Mathematical Girls - Gödel's Incompleteness Theorems

Seems like I gotta start some informatics study to understand that. That really was quite some abstract stuff going on. And I'm probably the only one who thought 'These last 2 pages were written in LaTex'.

Definitely looks like LaTeX. Wow.

Seems like I gotta start some informatics study to understand that. That really was quite some abstract stuff going on. And I'm probably the only one who thought 'These last 2 pages were written in LaTex'.

This was a very good manga but I can see why it would get canceled. Nobody likes Math. And I am sure barely anyone could keep up.

The characters are fresh and interesting.
The romance was amazing. Glasses Girl had a kiss with him so it kinda left me satisfied. But I am still very sad that such a good story ended like this.

The Mc is good but he's very childish. He falls off the ranking by a little and he acts like its the end of the world. And knowing Glasses girl would be leaving he acted very immature. He deserved that slap filled with love.

Again. That Kiss left me very happy and fulfilled. If it didn't end with a kiss I'd be mad at this Manga.
She's a very moe glasses girl indeed.

>nobody likes maths
Are you really that close minded?
Even following that line of reasoning, those that do like maths are a determinedly loyal fanbase, willing to put up with a little dramatic tension and manga fluff if they get what they want.
 

Nerd mangas are best mangas. We need more navel gazing mangas on the topics of math and science, bring us back to our roots as nerds and geeks.

Generally stories written to inform by exploiting literary devices are much more interesting than those written merely to entice the lowest common denominator with shallow dramatic tension.

Nerd mangas are best mangas. We need more navel gazing mangas on the topics of math and science, bring us back to our roots as nerds and geeks.

This was a very good manga but I can see why it would get canceled. Nobody likes Math. And I am sure barely anyone could keep up.

The characters are fresh and interesting.
The romance was amazing. Glasses Girl had a kiss with him so it kinda left me satisfied. But I am still very sad that such a good story ended like this.

The Mc is good but he's very childish. He falls off the ranking by a little and he acts like its the end of the world. And knowing Glasses girl would be leaving he acted very immature. He deserved that slap filled with love.

Again. That Kiss left me very happy and fulfilled. If it didn't end with a kiss I'd be mad at this Manga.
She's a very moe glasses girl indeed.

 

I don't believe it was cancelled. It said what it wanted to (and also it was released in volume format, which I wouldn't expect of a cancelled project.) 

 

And people read this for the romance?

This was a very good manga but I can see why it would get canceled. Nobody likes Math. And I am sure barely anyone could keep up.

The characters are fresh and interesting.
The romance was amazing. Glasses Girl had a kiss with him so it kinda left me satisfied. But I am still very sad that such a good story ended like this.

The Mc is good but he's very childish. He falls off the ranking by a little and he acts like its the end of the world. And knowing Glasses girl would be leaving he acted very immature. He deserved that slap filled with love.

Again. That Kiss left me very happy and fulfilled. If it didn't end with a kiss I'd be mad at this Manga.
She's a very moe glasses girl indeed.

Interestingly, set theory is used in cryptography and (indirectly) to classes in object-oriented programming through duck identification principle.

I'm learning this right now in my Principles of Programming Languages class. I can never escape.

Interestingly, set theory is used in cryptography and (indirectly) to classes in object-oriented programming through duck identification principle.

 

Interesting! Can you tell us about it please?

I learnt quite a bit of number theory when I was 15-16, but I think logic is nicer, more fundamental and more accessible, so these days I recommend this book:  Logic is usually (possibly outdated) classified into Proof Theory, Set Theory, Computability Theory, Model Theory and Category Theory, and all of these (with the possible exception of set theory) are fundamental to compsci.

Interestingly, set theory is used in cryptography and (indirectly) to classes in object-oriented programming through duck identification principle.

I read them on Forgotten Scans' website, but it looks like chapters 9 and 10 on batoto might need a reupload. A potato shows up every time I try to open them.

Yup, fixing it. A symbol in the files' name couldn't be read so it caused uploading issues.

I read them on Forgotten Scans' website, but it looks like chapters 9 and 10 on batoto might need a reupload. A potato shows up every time I try to open them.
 

Please note that with chapter 10 the series is not complete - one chapter remains.

I see; thanks for the info and the double release.

Please note that with chapter 10 the series is not complete - one chapter remains.

https://www.codecogs.com/latex/eqneditor.phpis good for generating pictures or links which can be embedded or downloaded of arbitrary mathematical expressions. Example at the end of the post.

