating the arguments in gödel first incompleteness in a not too crazy "fake" model of numbers. The devil is in the details of course.

This explanation was a bit confusing...

I think the biggest problem for intuitive understanding is the examination of counterfactual posibilities, a rather extreme example is "the blue eyed islander puzzle"  (link with elaboration: https://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-puzzle/).

Strictly speaking there is an additional assumption (they participants are always honest, perfect logicians and know the initial knowledge of the other participants, and they know that the others know all this, and they know that the others know that too, etc) this assumption is understandably hard to formalize, and it wouldn't surprise me if there is some self reference shenanigans as involved in the gödel incompleteness theorem.