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Mathematical Girls - Gödel's Incompleteness Theorems
| Alt Names: |  Mathematical Girls: Gödel's Incompleteness Theorems Suugaku Girl - Godel no Fukanzensei Teiri 数学少女-哥德尔不完备定理 数学ガール ゲーデルの不完全性定理 |
| Author: | Yuuki Hiroshi |
| Artist: | Matsuzaki Miyuki |
| Genres: |  Harem Romance School Life |
| Type: | Manga (Japanese) |
| Status: | Complete |
| Description: | "I" (Boku) love mathematics. Just after the high school entrance ceremony, "I" meet a beautiful girl: Miruka. Miruka is a mathematical genius. She gives me many math problems; she shows me many elegant solutions. Miruka and I spend a long time discussing math in the school library. One year later, I meet another mathematical girl: Tetra. Tetra is one year younger than me, and asks me to teach her math. While I teach her, she begins to understand math and to love its elegance gradually. In this third volume (series), we talk about logic puzzles, Peano arithmetic, epsilon-delta, Cantor's diagonal argument, Hilbert's program and Gödel's Incompleteness Theorems. Sequel of http://bato.to/comic/_/comics/suugaku-girl-r1550 |
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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." EDIT: also divide out the ->th power of the lowest prime left, ofc. 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?
Different field, really. In engineering you learn applied type math; emphasis is on areas that allow you to do particular things, and on what you can figure out using them, in a relatively concrete way. It's powerful, but concerns itself less with the "why" of it all than the "how".
This stuff is more pure math; you can find it in both math departments and, actually, philosophy departments. I took a course or two in both so I can follow somewhat. It seems impractical, but we should remember that much of what is now used for engineering or computer programming in very hardheaded practical ways, was originally dreamed up by impractical pure mathematicians working at the frontiers of concept-space.
All that said, I don't really get the point of this representation of logical operators with numbers schtick in ch. 7. I mean, just because you arbitrarily say you're going to represent "OR" with "3" and "NOT" with "7" or whatever doesn't mean you can suddenly do arithmetic adding "OR" to "NOT" and getting "10". I mean, in a sense it's not really a "3", it just looks like one; if you're making it mean "OR" then it doesn't currently mean "3". I don't understand why this is supposed to be something more than a gimmick, why they're saying you can somehow get meaningful results by doing computation on these "numbers".
U can do that?
woah that just nearly blew my mind.
the only vague analogy i can pull out of my head is compressing data, but that stops there.
I am most sorry to hear that. Is there any maths you'd like me to try to explain? The material here would mostly be covered in a 3rd year undergraduate Logic or Computability course, albeit of course more formally and in more detail.
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)
If it's the official term, I would definitely be happy to just translate it to "tends to".
The term "getting endlessly closer" in Japanese is the mathematical term used for "tends to" in limits. So if something gets endlessly closer to a number, this number is the limit.
Oh, no, no! You done brilliant job, i wasn't trying to say that you made some error in translation or something . It's only that it says "endlessly closer to 1" on page 13 in chapter 5. And just as Purple Library Guy and the rest said, it doesn't make the limit eqal to 1. I don't think even the author thought about that.
We'll re-release with corrections when the series is done. Do you have any suggestions about proper description?
Oh, now i understand. What i earlier said means that we have convergent sequence, but we still don't know the limit. And the way they describe it really lacks proper formalities.
Yup, all those love polygons...
Seeing as how the author also writes math textbooks, I'm not surprised, lol.
Sure, but again, by virtue of being less than 1, they are also less than 2 etc. Some additional element is needed to rigorously describe the situation.
For instance, in the case of .9, .99 and so on, each element is actually not just closer to 1 than the one before, but exponentially closer--the distance between any element and 1 is 1/10th as far as for the one before. This is not true for any number larger than 1 (and for any number smaller than 1, the sequence will eventually produce a number bigger than it).
In that sense, I suppose you could say the limit of that sort of sequence is the smallest (or largest, depending which direction the sequence is going) number that the sequence will never exceed.
The food porn and the math porn broke my mind. I can only do one of those at a time.