Thanks, good to know, but i always try to avoid pictures.
 

The stronger meaning that sjoe mentioned is enumerability, which is exactly what they talk about in the second-to-last panel here - http://vatoto.com/read/_/350574/mathematical-girls-gödels-incompleteness-theorems_v2_ch8_by_dkthias/7- what they are doing in the last panel is just setting up the table, which is a visual aid. Well-ordering is the right concept to think about here - it's what enables them to draw the start of the table.

There exists well-ordering for reals too (using ZFC + axiom of choice). Of course it's different than '≤', but it exists. So it doesn't mean we need countable set to arrange our table. Actually as you said we don't need well-order (or any order) at all.

https://www.codecogs.com/latex/eqneditor.phpis good for generating pictures or links which can be embedded or downloaded of arbitrary mathematical expressions. Example at the end of the post.

 

The stronger meaning that sjoe mentioned is enumerability, which is exactly what they talk about in the second-to-last panel here - http://vatoto.com/read/_/350574/mathematical-girls-g%C3%B6dels-incompleteness-theorems_v2_ch8_by_dkthias/7- what they are doing in the last panel is just setting up the table, which is a visual aid. Well-ordering is the right concept to think about here - it's what enables them to draw the start of the table.

 

Racky is right in the generalisation though -  Cantor's theorem that 2^X does not biject with X (which goes exactly along the lines of the argument presented here) doesn't need X to be orderable, it just uses the enumeration. 

 

(...) So if I get you right, you're saying the author/translation should've used a stronger word, which implies countability rather than just ordering?

I think it is just a language problem. Word "order" in this case is used in it's common meaning not an mathematical one.  Of course one could try to use synonymous words that are not "math related" but i don't think it's really needed.

Ah, I guess this is where my lack of pure math knowledge hinders me. I apologise if I stated something obvious. I had a look at the definition of ordering, (is this what you meant?). The statement "So if it were a countable set it should be possible to put it in order" is not wrong, like you say, but I see it's perhaps irrelevant since you can order ℝ too. So if I get you right, you're saying the author/translation should've used a stronger word, which implies countability rather than just ordering?

 

PS: I just googled for the symbols. Check this.

It's a true statement with the mathematical meaning - N is ordered quite naturally, so we order by the index from N. The point here is that you can enumerate it with natural indices, which is what was said in the previous panel. This panel is just setting up the illustration which you can visualise.

What i mean is that even if set wasn't countable, e.g. real line (or in this case (0,1) interval) it could be ordered (or in our case well-ordered). It's not something related to our proof. Countability and ordering are totally different things.
 

Hmm, I took ordering to mean "mapping of every element of the set in question, ℝ, to a corresponding (unique) element in the set ℕ" which is how countability is defined. In that sense, (or so the proof by contradiction goes) it's possible to assign an order to it, though that may not be the same as our intuitive notion of ordering in terms of how large or small the number is. Please correct me if I'm wrong.

I understand that, that's why i called myself a nitpicker. Order in mathematical meaning is something different. But, essentially what is written in manga is not wrong, just little out of place if you take word "order" as mathematical property.

PS: how you entered set symbols?

[...] on page 6 he says "So if it were a countable set it should be possible to put it in order". Ordering has nothing to do in this case, unless it was used as a common word without a mathematical meaning.

Hmm, I took ordering to mean "mapping of every element of the set in question, ℝ, to a corresponding (unique) element in the set ℕ" which is how countability is defined. In that sense, (or so the proof by contradiction goes) it's possible to assign an order to it, though that may not be the same as our intuitive notion of ordering in terms of how large or small the number is. Please correct me if I'm wrong.

Maybe i'm gonna be nitpicker but on page 6 he says "So if it were a countable set it should be possible to put it in order". Ordering has nothing to do in this case, unless it was used as a common word without a mathematical meaning. Also,  i don't know why but the same proof  with binary notation was much easier to understand for me.

 

It's a true statement with the mathematical meaning - N is ordered quite naturally, so we order by the index from N. The point here is that you can enumerate it with natural indices, which is what was said in the previous panel. This panel is just setting up the illustration which you can visualise.

Maybe i'm gonna be nitpicker but on page 6 he says "So if it were a countable set it should be possible to put it in order". Ordering has nothing to do in this case, unless it was used as a common word without a mathematical meaning. Also,  i don't know why but the same proof  with binary notation was much easier to understand for me.

