Thanks for watching everyone! Don't forget to submit your questions on my subreddit for a potential 10k Q&A video. www.reddit.com/r/anotherroof/ This concludes my series on defining numbers from the ground up, but there are plenty more videos to come.
@Spydragon Animations Simply use the definition with [(ac + bd, ad + bc)] for the integers. You can show that "negative * negative = positive". In particular, you can show that (- a) * (- b) = a * b if this is what you are unsure about
@Spydragon Animations The partition kind of "drops out" from the relation definition. Two elements are in the same partition if they are related. If you're being formal, you have to prove that (2,2) is related to (1,1) just as much as you have to prove that (-1,-1) is related to (1,1). I think, when he was at that part with the board and all the number pairs laid out in a grid, he was already being less formal - by that point he has shown that the definition is the same as our intuitive understanding of rational numbers, so he relied on our intuition to explain things visually without interrupting the rhythm with unnecessary proofs. I hope that helps, even if I'm a few months late :)
Why didn't you define the real numbers as an integer-subset with a highest element, representing the sum of 2^(each subset-member)? This allows you to avoid the rational numbers altogether. Also, why don't you consider quaternions numbers?
i gennuenly don't know if you are using that as an exemple of "really hard thing to explain" or of "nobody solved this yet". If it is the second option... Someone did.
As a graduate engineer I always thought I had a reasonable understanding of maths, now I see I was just given a box full of tools and shown how to use them, but with no explanation as to why the tools work. The biggest take for me from this video is that there is a fundamental difference between integers and naturals, the difference between the other sets being a bit easier to 'see'.
Yes, engineering curricula are designed that way because you want to go out into the real world and start solving real engineering problems (and earning money) as soon as possible! You don't want to spend 4 years learning all about the fundamentals of math used in engineering, then another 4 years learning about all the physics used in engineering, then another 4 years learning about all the chemistry used in engineering ... it's turtles all the way down, and you'll die before you graduate! Besides, if you want to learn more, you can always do so later, either in your spare time (like I'm doing now), or by going back to school.
In Icelandic, Imaginary numbers are called _"þvertölur"_ or "Across Numbers" (It sounds a lot less awkward in Icelandic) and complex numbers are broken down to _Vertical_ and _Horizontal_ components (Or just length and angle, if that's what you prefer); Because Imaginary numbers are just as real as any other..
I personally treat complex numbers more geometrically than algebraically, so I call them "spherical numbers" due to their intrinsic connection to spherical geometry. The name also hints at the possibility of numbers for other geometries, which do in fact exist, though they lack some of the nice features of the complex numbers.
Polish doubles down on the imaginary and the number "i" is called "jednostka urojona" or "Deluded Unit" as though we've lost our marbles, and gone fucking mad square rooting negatives.
@@Zaniahiononzenbei There are two main cousins of the spherical/complex numbers. One is called the "dual numbers," and the other is, depending on who you ask, either the "split-complex numbers," or the _"hyperbolic_ numbers." Knowing that the spherical unit squares to -1, I'll let you guess what its cousins square to.
This is such an incredible video massive props to you for taking on such a massive project and such difficult concepts and explaining them in a way that I actually feel like I understand :D
My favorite is the completion as a metric space. It requires a lot of conceptual overhead (if compared to dedekind cuts), but it’s so natural and gives a good intuition why we use reals to begin with
I don't know why it's said we can never fully write down pi when it's easy: π Sure, writing it down as a decimal number is hard, but if you pick the right base it's easy! In base π it's just 10. Now, sure, that's an absurd base for everything else but sometimes sacrifices need to be made! More seriously - I am loving your series so far. Thanks for the hard work you've put into it.
We can do the same tricks that we used to jump up a number system, to define pi, though it takes some steps. We need to define powers, using ordered pairs. Then we can define rotations, where powers of a complex number are equivalent if they give you the same complex number but scaled. That leads naturally to exp and ln.
Firstly, pi in base pi is not pi. It is "10". As that's how base systems work. But base pi doesn't really make sense, you end up with an unevenly spaced number line.
oh my goodness when I saw a new proof video was out I got so excited I could hardly wait! Your channel is my comfort channel and I really appreciate the work you do, and the jokes and the math and the bricks and everything!! Thank you so much
I'd love to see an appendix video of sorts on transcendentals, though that certainly sounds like a challenge to make. This series has been great to watch, thank you
Transcendentals are a subset of real numbers, (or maybe complex if you want) so they can be expressed in the same way. You just define it as the set of all rational numbers less than it paired with the set of rational numbers more than it.
@@thomaspeck4537 No, the transcendental numbers are a subset of the complex numbers, because the algebraic numbers are a subset of the complex numbers. The algebraic numbers are defined as the algebraic closure of the rational numbers, and this closure necessarily includes i. Therefore, they are not a subset of the real numbers.
@@Cammymoop To explain the surreal numbers would require an entire video series. An appendix certainly would not be sufficient. You would need to start by changing the axioms of set theory, since in reality, if we are being rigorous, the surreal numbers do not exist in the Zermelo-Fraenkel axioms, even if you include the axiom of choice or its negation. You need axioms of set theory that are equipped with the ability to speak about proper classes, in order to define the surreal numbers.
This is one of the most geeky, pedantic and "why does anyone care" videos I've found so far; but yet it is so beautiful (and nicely presented). The definition of negative numbers just drops out so easily. You did manage to refrain yourself from the obvious joke: "Z from the German for 'zee integers'" ;)
@@gracenc vaguely Yorkshire? I'm from Lancashire -- and you're all lucky I'm not one of those proud Lancastrians who actually care about this sort of thing!
Really hoping to see Dedekind cuts definition of R, but it might be a bit too advanced for TH-cam. Definitely gonna be a banger video though, we all know it
Me too. I don’t have the requisite knowledge to understand the formal definition, so I was hoping there would be a more intuitive way to understand them without getting rid of the complicated details; this channel seems very good at that.
Dedekind‘s construction is relatively simple compared to completion of Q as a metric space. Most of the ingredients are readily available at this point in the series. But the completion thing connects most naturally to what we’re actually doing with reals
@@AnotherRoof You could perhaps take a moment to emphasize on what you wrote for the "R" block at 1:11:31, which might not be obvious from the chalkboard: That the "mother set" in this case is power_set(Q). . This is important because (1) it is a conceptual leap from the previous cases (where the "mother set" is just a cartesian pdt), and (2) R is somewhat different than N Z and Q in "size", in that its construction asks you to start from something much "larger". . (Explaining (2), "size" of infinite sets, etc... = topic for another video maybe?)
I was curious about the proof that the real number are continuous, at 1:06:50 you hint at it but it seems to me that all you are proving is that the real numbers are dense, as in there's never an empty gap. But the rationals share that property, you even proved it, yet there were numbers that were missing. How can we be sure that there are no numbers hiding between the real numbers?
Agreed, it seems like he left out the key part of the proof. As I understand it the way to prove the real numbers are continuous is to do the dedekind cut procedure again using the real numbers, and then you can show that the resulting number is just another real and doesn't give you something new.
It's very subtle. In one sense, there are numbers between the real numbers, because you can put more numbers on the line with constructions like Robinson's Hyperreals or the Surreal Numbers, etc. But in another sense, the reals are gapless in a way those things are not: for instance, if you have a nonempty set of reals that doesn't grow arbitrarily big (e.g. maybe they're all less than a million), then there is a "cap" real that is greater than or equal to everything in the set, but smaller than any other candidate. (The cap may or may not lie in the original set. Think about intervals like (0,1) and [0,1], both of which have cap 1.) This "dedekind completeness" works for the reals and stops working if you try to stuff even more numbers inside.
The short answer is, it depends what you mean by a number. There are number systems (see hyperreal numbers or surreal numbers) in which there are other numbers hiding between the real numbers. What is meant in this context is that the real numbers are "complete" (this is a technical term). There are several equivalent definitions of what this means, but perhaps the simplest one to see (relative to this video) is that if we were to repeat the process of taking Dedekind cuts we wouldn't get any new numbers. That is to say, instead of the sets A and B consisting of rational numbers, say they consisted of real numbers. Then the set of Dedekind cuts of this form would just be the reals again and wouldn't contain any new numbers.
At 59:35, since we want to use the Dedekind cuts to define the real numbers, if r is not a rational number (1:00:49) and not a member of Q, how can we then prove that there exists another rational number between m and r? Also, how are we certain that in the Dedekind cut, there aren't two or more irrational numbers? (great video btw)
41:05 I never saw why q != 0. I can see that if we were to include q=0 then this relation is no longer transitive. (p,q) ~ (r,s) : ps=rq => (1,1) ~ (0,0) (0,0) ~ (2,3) But (1,1) !~ (2,3)
There is a slight generalization of this construction in the mathematical subject known as commutative algebra. It is called the localization. However, even in this context, including 0 results in a structure that is not very interesting. en.wikipedia.org/wiki/Field_of_fractions#Localization
Any chance of you covering the Surreal numbers? It seems like a good place to go off of this. I'd also just like to say that I really like that you're using physical blocks for your axioms/theorems and stacking them together as you go. I think it's a great visual metaphor for how math concepts build off of a few foundational concepts.
do you mean the trancendental numbers?, or do you actually mean the surreal number that include: the number greater than every real nubmber and the rnumber less than every real muber but greater than zero?
@@plopgoot5458 He means numbers greater and less than any real number. Unfortunately the surreal numbers' construction require the use of transfinite ordinals and thus cannot be a set as there is no "set of all transfinite ordinals" in ZFC. They are a proper class.