Make yourself comfortable!
...feels like I'm back in high school.
What a great manga, I just hope someday I will stop being lazy and actually pay attention to all this mathematical stuff. Whenever a manga has some kind of cool information/ practical knowledge to offer I just never pay attention, and it's kinda sad cause it could prove useful someday unlike all the shit I actually pay attention...
Actually if you think about it, they said something more, which makes their way of thinking valid. They said that each element of the sequence is lesser than 1, AND they are endlessly closer to 1. This means that we have bounded sequence which is also monotonically increasing.
A really weird representation of Zeno's Ahilles problem.
That's a good point. I tried to fit it in, but I couldn't work out how. The point is that 1 is the only number that sequence a) gets arbitrarily close to and
stays arbitrarily close to - which if you turn that into symbols becomes the epsilon-N definition of limit (X is the limit of x_n if for every epsilon > 0 there's an N such that for every n >= N X and x_n differ by no more than epsilon). If you say the reals ARE limits of (Cauchy, bounded, whatever) rational sequences up to equivalent limiting sequences, then you have by definition that every sequence has either no real limit or a unique real limit.
Seems to me there's something else fluffy about the explanation in ch 5. I mean, I guess it's a quibble, but all the stuff about .99, .999 etc being endlessly closer to 1 (and to .99 . . . ) is all fine and dandy, but it's also getting endlessly closer to, say, 2, or to anything else greater than 1--and nobody's claiming that makes them equivalent. There has to be something more precise they can say about the nature of this approach than just that it's an approach.
Chapter 5 was the last chapter of Volume 1. We're now moving onto Volume 2, with more logic and less arithmetic. Look forward to it!
It doesn't matter how you write 0.3... so long as you know it means
.
I agree the presentation was a bit waffly. Maybe one should start by saying what the reals are, but precise answers to that can vary. I actually like working with decimals up to equivalence of limits, or the (bounded) order completion of the rationals. Others prefer to think of the reals as the power set of the integers (or your favourite countable set), but the argument will be slightly different for each coding. In short, you can have a formal background for the reals, or you can establish them here as the limits of convergent rational sequences, e.g. decimal expansions.
Basically, for this presentation, the precise coding does not matter - you can think about things however you like. That comes up later as well when we discuss Goedel coding. It's easy to specify a code, but turning prepositional computations into arithmetic computations along those lines, while good for the soul, isn't a useful consideration for your typical logician, though Goedel coding itself is.
Of course, one point of the audience is that it shouldn't know everything - why would you attend a lecture if you knew the content and the underlying thinking?
whh, maybe if you'd like to ask some questions we can do our best to address them.
This manga confuses me. As someone whose strongest subject at school was Maths (so much so that I skipped a grade for it), I feel really weird seeing a manga about this and making such a big deal about these mathematical theorems.
....Or maybe I'm uncomfortable seeing maths in my manga.
yeah it was a bit confusing to read, but i think the point that they were tryin to put out is still reached, atleast the point i was thinking about, that the equal sign when applied to the limit system can represent the limit and that the value of the two are the same. so even with 1/n, if n approaches infinity, then 1/n approaches zero and when applied with a = sign, they are effectively of equivalent value, so 1/infinity = 0, atleast in the natural number system.
also on the .999... thing, i think its a preference by country, cause i know people are use to use it in mine, it does look really weird and it always feel incomplete, but u get use to it i guess, i use to do it by the recurring dot but for some reason i feel now the ... makes it feel more correct somehow.
Looks like my guess is off the mark and I'm confused by this chapter.
I have to say i didn't like this chapter. The explanation was hazy. They tried to talk about limits without proper formal background. This is why students in the audience didn't know that greater than relation doesn't have to be kept when taking the limit. Example 1/n will be always greater than zero but limit is equal to 0. So in our case 0.9<1, 0.99<1... 0.(9)<=1, and nothing is wrong.
Also, i dont really understand writing 0.(9) as 0.99... it isn't as clear as other notations of repeating fractions (or what they call it in english). If you want to write for example 0.(123235512), writing it as 0.123235512... looks weird.