That ending

 

Wow. Epsilon-delta, mathematical induction, Cantor's diagonal argument, those references to the Klein bottle and fractals... How had I not heard of this manga so far?
 

i dont even know any of the shit they show here, and im finishing engineering college this year.
do they really teach this kind of stuff in Japan? cause i understood none of it.
not to mention the author sucks at explaining so i kind of just skip the math(now this is a lot to bear considering i really love math)

Oh, I don't know. I'm a mechanical engineer who just graduated this year and I was perfectly fine with the math. I positively enjoyed it, but again that might be because I've always liked mathematics :/ Also, just my opinion but I don't really think the author sucks at explaining at all. Or at least the translations are well done, and some of the explanations/comments on the last page were interesting. Thanks a lot, DKThias & Norway Scan & Forgotten Scans.

yeah, definitely something u wanna learn in high school in Japan i guess

 

I learnt quite a bit of number theory when I was 15-16, but I think logic is nicer, more fundamental and more accessible, so these days I recommend this book: https://en.wikipedia.org/wiki/Naive_Set_Theory_(book)to secondary schoolchildren who want to learn more interesting mathematics. I recommend it to you too.

 

 

So in summary, its using an established system that has limitless ability to generate variables and outcomes (arithmetic) and transform logic statements and systems into them so that u can process them easier and since many of these things uses arithmetic in them, it makes them consistent within themselves?

 

thinking about it, that is wat computers do essentially and i never thought u can do it in such a basic level things, i mean physical input is one thing, but transforming logic was a bit out of my imagination.

 

They are consistent to begin with (hopefully) and this actually leads to showing that they can't prove themselves consistent (which is much of the point of the Incompleteness Theorems), but yes the manipulation is what computers do and it happens on a very fundamental level. This is why there's massive overlap between Theoretical Computer Science and Logic. Logic is usually (possibly outdated) classified into Proof Theory, Set Theory, Computability Theory, Model Theory and Category Theory, and all of these (with the possible exception of set theory) are fundamental to compsci.

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So in summary, its using an established system that has limitless ability to generate variables and outcomes (arithmetic) and transform logic statements and systems into them so that u can process them easier and since many of these things uses arithmetic in them, it makes them consistent within themselves?

 

thinking about it, that is wat computers do essentially and i never thought u can do it in such a basic level things, i mean physical input is one thing, but transforming logic was a bit out of my imagination.

Short version: in a sense your comment is a number, and my computer can parse it, I can read it, and I can type this number back to you. If I just copy and paste bits of it, those are also numbers and I don't have to have read it all to copy and paste bits of it.

 

Long version: because you can. I mean when it comes down to it these formal manipulations are string operations, which can be done by computer. The prime number implementation is a bit awkward to work with, but e.g. let's say the implies symbol (->) is represented by a power n. Then Modus Ponens (from X -> Y and X, deduce Y) is simply "if b is a factor of a and the least prime factor of a/b has power n, then deduce the reduction of a/b." In that line, to reduce a number take its least prime factor, if that's not the least prime (2) then make it 2, then if its second least prime factor isn't the second least prime then make it 3, etc.

 

To work with the coding they describe in the chapter and not want to kill yourself, you'll want to define a notion of reduction-equivalence (at least for substrings) which says "these are the same substring" (but for the actual rules the substrings should be initial substrings one way or another, so you can just do division followed by reduction). Doing such mechanically is of spiritual rather than practical benefit, but I have been sinful of late so let's just say a substring T is convex if its subset of prime factors is convex (for every prime p, q dividing T, if r is between p and q and also prime, it also divides T) and two convex substrings are reduction-equivalent if they have the same reduction.

 

Goedel coding is important as a point of view. It's thinking of proofs as numbers, which is how they're treated when you do automatic proof verification. It makes arithmetic special - without arithmetic, you have to go to second-order logic to talk about first-order statements, which prevents statements from referring to themselves. That's part of what we need to write "This statement is unprovable." Goedel coding says that there are coherent ways to actually do that once we have arithmetic at our disposal. Finally and most importantly, it internalises the external logic of your system into the arithmetic of that system. Externally of course we can say "clearly the system has only countably many symbols and variables, so it has a countable set of statements, let f: N -> be an enumeration of the same", but this doesn't say f is internally definable. Making f internally definable lets us do so much more.

 

Does the above help?

yeah, definitely something u wanna learn in high school in Japan i guess