@@crosseyedcat1183 Is that an issue, that they are a proper class as opposed to a set? It is an important caveat, yes, but I fail to see what is so "unfortunate" about it.
@@plopgoot5458 He is talking about the Conway surreal numbers, which extend the real numbers, and they are the "largest" ordered field, in the category theoretic sense.
You have an amazing teaching style. I really appreciate your approach of laying out a few axioms and incrementally building up to every topic covered step by step. One thing that shows up in a number of math explainer videos is the Monster group, but without a background in group theory I've never had a foundation to be able to make sese of any of the writing about it, besides "big number interesting". Given your incremental approach to explaining things a series of videos on group theory that build up to the point of having the context and understanding to make sense of statements like "monster group M is the largest sporadic simple group", would be super nifty.
I've mentioned this in other comments and maybe I'll go into more detail in my 10k Q&A video, but my PhD was in group theory and I would love to do a series on the various families of finite groups. That said, because it a topic so dear to me, I want to gain more experience to make sure I do the topic justice!
Loved the series! I don’t know how useful it is for someone who isn’t already familiar with this stuff, but I like the balance between intuition and rigor here
Well, I was familiar with all the building blocks (relations, power sets etc.). The definitions of each class of numbers were completely new to me, though, but the explanations were impeccable and I understood all of them. So definitely very useful for me at least:)
Here's a Dedekind cut definition of π: Lower set A = Union over all n in N of the sets: { r in Q : r < 4 * the sum from k = 0 to (2n + 1) of [ (-1)^k / (2k + 1) ] } Upper set B = Union over all n in N of the sets: { r in Q : r > 4 * the sum from k = 0 to (2n) of [ (-1)^k / (2k + 1) ] } You're welcome! :)
@@acrm-sjork That's because the rationals are not complete. Real numbers have the completeness property saying that every converging series has a limit point in the set.
@yanntal954 not sure that I got why infinite sum of rationals MUST or even CAN be irrational? Why not all sums of rationals converge to rationals? It is not contradict your statement
I was hoping you might explore how C starts to lose properties compared to R. In a sense R is the top of the "number pyramid". Because you lose ordering in C, and the other way, you lose completeness in Q. If you go to quaternions you lose more (commutativity), and octonions lose even more (associativity).
You lose any natural or intuitive ordering in C, but it would certainly be possible to define an arbitrary ordering. Start by saying that anything that is closer to the origin on the imaginary plane than x+iy is less than x+iy, and then define an arbitrary ordering for all the points that are equidistant from the origin (e.g. saying that x is the highest within that set, and then going clockwise around the circle).
@@stephengray1344 Yes, but no. In the context of fields, when we talk about ordered fields, we talk about fields with an ordering such that the field operations are isotonic (in a standard sense) with respect to the ordering. The real numbers are an order field. The complex numbers are not.
@@stephengray1344 Also, specifying that an ordering does exist is sort of redundant, since by the well-ordering theorem, all sets can be well-ordered. I know the theorem requires the axioms of choice, but if you relax the conditions so that you only need a total order, then you can significantly weaken the axioms needed. Also, most mathematicians do accept the axiom of choice, despite it being controversial.
In a very real respect, you're single-handedly undertaking what PBS Infinite Series was doing. :) I greatly appreciated that channel while they were active, but it turns out there are just not that many channels engagingly tackling topics at the foundations of mathematics outside of "headline-friendly" ones like trying to quickly explain Goedel's theorems ad nauseum. (I'm sure you'll get to those theorems, too, but if it's anything like this presentation, I expect it will be thorough and cohesive!)
It's funny you mention PBS infinite series, I was literally just thinking about that channel because I got a question asking if I'll ever make a video on the axiom of choice and I thought their video was good. I'm glad your enjoying my videos! A slight disclaimer: I'm not planning for every single one of my videos to be a deep dive into foundational topics, but I'll always try to approach ideas with my own style and an emphasis on rigour. Anyway thanks for watching :)
@@AnotherRoof Of course! I'm enjoying it, and it's been long enough since my university coursework that I appreciate the refresher either way. I find this stuff fascinating, even if it has very little bearing on my day to day life, but it's not at all like a bicycle; the gist of it may stick for ages, the details and practice not so much.
At 59:00, you explain how to construct a Dedekind set, defining it as being all the elements of the rationals which are smaller than a given number (r). You then prove that this set will not have a maximum value, because of the density property. This implies (as you say in the video) that both m and r (in your demonstration) are part of the rationals. However, you then says that "this is how we define the real value r". But wasn't r supposed to be a rational? Otherwise we cannot use the density we used to construct the Dedekind set
Let's say that we have some rational number m that is smaller than r (which is assumed to be irrational). Then we can add a positive rational number n to m to make a new rational number that is larger than m but still smaller than r. And we can make n as small as needed because there is always another rational number between 0 and n. That's as far as I got, but I'm not convinced it's a complete proof since I haven't ruled out the possibility of there being a real number that is "smaller than" all the positive rational numbers but larger than 0
I've always loved educational videos, but this gets me some different levels of joy. I'm actually having popcorns right now while watching. Thanks so much for this series of videos!!
@@cadekachelmeier7251 I'm sure you're right, but that seems weird to me. It's just 2^(countable infinity), right? Why doesn't that just return countable infinity again?
@@vanderkarl3927 For reference, it's Cantor's Theorem. Honestly, it's pretty much the same logic as Cantor's diagonal argument. It's just more abstract because instead of reasoning about lists and numbers, you have to reason about any potential set and a set containing an infinite number of other sets. Other than that, it's just constructing an element that can't exist in any potential mapping, same as the diagonal argument. en.wikipedia.org/wiki/Cantor%27s_theorem
Complex numbers are just the algebratic closure of the topologic closure of the groupification under multiplication of the groupification under addition of the monoidisation of natural numbers provided with the standart z-module stracture.
17:00 I'm curious... I notice that only 4 of the 8 potential combinations of reflexive/symmetric/transitive are represented here. Do relations exist for all of the other possibilities, or are any excluded? In other words, are the three properties completely independent?
2:15 Illustrating the gap between the whole numbers as being foamy is such a splendidly intuitive way of conveying that only a proportion of that space can be represented with fractions.
Actually you have skipped redefining operations on real numbers. But some of them are not trivial like multiplication or division, so it would be interesting to have it in the video. But I understand that the video is quite long. Great series!
Yeah I just kind of glossed over that and said "it can be done" haha -- probably should have just displayed them at least for a few seconds but yeah, 80 minutes was long enough!
The most beautiful thing I saw in this video was that the question "why minus times minus is plus" never arises! It just is baked on the rules of the definition for the integers. Great stuff, marvelous presentation.
Your proof of the denseness of Q made me realise how denseness is related to division. It's so obvious now! Of course if a set of numbers (containing at least two members) is closed under averaging, it is dense! And if it's not closed under division of two, it's not going to be closed under averaging.
Wow this was an awesome culmination to an awesome series. I am not a mathematician, but I have taken some advanced courses (Real Analysis, Measure Theory, etc), so I knew more or less what you were talking about with all the "hand waving". I had never seen this conception of numbers, and I loved it! Keep up the great work!
I feel like you keep bumping right up against group theory and algebra. Hopeful to see some vids on those topics some day from you! Fantastic video as usual.
@@AnotherRoof it's also my favorite area of mathematics, and I am REALLY enjoying your videos and style and just everything, hence my yearning for the topic. Whenever you do get to it, I'm sure it will be fantastic. Thanks again!
I've not looked at group theory at all but I kept feeling like category theory was around the corner. Not that I have a much of an understanding of that either :). I'm sure I'd enjoy your take on either/both.
@@brendanobrien4095 I only have an undergrad degree in mathematics so I haven't studied category theory at all (that's a graduate topic, 100 percent), but my understanding of it is that it is essentially another layer of abstraction on top of group theory/algebra (those terms are not at all interchangable, but whatever; it's fine for our purpose here.) Like, my understanding is that category theory is essentially an algebra of algebras. But again: I don't have any formal training in category theory. So if I'm wrong, that is not unexpected. Also, you should look at group theory. If you know what category theory is, you ought to know what group theory is :p. No judgment here, just a recommendation. It would be a cool video idea for sure though. Not sure how much another roof knows about category theory, but any video he wants to make is fine by me, cause he's killing it. I've rewatched every video at least twice and just loved every second of it.
@@kruksog You should understand that my path to this stuff is atypical. I am a software engineer and I took none of this in college. I came across category theory by way of lambda calculus which I was studying for applications in functional programming paradigms. This is the first I've even heard of group theory and I'm very interested if it relates to these topics.
The video omits defining the < relation for integers. It defines it for natural numbers: a < b is the same as a ∈ b, but it doesn't define it for integers. Yet, when defining < for rational numbers, it states it using < for integers.
I've watched the real part of this several times, and I think I get the process, but why is it so different than the rest of the parts? Every other step takes a pair from the set made before, and than the real numbers seem to need to do this dedican cut thing. If you try to skip it, and just try to go from rational to complex, can you get something sensible or does it just always break?
I mean, you can define the rational complex numbers (numbers a + bi where a, b are rational), and those are useful in some contexts. However, you don't really gain much by doing so. The real numbers are difficult to define because they are "uncountable" (a technical term, worth looking into if you are unfamiliar with the concept). The operation of taking ordered pairs and then partitioning can never produce an uncountable set from a countable one.
There are other ways to define R. One way-which is more like the previous ones-is the following: Given two Cauchy sequences (a_n) and (b_n) of rational numbers we say that (a_n) ~ (b_n) if and only if the sequence of differences (b_n - a_n) has limit 0. This defines an equivalent relation on the set of all Cauchy sequences over Q, therefore partitioning this set. A real number r is now defined as an equivalent class of those sequences, that is r=[(a_n)]. Keep in mind that we don't need irrational numbers or the concept of limits (expect for 0 which is no problem) for this definition. The term Cauchy sequence can easily be defined for rational numbers.
30:45 “it’s like upgrading the operating system; it does everything the old one did, and more.” Little did he know that now upgrading your OS will remove features.
Thanks this was very thorough, and I can see how various ideas take the 'blocks' of number theory that you explain so well, and build a whole set of new theories out of them... i find this actually useful as a presrequisite to the PBS infinity series... Thanks!
you say not to worry about tagging ordered pairs with 1 and 2 because we can use different numbers, but what if we want to use the set of all natural numbers? We’d need new tags to avoid duplicates. And then what if we use the set of all of those tags, and so on. There must be a cleaner solution
Yes, I wish I'd never used the Hausdorff ordered pair but I felt it was more intuitive to non-mathematicians. Anyway, for a set A it is always possible to find a thing which is not contained in A, namely A itself by the axiom of regularity. So if worse comes to worst we can use the set A itself to tag the elements of A. So an ordered pair (x,y) with x in A and y in B could be encoded {{a,A},{b,B}}. Hope this helps!
There are sets not in Naturals for example sets {0,2} and {1,2} you can use this two for tagging. This set are distinct from {},{0},{0,1},{0,1,2} which are the set represented by 0,1,2 and 3. Sets {0,2} and {1,2} are not in Naturals.
@@AnotherRoof this works except that a lot of the cartiesian products you use in this video are of the form A×A. So you probably want something like A and {B} as the two tags (which should work for this video/most applications in math).
I was fairly familiar with the construction of the rationals from integers and complex numbers from rationals, but had never seen/considered the similar construction of integers from naturals! In hindsight, the parallels seem obvious, but I guess I'd never really considered "negativeness" with enough of a critical, "what _are_ you, really?" attitude.
I don't know if he took my suggestion of laying down the bricks like a wall in his outro, but regardless of whether he did or didn't, it looks cool. It feels like we all laid the bricks together. Though with math, it took us quite a few hours to get there...
I'm actually proud of myself because I barely understood the information presented here. I'm going to have to binge watch your videos, I want to be sure I understand things of this nature.
0:25 Temperature and altitude are bad examples for negative numbers. They have an arbitrary zero value. Imagine it like this: Define 0 °Apples as 10 Apples. A basket with 3 Apples in it contains −7 °Apples. It that sounds like nonsense to you, it’s because it is, but temperature and altitude work exactly like that, it only feels natural because of exposure. There is absolute zero of temperature (0 Kelvin) and altitude (the center of the earth). Negatives only exist in differences; if you ask me “how much taller is your brother than you are” the answer is −2 cm because he’s actually the smaller one, still body height can’t be negative.
43:58 I thought the rocks were real the whole time. Edit: 45:52 What happened to the rocks?! Edit 2: 46:13 Rocks are back again :) Edit 3: 1:11:28 Now the rocks are real?! 1:16:09. Edit 4: Now they are changing colors? 1:17:58 - red, 1:18:13 - blue, 1:18:34 - green, 1:18:55 - yellow, 1:19:10 - pink. Edit 5: Watching the video again and it turns out the rocks were real 4:58, 10:37 except that one time when they weren't. Edit 6: 9:13 And what's up with the paper? 9:32, 9:46, 9:59.
1:02:32 - "we can describe all of the irrational numbers in this way". This kinda made me wince. I'm not sure, is it actually technically true, or just an inaccuracy in the way you describe things? I'm not intentially trying to nitpick, I'm actually confused. Sure, you can come up with a Dedekind cut for each real number *you know of,* but is this *every* real number? If it is, it would imply there are only ℵ₀ real numbers, since the set of formulas you can write is enumerable. Which, as we know, isn't true. So, did we actually define *all* real numbers in this video, or just *some* real numbers?
How would you come to define a real number with this cut thing if you have a uncomputable number (a number with infinite random digits that cannot be expressed as an equation)?
At 52:46, when the expression is multiplied by 'q' is important to notice that q > 0, otherwise the ''. But we know that q is positive because we chose the right representative of the rational number.
1. There are operations you can’t perform on the complex numbers, like division. Division is not a total function; z1/ z2 is not always defined. 2. I think you cut out a bit too many of the details of real completeness; as described in the video it sounds like the same property as the density of the rationals.
except a method for identifying what is in between 2 real numbers is identical to the method for identifying the real numbers in the first place and thus anything in-between them is itself a real number. This fact means necessarily that the real numbers are complete as that term is defined in the video.
Division is never a total function. There are no rings in which you can divide by zero. The closest you can ever get is to say that you can divide by everything except zero, and the complex numbers have this property
I would like to hear what he cut out with regards to completeness. I left another comment asking for such because I still don't know why the reals are complete and not merely dense.
The method you use for constructing pairs is quite unaxiomatic, as you there is already an infinite set with a full ordering. The more common definition of an ordered pair is {{a}, {a, b}} for (a, b)
one thing isn't clear to me: when defining the cartesian product, you said we can't just apply the set rules infinitely many times to construct the product of infinite sets (which i'm not sure is too different from using recursive definitions or proofs by induction, but that's another story), but to work around it you used power sets of potentially infinite sets without first defining them. does that mean the regular definition applies? but that would mean applying infinitely many times the union rule, which shouldn't be possible?
Gonna be honest, I'm still a little lost on the part about completeness of reals. You proved that there's always a rational between any two real numbers; how does that prove that there are no holes? The proof that there's always a rational between two rationals doesn't prove the rational numbers are complete. I would think a proof would go sort of like how you proved √2 isn't in the rationals, but instead some general function f such that f(R) outputs a number between two real numbers and is not contained in R, cannot exist.
I'm 10 years separated from my compsci BSci, and this video is an entertaining bit of bringing it back to the fundamental axioms of the inductive power of diagonalization. Great stuff.
I knew you couldn't possibly be defining every number ever, but I could not imagine you would speak so fondly of Conway after leaving his surreal numbers out!
I think to make the dedicind cut more concrete, a set with no greatest numbers could be the set of all numbers less than 2. is there a greatest number less than 2? no, cause you can always stick more nines behind the previous answer.
He didn't specify whether or not r is in Q. He just said, "pick a number r". The argument is just to show that such open sets, with no largest member, actually exist. You can construct such sets out of rational numbers, as shown in the proof. He later shows that the reals are _also_ dense in the same way, and therefore you can construct such sets using them as well.
So at 35:07, it looks like you're assuming that a negative times a negative is positive. Is that an assumption or is that proven another way and just left out here? Similarly, at 48:02 you're defining all the operations for the rationals. But is that proven in this system off camera or just assumed as a definition?
The former can be proven from the definition of multiplication for the integers or rationals. The latter is just a definition chosen so that the numbers work the way we expect. You could prove those operations agree with the integers' operations when all the rationals are like z/1 for some integer z, though.
I had to pause this several times to revel in the mind blasts this gave me. It's insane how much sense it will makes now. Thank you for the concise explanation!
Great for a start. Examples of additional stuff: 1)Quotient systems of numbers, for example Z modulo an integer n. In particular, if n= a prime p, the only interesting case to be honest, then Z modulo p is a field, i.e. it satisfies all axioms satisfied by Q and beyond. It's an example of a finite field; it's incomparable to N, Z, Q, R, C etc. It can be shown that ANY finite field is essentially unique (i.e. unique up to isomorphism) and has q=p^n elements, where p a prime and n any integer from 1 and up. It's symbolised F_q. The prime p is called the Characteristic. Every field has a characteristic which is defined to be the smallest positive number that annihilates (i.e. if multiplied with, it zeroes out) all elements, equivalently the multiplicative unit, if such a positive number exists; otherwise the characteristic is 0. So, characteristic p means that 1+...+1 (p times) is 0, and is the first time we get 0 if we add 1 a non-zero number of times to itself, equivalently all other numbers in the list of 1+...+1 etc. are distinct. Likewise if we add any other element to itself consecutively. Every field has characteristic 0 or a prime number p. In the first case it contains a copy of Z, thus of Q (considering fractions), thus it's an extension of Q. In the latter it contains a copy of F_p. So basically, if we stick to only Q, R, C we miss out on plenty of action. 2) Transcendental extensions: If F is a field and V a set of "variables" (could be finite or infinite) we can form the set of polynomials on the variables of V and call it F[V]. The elements of F[V] are called polynomials in V with coefficients in F. They are (including 0) the finite sums of monomials of F[V], who are in turn finite products of elements of V (could be empty, which gives 1), multiplied with an element of F (could be 1). If we copy the construction that gives Q from Z (it's called the "Field of Fractions" construction), we get the set of rational "functions" in F[V], symbolised as F(V). The idea is that F(V) is the smallest field containing F and the elements of V, which are considered ALGEBRAICALLY INDEPENDENT over F. This means they satisfy no non-obvious algebraic relation. In case V={x_1,...,x_n} this is just the sets of polynomials and rational "functions" in x_1,...,x_n repsectively. If furthermore n=1, then we get V={x} and we write F[x] and F(x) for the set of polynomials in x with coefficients in F and rational "functions" in x with coeffs in F. F(x) is called a simple transcendental extension. Basically, it behaves like π and e in R with rational coefficients, every algebraic relations using x and elements of F are trivial, i.e. they merely follow from the axioms of arithmetic, equivalently: a_nx^n+...a_0=0 => all coefficients are 0. 3) Algebraic Extensions: We can add "algebraic relations" to F(V) above using quotient constructions (an example below), so that the images of V are no longer algebraically independent. Alternatively, if F, L fields such that F is a subfield of L (like Q in R and R in C) then L is algebraic over F if every element a in L satisfies a polynomial equation with coefficients in F. C is an algebraic extension of R since every complex number has a polynomial (more below), R is DEFINITELY NOT an algebraic extension of Q, in fact almost everything is transcendental over Q. So, in the example of F[x] x a variable over F, we can take any polynomial m(x) (a prime/irreducible polynomial for good results) and form F[x] modulo m(x). This isn't a field if m(x) isn't irreducible, in fact if m(x)=p(x)q(x) then p(x) and q(x) modulo m(x) would be non-zero, but their product m(x) is 0 mod m(x) by definition. If m(x) is irreducible, then F[x] modulo m(x) is a field, similar to 1), and is the "smallest" field containing F such that m(x) has a root; clearly x mod m(x) satisfies m(x) itself. If we define x mod m(x)=: ρ, then we call the above set F[ρ]=F(ρ) the simple extension defined by a root of m(x). The reason the above equality holds is that inverses of non-zero elements already exist, due to the following: F[ρ] is the set of all (unique) linear combinations of 1, ρ, ρ^2,..., ρ^(n-1) with coefficients in F. This is being proven by Euclidean division of polynomials. So, any such non-zero combination, corresponds to a polynomial g with a degree smaller than the one of m(x), thus it isn't being divided by m(x), and since m(x) is prime it must be relatively prime, thus there exist polynomials s,t such that sg+tm=1, which modulo m(x) gives sg=1 modulo m(x) (same proof works in 1) ). So we could define "square root of 2", and any other algebraic number in general, without dedekind cuts necessarily (if we aren't interested in ordering numbers on a line) as follows: Start with Q, form Q[x], then consider m(x)=x^2-2 which can be shown to be irreducible, and form Q[x] mod m(x). We could call it Q[j]=Q(j), where j is an "imaginary number" such that j^2=2 (note it could also be minus square root of 2). It contains all elements a+bj where a,b are rationals; any such writing is unique and if we have a non-zero element, we can find the inverse by multiplying and dividing with the conjugate element a-bj. Same construction can give us C from R, just take x^2+1 as m(x). In general if L is an algebraic extension of F, and a an element of L, then the polynomial of F[x] of smallest degree having a as a root is called the MINIMAL POLYNOMIAL, and it also happens to be the unique polynomial which divides all such polynomials. If it is defined to be m(x), then the smallest field containing F and a, F(a) is going to be essentially the same thing as the above construction F[x] modulo m(x). The degree of m(x) is called the degree of a. In particular F(a)=F[a]. Note that if F=L then the minimal polynomial of a is x-a which has degree 1. We can keep adjoining elements to a field F. If a is a root of a (w.l.o.g. irreducible) polynomial p, then we form F[a]=F(a), and then if b is a root of the (irreducible over F[a] ) polynomial q, we can form F[a,b]:=(F[a])[b]. In theory, every algebraic extension can be obtained by simple extensions, if we "take the limit" for infinite-dimensional extensions. So we can have Q[sqrt(2), sqrt(3)] etc. Theorem: Any field extension can be seen as a transcendental extension followed by an algebraic one. So, wlog for more "numbers" we can take, for example, C(x,y), C with two extra algebraically independent elements, identified by a "surface" over C if you do algebraic geometry, then you add the relation, say y^2-x=0 and you get... something. 4) Algebraic closure. If you take the subfield of all complex numbers who are algebraic (over Q) you get the Algebraic Numbers. This is called the algebraic closure of Q, since every polynomial over Q has a root now (but without the additional transcendental elements of R and Q). You can always take the algebraic closure of any field! So, you can take e.g. F_3, the finite field with 3 elements, and form the algebraic closure of it, the smallest field with characteristic 3 s.t. every polynomial in it splits into linear factors. This is endless. You can form F_5[x,y,z], take field of fractions, then algebraic closure. The algebraic closure of C(x) deserves to be called the field of Algebraic Functions over C. 5) p-adics. He mentions them in another video I think? You can combine them with all the other examples. "p-adics" in F(x) will end up being Laurent series in x, another example of field. 6) Power series, Laurent series, Puiseux series, etc. 7) Infinitesimals, that is, Levi-civita fields, Hyperreals, Surreals, etc. I hope I didn't forget anything! PS: Of course I forgot; Quaternions, Octonions and more exotic constructions like that.
John Conway is amazing, although I'll admit I mostly know him because of the Game of Life. As I recall, in an interview he expressed an annoyance that that was the thing for which he was known to many people (I believe it was a Numberphile video), but it's still a beautiful piece of maths (it's Turing complete too, as I recall), and exposed many people to cellular automata as a subject. Langton's Ant is another interesting automaton, and I got part way through trying to knit a scarf in the pattern of Rule 135, but never finished it.
Operations in whole numbers and in rationals are defined in terms of representatives of the classes, so it should be mention that one should prove that the result doesn't depend on the choice of representatives. I think that wasn't emphasized in the video.
Thanks for watching everyone! Don't forget to submit your questions on my subreddit for a potential 10k Q&A video. www.reddit.com/r/anotherroof/
This concludes my series on defining numbers from the ground up, but there are plenty more videos to come.
@Spydragon Animations Simply use the definition with [(ac + bd, ad + bc)] for the integers. You can show that "negative * negative = positive". In particular, you can show that (- a) * (- b) = a * b if this is what you are unsure about
@Spydragon Animations The partition kind of "drops out" from the relation definition. Two elements are in the same partition if they are related. If you're being formal, you have to prove that (2,2) is related to (1,1) just as much as you have to prove that (-1,-1) is related to (1,1).
I think, when he was at that part with the board and all the number pairs laid out in a grid, he was already being less formal - by that point he has shown that the definition is the same as our intuitive understanding of rational numbers, so he relied on our intuition to explain things visually without interrupting the rhythm with unnecessary proofs.
I hope that helps, even if I'm a few months late :)
I have learned *_e v e r y t h i n g !_*
Why didn't you define the real numbers as an integer-subset with a highest element, representing the sum of 2^(each subset-member)? This allows you to avoid the rational numbers altogether. Also, why don't you consider quaternions numbers?
I prefer (a,b)={a,{a,b}}
"Oh, so you like math? Name every number"
This guy:
ℂ
Well... Have you heard of quaternion-surreal numbers?
he didn't just name all the numbers. he defined all of them with axioms too
Yunno I kinda think someone stopped at one hundred and fifty-two meameamealokkapoowa oompa, seven googolplex and twenty-four
:/
@@FireyDeath4 Lmao
cant wait for 30 years down the line when after reviewing all of the foundations of mathematics he drops a 19 hr video proving fermats last theorem
i gennuenly don't know if you are using that as an exemple of "really hard thing to explain" or of "nobody solved this yet". If it is the second option... Someone did.
@@poketoscoparentesesloparen7648 the nature of his content is walking through proofs to explain really hard things.
I soooo hope he does!!!!
@@jongy that, and you need LOTS of math to do the proofing of Fermat's last theorem
@@radbarij oh absolutely. I wad talking more about the requirements to even understand the proof, and only in a wishy washy, approximated way
As a graduate engineer I always thought I had a reasonable understanding of maths, now I see I was just given a box full of tools and shown how to use them, but with no explanation as to why the tools work. The biggest take for me from this video is that there is a fundamental difference between integers and naturals, the difference between the other sets being a bit easier to 'see'.
Yes we usually use Z+ as a replacement for N as to unambiguously account for 0, even though these two groups just happen to represent the same numbers
Yes, engineering curricula are designed that way because you want to go out into the real world and start solving real engineering problems (and earning money) as soon as possible! You don't want to spend 4 years learning all about the fundamentals of math used in engineering, then another 4 years learning about all the physics used in engineering, then another 4 years learning about all the chemistry used in engineering ... it's turtles all the way down, and you'll die before you graduate! Besides, if you want to learn more, you can always do so later, either in your spare time (like I'm doing now), or by going back to school.
In Icelandic, Imaginary numbers are called _"þvertölur"_ or "Across Numbers" (It sounds a lot less awkward in Icelandic) and complex numbers are broken down to _Vertical_ and _Horizontal_ components (Or just length and angle, if that's what you prefer); Because Imaginary numbers are just as real as any other..
I like the term "lateral number" for English.
I personally treat complex numbers more geometrically than algebraically, so I call them "spherical numbers" due to their intrinsic connection to spherical geometry. The name also hints at the possibility of numbers for other geometries, which do in fact exist, though they lack some of the nice features of the complex numbers.
Polish doubles down on the imaginary and the number "i" is called "jednostka urojona" or "Deluded Unit"
as though we've lost our marbles, and gone fucking mad square rooting negatives.
@@angeldude101that's really interesting, do you have a name for these other numbers? I've never heard of that, but it's really cool!
@@Zaniahiononzenbei There are two main cousins of the spherical/complex numbers. One is called the "dual numbers," and the other is, depending on who you ask, either the "split-complex numbers," or the _"hyperbolic_ numbers." Knowing that the spherical unit squares to -1, I'll let you guess what its cousins square to.
This is such an incredible video massive props to you for taking on such a massive project and such difficult concepts and explaining them in a way that I actually feel like I understand :D
these videos are actually some of the best educational content i've ever seen, please don't stop making content
Okay, N to Z, Z to Q and to a lesser extent R to C have very standard definitions. I‘m really excited for what version of R you choose tho.
My favorite is the completion as a metric space. It requires a lot of conceptual overhead (if compared to dedekind cuts), but it’s so natural and gives a good intuition why we use reals to begin with
I like the Surreal Number construction of real numbers. I find it easier than dedekind cuts.
@@louisreinitz5642 surreal numbers define reals in pretty much the same way though;
Honestly opened up the video saying to myself "surely it isn't dedekind cuts", and was pleasently surprised
I don't know why it's said we can never fully write down pi when it's easy: π
Sure, writing it down as a decimal number is hard, but if you pick the right base it's easy! In base π it's just 10. Now, sure, that's an absurd base for everything else but sometimes sacrifices need to be made!
More seriously - I am loving your series so far. Thanks for the hard work you've put into it.
You can also just write down the quantity without using the symbol for it. π is _defined_ as the integral of 1/sqrt(1 - x^2) on (-1, 1).
@@angelmendez-rivera351
π = ∫ [ ̠₁ ¹] 1/√(1-x²)
?
@@xXJ4FARGAMERXx Yes
We can do the same tricks that we used to jump up a number system, to define pi, though it takes some steps. We need to define powers, using ordered pairs. Then we can define rotations, where powers of a complex number are equivalent if they give you the same complex number but scaled. That leads naturally to exp and ln.
Firstly, pi in base pi is not pi. It is "10". As that's how base systems work. But base pi doesn't really make sense, you end up with an unevenly spaced number line.
oh my goodness when I saw a new proof video was out I got so excited I could hardly wait! Your channel is my comfort channel and I really appreciate the work you do, and the jokes and the math and the bricks and everything!! Thank you so much
Comments like these make my day, thanks so much for watching and sharing :)
I'd love to see an appendix video of sorts on transcendentals, though that certainly sounds like a challenge to make. This series has been great to watch, thank you
Transcendentals are a subset of real numbers, (or maybe complex if you want) so they can be expressed in the same way. You just define it as the set of all rational numbers less than it paired with the set of rational numbers more than it.
Just realized I said transcendentals but I meant surreals. these names are getting out of hand
@@thomaspeck4537 No, the transcendental numbers are a subset of the complex numbers, because the algebraic numbers are a subset of the complex numbers. The algebraic numbers are defined as the algebraic closure of the rational numbers, and this closure necessarily includes i. Therefore, they are not a subset of the real numbers.
@@Cammymoop To explain the surreal numbers would require an entire video series. An appendix certainly would not be sufficient. You would need to start by changing the axioms of set theory, since in reality, if we are being rigorous, the surreal numbers do not exist in the Zermelo-Fraenkel axioms, even if you include the axiom of choice or its negation. You need axioms of set theory that are equipped with the ability to speak about proper classes, in order to define the surreal numbers.
"a super Saiyan set is one with cardinality greater than 9000"
You're a fakkin mad genius
This is one of the most geeky, pedantic and "why does anyone care" videos I've found so far; but yet it is so beautiful (and nicely presented). The definition of negative numbers just drops out so easily.
You did manage to refrain yourself from the obvious joke: "Z from the German for 'zee integers'" ;)
“Geeky, pedantic and ’why does anyone care’” - Welcome to pure maths
Maybe it is because, as a British person, he pronounces z as "zed" instead of "zee"
@@egwenealvereiscool7726 and germans dont say zee they say die
For anyone who wants to now the actual answer: Z stands for Zahlen ich just means numbers
Yorkshire man invents numbers. More news at 7 (which has just been invented)
More news at the set containing ...
Ey up! Ain't from Yorkshire pal.
*Man with an accent vaguely resembling the Yorkshire accent
@@gracenc vaguely Yorkshire? I'm from Lancashire -- and you're all lucky I'm not one of those proud Lancastrians who actually care about this sort of thing!
@@AnotherRoof haha that's what my friend characterized your accent as, I'm not a native speaker so 🤷♀
"Dedekind dedicated his kind name"... lol
Really hoping to see Dedekind cuts definition of R, but it might be a bit too advanced for TH-cam. Definitely gonna be a banger video though, we all know it
Me too. I don’t have the requisite knowledge to understand the formal definition, so I was hoping there would be a more intuitive way to understand them without getting rid of the complicated details; this channel seems very good at that.
@@gracenc Let me know how I do.
Dedekind‘s construction is relatively simple compared to completion of Q as a metric space. Most of the ingredients are readily available at this point in the series. But the completion thing connects most naturally to what we’re actually doing with reals
@@AnotherRoof You could perhaps take a moment to emphasize on what you wrote for the "R" block at 1:11:31, which might not be obvious from the chalkboard: That the "mother set" in this case is power_set(Q).
.
This is important because (1) it is a conceptual leap from the previous cases (where the "mother set" is just a cartesian pdt), and (2) R is somewhat different than N Z and Q in "size", in that its construction asks you to start from something much "larger".
.
(Explaining (2), "size" of infinite sets, etc... = topic for another video maybe?)
@@strikeemblem2886 The power set is not a Cartesian product.
Yessss I love your videos, nobody can explain math as good and simple as you!
Well that title explains why it's a longer video
How do you see the video length before it's out?
OP has said it to his community
I was curious about the proof that the real number are continuous, at 1:06:50 you hint at it but it seems to me that all you are proving is that the real numbers are dense, as in there's never an empty gap. But the rationals share that property, you even proved it, yet there were numbers that were missing. How can we be sure that there are no numbers hiding between the real numbers?
Agreed, it seems like he left out the key part of the proof. As I understand it the way to prove the real numbers are continuous is to do the dedekind cut procedure again using the real numbers, and then you can show that the resulting number is just another real and doesn't give you something new.
It's very subtle. In one sense, there are numbers between the real numbers, because you can put more numbers on the line with constructions like Robinson's Hyperreals or the Surreal Numbers, etc. But in another sense, the reals are gapless in a way those things are not: for instance, if you have a nonempty set of reals that doesn't grow arbitrarily big (e.g. maybe they're all less than a million), then there is a "cap" real that is greater than or equal to everything in the set, but smaller than any other candidate. (The cap may or may not lie in the original set. Think about intervals like (0,1) and [0,1], both of which have cap 1.) This "dedekind completeness" works for the reals and stops working if you try to stuff even more numbers inside.
The short answer is, it depends what you mean by a number. There are number systems (see hyperreal numbers or surreal numbers) in which there are other numbers hiding between the real numbers. What is meant in this context is that the real numbers are "complete" (this is a technical term). There are several equivalent definitions of what this means, but perhaps the simplest one to see (relative to this video) is that if we were to repeat the process of taking Dedekind cuts we wouldn't get any new numbers. That is to say, instead of the sets A and B consisting of rational numbers, say they consisted of real numbers. Then the set of Dedekind cuts of this form would just be the reals again and wouldn't contain any new numbers.
At 59:35, since we want to use the Dedekind cuts to define the real numbers, if r is not a rational number (1:00:49) and not a member of Q, how can we then prove that there exists another rational number between m and r? Also, how are we certain that in the Dedekind cut, there aren't two or more irrational numbers? (great video btw)
41:05
I never saw why q != 0. I can see that if we were to include q=0 then this relation is no longer transitive.
(p,q) ~ (r,s) : ps=rq
=> (1,1) ~ (0,0)
(0,0) ~ (2,3)
But
(1,1) !~ (2,3)
There is a slight generalization of this construction in the mathematical subject known as commutative algebra. It is called the localization. However, even in this context, including 0 results in a structure that is not very interesting. en.wikipedia.org/wiki/Field_of_fractions#Localization
Any chance of you covering the Surreal numbers? It seems like a good place to go off of this.
I'd also just like to say that I really like that you're using physical blocks for your axioms/theorems and stacking them together as you go. I think it's a great visual metaphor for how math concepts build off of a few foundational concepts.
do you mean the trancendental numbers?, or do you actually mean the surreal number that include: the number greater than every real nubmber and the rnumber less than every real muber but greater than zero?
@@plopgoot5458 He means numbers greater and less than any real number. Unfortunately the surreal numbers' construction require the use of transfinite ordinals and thus cannot be a set as there is no "set of all transfinite ordinals" in ZFC. They are a proper class.
@@crosseyedcat1183 Is that an issue, that they are a proper class as opposed to a set? It is an important caveat, yes, but I fail to see what is so "unfortunate" about it.
@@plopgoot5458 He is talking about the Conway surreal numbers, which extend the real numbers, and they are the "largest" ordered field, in the category theoretic sense.
@@angelmendez-rivera351 The non-self-containing-set collection isn't a set as it neither contains, nor excludes itself
You have an amazing teaching style. I really appreciate your approach of laying out a few axioms and incrementally building up to every topic covered step by step. One thing that shows up in a number of math explainer videos is the Monster group, but without a background in group theory I've never had a foundation to be able to make sese of any of the writing about it, besides "big number interesting". Given your incremental approach to explaining things a series of videos on group theory that build up to the point of having the context and understanding to make sense of statements like "monster group M is the largest sporadic simple group", would be super nifty.
I've mentioned this in other comments and maybe I'll go into more detail in my 10k Q&A video, but my PhD was in group theory and I would love to do a series on the various families of finite groups. That said, because it a topic so dear to me, I want to gain more experience to make sure I do the topic justice!
That desire to do a topic justice is seen in your work. Regardless of topic I'm excited to see what you'll work on next. :)
Loved the series! I don’t know how useful it is for someone who isn’t already familiar with this stuff, but I like the balance between intuition and rigor here
No intuition, all rigor
@@spencersivertson9321 that would be coq, but without comments
Well, I was familiar with all the building blocks (relations, power sets etc.). The definitions of each class of numbers were completely new to me, though, but the explanations were impeccable and I understood all of them. So definitely very useful for me at least:)
When will you do the surreal numbers? I just learned about the (mostly) number "up the second" yesterday, and it's been a mind altering experience.
As you pointed out, something like that is a game, but doesn't have enough nice properties to even be called a number. But the surreal 1/ω^2, sure.
Here's a Dedekind cut definition of π:
Lower set A = Union over all n in N of the sets: { r in Q : r < 4 * the sum from k = 0 to (2n + 1) of [ (-1)^k / (2k + 1) ] }
Upper set B = Union over all n in N of the sets: { r in Q : r > 4 * the sum from k = 0 to (2n) of [ (-1)^k / (2k + 1) ] }
You're welcome! :)
I'm just curious: if any finite sum of a series is a rational number, how comes that infinite sum jumps to irrationals?
@@acrm-sjork That's because the rationals are not complete. Real numbers have the completeness property saying that every converging series has a limit point in the set.
@yanntal954 not sure that I got why infinite sum of rationals MUST or even CAN be irrational? Why not all sums of rationals converge to rationals? It is not contradict your statement
Love me some numbers.
I was hoping you might explore how C starts to lose properties compared to R. In a sense R is the top of the "number pyramid". Because you lose ordering in C, and the other way, you lose completeness in Q. If you go to quaternions you lose more (commutativity), and octonions lose even more (associativity).
You lose any natural or intuitive ordering in C, but it would certainly be possible to define an arbitrary ordering. Start by saying that anything that is closer to the origin on the imaginary plane than x+iy is less than x+iy, and then define an arbitrary ordering for all the points that are equidistant from the origin (e.g. saying that x is the highest within that set, and then going clockwise around the circle).
@@stephengray1344 Or tile the plane with a Hilbert Curve.
You lose ordering that respects addition, sure. However, you gain algebraic closure, which is arguably a much stronger and more important property.
@@stephengray1344 Yes, but no. In the context of fields, when we talk about ordered fields, we talk about fields with an ordering such that the field operations are isotonic (in a standard sense) with respect to the ordering. The real numbers are an order field. The complex numbers are not.
@@stephengray1344 Also, specifying that an ordering does exist is sort of redundant, since by the well-ordering theorem, all sets can be well-ordered. I know the theorem requires the axioms of choice, but if you relax the conditions so that you only need a total order, then you can significantly weaken the axioms needed. Also, most mathematicians do accept the axiom of choice, despite it being controversial.
In a very real respect, you're single-handedly undertaking what PBS Infinite Series was doing. :) I greatly appreciated that channel while they were active, but it turns out there are just not that many channels engagingly tackling topics at the foundations of mathematics outside of "headline-friendly" ones like trying to quickly explain Goedel's theorems ad nauseum. (I'm sure you'll get to those theorems, too, but if it's anything like this presentation, I expect it will be thorough and cohesive!)
It's funny you mention PBS infinite series, I was literally just thinking about that channel because I got a question asking if I'll ever make a video on the axiom of choice and I thought their video was good. I'm glad your enjoying my videos! A slight disclaimer: I'm not planning for every single one of my videos to be a deep dive into foundational topics, but I'll always try to approach ideas with my own style and an emphasis on rigour. Anyway thanks for watching :)
@@AnotherRoof Of course! I'm enjoying it, and it's been long enough since my university coursework that I appreciate the refresher either way. I find this stuff fascinating, even if it has very little bearing on my day to day life, but it's not at all like a bicycle; the gist of it may stick for ages, the details and practice not so much.
Why did they stop?
This is incredible, you are the first person in my life that I help with patreon.
At 59:00, you explain how to construct a Dedekind set, defining it as being all the elements of the rationals which are smaller than a given number (r). You then prove that this set will not have a maximum value, because of the density property. This implies (as you say in the video) that both m and r (in your demonstration) are part of the rationals.
However, you then says that "this is how we define the real value r". But wasn't r supposed to be a rational? Otherwise we cannot use the density we used to construct the Dedekind set
Let's say that we have some rational number m that is smaller than r (which is assumed to be irrational). Then we can add a positive rational number n to m to make a new rational number that is larger than m but still smaller than r. And we can make n as small as needed because there is always another rational number between 0 and n.
That's as far as I got, but I'm not convinced it's a complete proof since I haven't ruled out the possibility of there being a real number that is "smaller than" all the positive rational numbers but larger than 0
This comment was made during the stream
I wonder if we’ll have transcendentals… or even quaternions! Or haha split complex numbers! Yeah I doubt it… man there are a lot of numbers
I've always loved educational videos, but this gets me some different levels of joy. I'm actually having popcorns right now while watching. Thanks so much for this series of videos!!
Hoping that when going from Q to R we will also visit some other countable sets first- constructible, algebraic, computable, and definable numbers
Quality over quanity
I would have liked to see the proof that an equivalence relation always induces a partition of a set
Maybe I'll add it to the notes but when I construct the relatives of 1 in the parity relation I outline the key idea!
It is quite clear, yes. It was just the first thing I noticed that you neither proved nor explicitly left for us (the viewers) to prove ourselves
Hm, to get from a countable set to an uncountable set seems like a monumental challenge. I'm looking forward to the video!
He actually does it already at 5:00. The power set of an infinite set always has a greater cardinality than the original set.
@@cadekachelmeier7251
I'm sure you're right, but that seems weird to me. It's just 2^(countable infinity), right? Why doesn't that just return countable infinity again?
@@vanderkarl3927 For reference, it's Cantor's Theorem. Honestly, it's pretty much the same logic as Cantor's diagonal argument. It's just more abstract because instead of reasoning about lists and numbers, you have to reason about any potential set and a set containing an infinite number of other sets. Other than that, it's just constructing an element that can't exist in any potential mapping, same as the diagonal argument.
en.wikipedia.org/wiki/Cantor%27s_theorem
@@vanderkarl3927 Infinite Series on TH-cam has a video on Cantor's theorem about why power sets get bigger, even in the infinite case.
@@vanderkarl3927 2^Aleph(0) is greater than Aleph(0). Cantor's theorem states that for all cardinal numbers κ, κ < 2^κ.
Complex numbers are just the algebratic closure of the topologic closure of the groupification under multiplication of the groupification under addition of the monoidisation of natural numbers provided with the standart z-module stracture.
I wonder if Star Trek wanted to troll us when they introduced the Q continuum
I think Star Trek chooses that name to describe something paradoxical. Q continuum are both in Q but also a continuum.
I can't believe I never made that connection in 35 years.
3:14 Now i see why you included 0 in the natural numbers.
So the center of gravity stayed over the natural numbers when you defined the other numbers.
Petition to rename "imaginary numbers" to Lateral Numbers, and rename "Real numbers" to Medial Numbers when?
I wholeheartedly agree that "real" and "imaginary" are terrible names. Not sure I'm with you on the suggested alternatives though...
@@AnotherRoof because it sounds nicer
cute line
17:00 I'm curious... I notice that only 4 of the 8 potential combinations of reflexive/symmetric/transitive are represented here. Do relations exist for all of the other possibilities, or are any excluded? In other words, are the three properties completely independent?
Any comment about how much I love your videos would be an understatement
2:15 Illustrating the gap between the whole numbers as being foamy is such a splendidly intuitive way of conveying that only a proportion of that space can be represented with fractions.
Actually you have skipped redefining operations on real numbers. But some of them are not trivial like multiplication or division, so it would be interesting to have it in the video. But I understand that the video is quite long. Great series!
Yeah I just kind of glossed over that and said "it can be done" haha -- probably should have just displayed them at least for a few seconds but yeah, 80 minutes was long enough!
@@AnotherRoof You should have made it a two-part video
The most beautiful thing I saw in this video was that the question "why minus times minus is plus" never arises! It just is baked on the rules of the definition for the integers. Great stuff, marvelous presentation.
ah yes my favorite series: "tricking people into learning discreet mathematics"
This is literally the best video i have ever watched. Please keep on making beautiful content like this!
Your proof of the denseness of Q made me realise how denseness is related to division. It's so obvious now! Of course if a set of numbers (containing at least two members) is closed under averaging, it is dense! And if it's not closed under division of two, it's not going to be closed under averaging.
44:17 the bricks are a lie :o
Wow this was an awesome culmination to an awesome series. I am not a mathematician, but I have taken some advanced courses (Real Analysis, Measure Theory, etc), so I knew more or less what you were talking about with all the "hand waving". I had never seen this conception of numbers, and I loved it! Keep up the great work!
I feel like you keep bumping right up against group theory and algebra. Hopeful to see some vids on those topics some day from you!
Fantastic video as usual.
I mention here that my PhD was in group theory. I want to get more experience making videos before I make one on the topic I love the most!
@@AnotherRoof it's also my favorite area of mathematics, and I am REALLY enjoying your videos and style and just everything, hence my yearning for the topic.
Whenever you do get to it, I'm sure it will be fantastic. Thanks again!
I've not looked at group theory at all but I kept feeling like category theory was around the corner. Not that I have a much of an understanding of that either :). I'm sure I'd enjoy your take on either/both.
@@brendanobrien4095 I only have an undergrad degree in mathematics so I haven't studied category theory at all (that's a graduate topic, 100 percent), but my understanding of it is that it is essentially another layer of abstraction on top of group theory/algebra (those terms are not at all interchangable, but whatever; it's fine for our purpose here.) Like, my understanding is that category theory is essentially an algebra of algebras. But again: I don't have any formal training in category theory. So if I'm wrong, that is not unexpected.
Also, you should look at group theory. If you know what category theory is, you ought to know what group theory is :p. No judgment here, just a recommendation.
It would be a cool video idea for sure though. Not sure how much another roof knows about category theory, but any video he wants to make is fine by me, cause he's killing it. I've rewatched every video at least twice and just loved every second of it.
@@kruksog You should understand that my path to this stuff is atypical. I am a software engineer and I took none of this in college. I came across category theory by way of lambda calculus which I was studying for applications in functional programming paradigms. This is the first I've even heard of group theory and I'm very interested if it relates to these topics.
wow this series was fantastic, but i was shocked at the end to see how many bricks you built up over the whole thing... amazing work
The video omits defining the < relation for integers. It defines it for natural numbers: a < b is the same as a ∈ b, but it doesn't define it for integers. Yet, when defining < for rational numbers, it states it using < for integers.
That's right! My original cut of this video was much longer but I wound up cutting out some stuff like this because it got a bit repetitive.
I've watched the real part of this several times, and I think I get the process, but why is it so different than the rest of the parts? Every other step takes a pair from the set made before, and than the real numbers seem to need to do this dedican cut thing. If you try to skip it, and just try to go from rational to complex, can you get something sensible or does it just always break?
I mean, you can define the rational complex numbers (numbers a + bi where a, b are rational), and those are useful in some contexts. However, you don't really gain much by doing so. The real numbers are difficult to define because they are "uncountable" (a technical term, worth looking into if you are unfamiliar with the concept). The operation of taking ordered pairs and then partitioning can never produce an uncountable set from a countable one.
There are other ways to define R. One way-which is more like the previous ones-is the following: Given two Cauchy sequences (a_n) and (b_n) of rational numbers we say that (a_n) ~ (b_n) if and only if the sequence of differences (b_n - a_n) has limit 0. This defines an equivalent relation on the set of all Cauchy sequences over Q, therefore partitioning this set. A real number r is now defined as an equivalent class of those sequences, that is r=[(a_n)].
Keep in mind that we don't need irrational numbers or the concept of limits (expect for 0 which is no problem) for this definition. The term Cauchy sequence can easily be defined for rational numbers.
30:45 “it’s like upgrading the operating system; it does everything the old one did, and more.”
Little did he know that now upgrading your OS will remove features.
Thanks this was very thorough, and I can see how various ideas take the 'blocks' of number theory that you explain so well, and build a whole set of new theories out of them... i find this actually useful as a presrequisite to the PBS infinity series... Thanks!
you say not to worry about tagging ordered pairs with 1 and 2 because we can use different numbers, but what if we want to use the set of all natural numbers? We’d need new tags to avoid duplicates. And then what if we use the set of all of those tags, and so on. There must be a cleaner solution
I think this should work { {x,{x}} , {y,{{y}}} }
Maybe there’s a case where this runs into a duplicate but I dont see how
Yes, I wish I'd never used the Hausdorff ordered pair but I felt it was more intuitive to non-mathematicians. Anyway, for a set A it is always possible to find a thing which is not contained in A, namely A itself by the axiom of regularity. So if worse comes to worst we can use the set A itself to tag the elements of A. So an ordered pair (x,y) with x in A and y in B could be encoded {{a,A},{b,B}}. Hope this helps!
There are sets not in Naturals for example sets {0,2} and {1,2} you can use this two for tagging. This set are distinct from {},{0},{0,1},{0,1,2} which are the set represented by 0,1,2 and 3. Sets {0,2} and {1,2} are not in Naturals.
@@AnotherRoof this works except that a lot of the cartiesian products you use in this video are of the form A×A. So you probably want something like A and {B} as the two tags (which should work for this video/most applications in math).
I was fairly familiar with the construction of the rationals from integers and complex numbers from rationals, but had never seen/considered the similar construction of integers from naturals! In hindsight, the parallels seem obvious, but I guess I'd never really considered "negativeness" with enough of a critical, "what _are_ you, really?" attitude.
I don't know if he took my suggestion of laying down the bricks like a wall in his outro, but regardless of whether he did or didn't, it looks cool. It feels like we all laid the bricks together. Though with math, it took us quite a few hours to get there...
how are you going to compare these?
Great. Please also do Surreal, p-adic, hyper real, hyperbolic, quaternion, octonion, sedenion and beyond.
"Complex Literally" is somehow the boring type of complex on this kind of channels...
I'm actually proud of myself because I barely understood the information presented here. I'm going to have to binge watch your videos, I want to be sure I understand things of this nature.
0:25 Temperature and altitude are bad examples for negative numbers. They have an arbitrary zero value. Imagine it like this: Define 0 °Apples as 10 Apples. A basket with 3 Apples in it contains −7 °Apples. It that sounds like nonsense to you, it’s because it is, but temperature and altitude work exactly like that, it only feels natural because of exposure.
There is absolute zero of temperature (0 Kelvin) and altitude (the center of the earth). Negatives only exist in differences; if you ask me “how much taller is your brother than you are” the answer is −2 cm because he’s actually the smaller one, still body height can’t be negative.
43:58 I thought the rocks were real the whole time.
Edit:
45:52 What happened to the rocks?!
Edit 2:
46:13 Rocks are back again :)
Edit 3:
1:11:28 Now the rocks are real?! 1:16:09.
Edit 4:
Now they are changing colors? 1:17:58 - red, 1:18:13 - blue, 1:18:34 - green, 1:18:55 - yellow, 1:19:10 - pink.
Edit 5:
Watching the video again and it turns out the rocks were real 4:58, 10:37 except that one time when they weren't.
Edit 6:
9:13 And what's up with the paper? 9:32, 9:46, 9:59.
1:02:32 - "we can describe all of the irrational numbers in this way". This kinda made me wince. I'm not sure, is it actually technically true, or just an inaccuracy in the way you describe things? I'm not intentially trying to nitpick, I'm actually confused. Sure, you can come up with a Dedekind cut for each real number *you know of,* but is this *every* real number? If it is, it would imply there are only ℵ₀ real numbers, since the set of formulas you can write is enumerable. Which, as we know, isn't true. So, did we actually define *all* real numbers in this video, or just *some* real numbers?
How would you come to define a real number with this cut thing if you have a uncomputable number (a number with infinite random digits that cannot be expressed as an equation)?
Your slight of quaternions has awakened the ghost of Sir Hamilton, who is now coming to haunt you until you make a video about hypercomplex numbers.
At 52:46, when the expression is multiplied by 'q' is important to notice that q > 0, otherwise the ''.
But we know that q is positive because we chose the right representative of the rational number.
1. There are operations you can’t perform on the complex numbers, like division. Division is not a total function; z1/ z2 is not always defined.
2. I think you cut out a bit too many of the details of real completeness; as described in the video it sounds like the same property as the density of the rationals.
except a method for identifying what is in between 2 real numbers is identical to the method for identifying the real numbers in the first place and thus anything in-between them is itself a real number. This fact means necessarily that the real numbers are complete as that term is defined in the video.
Division is never a total function. There are no rings in which you can divide by zero. The closest you can ever get is to say that you can divide by everything except zero, and the complex numbers have this property
I agree that the proof in the video describes density, not completeness.
I would like to hear what he cut out with regards to completeness. I left another comment asking for such because I still don't know why the reals are complete and not merely dense.
The method you use for constructing pairs is quite unaxiomatic, as you there is already an infinite set with a full ordering.
The more common definition of an ordered pair is {{a}, {a, b}} for (a, b)
one thing isn't clear to me: when defining the cartesian product, you said we can't just apply the set rules infinitely many times to construct the product of infinite sets (which i'm not sure is too different from using recursive definitions or proofs by induction, but that's another story), but to work around it you used power sets of potentially infinite sets without first defining them. does that mean the regular definition applies? but that would mean applying infinitely many times the union rule, which shouldn't be possible?
Definition of addition, definition of addition, definition of addition, definition of addition, definition of addition...
Gonna be honest, I'm still a little lost on the part about completeness of reals. You proved that there's always a rational between any two real numbers; how does that prove that there are no holes? The proof that there's always a rational between two rationals doesn't prove the rational numbers are complete.
I would think a proof would go sort of like how you proved √2 isn't in the rationals, but instead some general function f such that f(R) outputs a number between two real numbers and is not contained in R, cannot exist.
I'm 10 years separated from my compsci BSci, and this video is an entertaining bit of bringing it back to the fundamental axioms of the inductive power of diagonalization. Great stuff.
I feel simultaneously smarter and dumber after watching this video
im a last year student on physics and still find your channel so amazing and illustrative. keep on with this!
I knew you couldn't possibly be defining every number ever, but I could not imagine you would speak so fondly of Conway after leaving his surreal numbers out!
Maybe one day!
In his defense, defining the surreal numbers would require an entire video series of its own.
This is like watching the channel “primitive technologies” but for math heads
I think to make the dedicind cut more concrete, a set with no greatest numbers could be the set of all numbers less than 2. is there a greatest number less than 2? no, cause you can always stick more nines behind the previous answer.
great video!
also nice glitch in the matrix at 44:00
at 45:53 also
At 1:00:05 r isnt in Q, why u can apply the density rule?
He didn't specify whether or not r is in Q. He just said, "pick a number r".
The argument is just to show that such open sets, with no largest member, actually exist. You can construct such sets out of rational numbers, as shown in the proof. He later shows that the reals are _also_ dense in the same way, and therefore you can construct such sets using them as well.
8hour slate
hithings
This is the Assembly language of Mathematics, Similar to what we have in Computing!
So at 35:07, it looks like you're assuming that a negative times a negative is positive. Is that an assumption or is that proven another way and just left out here?
Similarly, at 48:02 you're defining all the operations for the rationals. But is that proven in this system off camera or just assumed as a definition?
The former can be proven from the definition of multiplication for the integers or rationals. The latter is just a definition chosen so that the numbers work the way we expect. You could prove those operations agree with the integers' operations when all the rationals are like z/1 for some integer z, though.
Howcome you chose to use that definition of ordered pairs rather than the more common one?
(a,b) = { {a}, {a,b} }
I had to pause this several times to revel in the mind blasts this gave me. It's insane how much sense it will makes now. Thank you for the concise explanation!
YOOOOOO LET'S GOOOOO I'M WAITING
I always thought Z for Integer means Zero since Zero is literally the center
Rationals be like: infinite set of infinite sets of ordered pairs of infinite sets of ordered pairs of sets of [some stuff and also empty sets].
Great for a start. Examples of additional stuff:
1)Quotient systems of numbers, for example Z modulo an integer n. In particular, if n= a prime p, the only interesting case to be honest, then Z modulo p is a field, i.e. it satisfies all axioms satisfied by Q and beyond. It's an example of a finite field; it's incomparable to N, Z, Q, R, C etc. It can be shown that ANY finite field is essentially unique (i.e. unique up to isomorphism) and has q=p^n elements, where p a prime and n any integer from 1 and up. It's symbolised F_q. The prime p is called the Characteristic. Every field has a characteristic which is defined to be the smallest positive number that annihilates (i.e. if multiplied with, it zeroes out) all elements, equivalently the multiplicative unit, if such a positive number exists; otherwise the characteristic is 0. So, characteristic p means that 1+...+1 (p times) is 0, and is the first time we get 0 if we add 1 a non-zero number of times to itself, equivalently all other numbers in the list of 1+...+1 etc. are distinct. Likewise if we add any other element to itself consecutively.
Every field has characteristic 0 or a prime number p. In the first case it contains a copy of Z, thus of Q (considering fractions), thus it's an extension of Q. In the latter it contains a copy of F_p. So basically, if we stick to only Q, R, C we miss out on plenty of action.
2) Transcendental extensions: If F is a field and V a set of "variables" (could be finite or infinite) we can form the set of polynomials on the variables of V and call it F[V]. The elements of F[V] are called polynomials in V with coefficients in F. They are (including 0) the finite sums of monomials of F[V], who are in turn finite products of elements of V (could be empty, which gives 1), multiplied with an element of F (could be 1). If we copy the construction that gives Q from Z (it's called the "Field of Fractions" construction), we get the set of rational "functions" in F[V], symbolised as F(V). The idea is that F(V) is the smallest field containing F and the elements of V, which are considered ALGEBRAICALLY INDEPENDENT over F. This means they satisfy no non-obvious algebraic relation. In case V={x_1,...,x_n} this is just the sets of polynomials and rational "functions" in x_1,...,x_n repsectively. If furthermore n=1, then we get V={x} and we write F[x] and F(x) for the set of polynomials in x with coefficients in F and rational "functions" in x with coeffs in F. F(x) is called a simple transcendental extension. Basically, it behaves like π and e in R with rational coefficients, every algebraic relations using x and elements of F are trivial, i.e. they merely follow from the axioms of arithmetic, equivalently:
a_nx^n+...a_0=0 => all coefficients are 0.
3) Algebraic Extensions: We can add "algebraic relations" to F(V) above using quotient constructions (an example below), so that the images of V are no longer algebraically independent. Alternatively, if F, L fields such that F is a subfield of L (like Q in R and R in C) then L is algebraic over F if every element a in L satisfies a polynomial equation with coefficients in F.
C is an algebraic extension of R since every complex number has a polynomial (more below), R is DEFINITELY NOT an algebraic extension of Q, in fact almost everything is transcendental over Q.
So, in the example of F[x] x a variable over F, we can take any polynomial m(x) (a prime/irreducible polynomial for good results) and form F[x] modulo m(x). This isn't a field if m(x) isn't irreducible, in fact if m(x)=p(x)q(x) then p(x) and q(x) modulo m(x) would be non-zero, but their product m(x) is 0 mod m(x) by definition.
If m(x) is irreducible, then F[x] modulo m(x) is a field, similar to 1), and is the "smallest" field containing F such that m(x) has a root; clearly x mod m(x) satisfies m(x) itself. If we define x mod m(x)=: ρ, then we call the above set F[ρ]=F(ρ) the simple extension defined by a root of m(x). The reason the above equality holds is that inverses of non-zero elements already exist, due to the following:
F[ρ] is the set of all (unique) linear combinations of 1, ρ, ρ^2,..., ρ^(n-1) with coefficients in F. This is being proven by Euclidean division of polynomials. So, any such non-zero combination, corresponds to a polynomial g with a degree smaller than the one of m(x), thus it isn't being divided by m(x), and since m(x) is prime it must be relatively prime, thus there exist polynomials s,t such that sg+tm=1, which modulo m(x) gives sg=1 modulo m(x) (same proof works in 1) ).
So we could define "square root of 2", and any other algebraic number in general, without dedekind cuts necessarily (if we aren't interested in ordering numbers on a line) as follows: Start with Q, form Q[x], then consider m(x)=x^2-2 which can be shown to be irreducible, and form Q[x] mod m(x). We could call it Q[j]=Q(j), where j is an "imaginary number" such that j^2=2 (note it could also be minus square root of 2). It contains all elements a+bj where a,b are rationals; any such writing is unique and if we have a non-zero element, we can find the inverse by multiplying and dividing with the conjugate element a-bj.
Same construction can give us C from R, just take x^2+1 as m(x).
In general if L is an algebraic extension of F, and a an element of L, then the polynomial of F[x] of smallest degree having a as a root is called the MINIMAL POLYNOMIAL, and it also happens to be the unique polynomial which divides all such polynomials. If it is defined to be m(x), then the smallest field containing F and a, F(a) is going to be essentially the same thing as the above construction F[x] modulo m(x). The degree of m(x) is called the degree of a. In particular F(a)=F[a]. Note that if F=L then the minimal polynomial of a is x-a which has degree 1.
We can keep adjoining elements to a field F. If a is a root of a (w.l.o.g. irreducible) polynomial p, then we form F[a]=F(a), and then if b is a root of the (irreducible over F[a] ) polynomial q, we can form F[a,b]:=(F[a])[b]. In theory, every algebraic extension can be obtained by simple extensions, if we "take the limit" for infinite-dimensional extensions. So we can have Q[sqrt(2), sqrt(3)] etc.
Theorem: Any field extension can be seen as a transcendental extension followed by an algebraic one.
So, wlog for more "numbers" we can take, for example, C(x,y), C with two extra algebraically independent elements, identified by a "surface" over C if you do algebraic geometry, then you add the relation, say y^2-x=0 and you get... something.
4) Algebraic closure. If you take the subfield of all complex numbers who are algebraic (over Q) you get the Algebraic Numbers. This is called the algebraic closure of Q, since every polynomial over Q has a root now (but without the additional transcendental elements of R and Q). You can always take the algebraic closure of any field! So, you can take e.g. F_3, the finite field with 3 elements, and form the algebraic closure of it, the smallest field with characteristic 3 s.t. every polynomial in it splits into linear factors. This is endless. You can form F_5[x,y,z], take field of fractions, then algebraic closure. The algebraic closure of C(x) deserves to be called the field of Algebraic Functions over C.
5) p-adics. He mentions them in another video I think? You can combine them with all the other examples. "p-adics" in F(x) will end up being Laurent series in x, another example of field.
6) Power series, Laurent series, Puiseux series, etc.
7) Infinitesimals, that is, Levi-civita fields, Hyperreals, Surreals, etc.
I hope I didn't forget anything!
PS: Of course I forgot; Quaternions, Octonions and more exotic constructions like that.
this is becoming my go to channel on "seemingly easy things explained super thorough" section.
John Conway is amazing, although I'll admit I mostly know him because of the Game of Life. As I recall, in an interview he expressed an annoyance that that was the thing for which he was known to many people (I believe it was a Numberphile video), but it's still a beautiful piece of maths (it's Turing complete too, as I recall), and exposed many people to cellular automata as a subject. Langton's Ant is another interesting automaton, and I got part way through trying to knit a scarf in the pattern of Rule 135, but never finished it.
5:04 random definition of super Saiyan set...
Yay, Dedekind cuts instead of Cauchy sequences!
Operations in whole numbers and in rationals are defined in terms of representatives of the classes, so it should be mention that one should prove that the result doesn't depend on the choice of representatives. I think that wasn't emphasized in the video.
I thought the same.
Oh no… oh no…