 I promised earlier that in lecture 11, we would explain why an order geometry is called ordered. What order are we referring to? First of all, we have to make sure we understand what an order is in the first place. It's an order, it's more specifically a strict order to not be confused with a partial order, although the two notions are related. I'll explain the relation just a second. We say that a strict order or a strict ordering, it's a relation on a set X with itself. By definition, a relation is a subset of X cross X there. And a strict order satisfies two assumptions, two axioms. The first one is the irreflexive property. That is, we never have that an element X is related to itself. So this is like the antithesis of the reflexive property that we see with equivalence relations. We don't allow an element to be related to itself. We also require that it be transitive. And this is the same transitivity axiom that you see with an equivalence relation. If X is related to Y and Y is related to Z, then it must be the case that X is related to Z. And so this gives us what we call a strict order. And the way we write it, we often think of it as like less than, right? X is, if X is less than Y and Y is less than Z, then that implies that X is less than Z, all right? And you can't be less than yourself. I mean, it's like, what, you shorter than yourself or younger than yourself? It doesn't really make any sense with that. Now, whenever you construct a strict order, there are other relations related to it that are also important to remember. So the way we've drawn this, this little symbol here, we're saying that it's less than, we're generalizing the notion of the less than symbol, it's axiomatics there, is that it's irreflexive and transitive. But we could also talk about greater than, right? So we can define a new relationship, call it greater than if this one's called less than, in which case we mean that X is greater than Y if and only if Y is less than X. So basically when it comes to an ordering, there's a direction to it, the direction matters, we can define its inverse relation by flipping it around. So we can talk about less than or greater than, okay? And so when one talks about a strict order, we're talking either about the less than or the greater than, you can call both of them the strict order because you have one, you have the other. Associated to every strict order, there's also the related partial order. So you see a symbol like this, it kind of looks like the less than or equal to symbol and that's what we're trying to do here. So we think of this as less than or equal to. So an element X is less than or equal to Y if either X is less than Y or X is equal to Y. Now notice it can't be both, this is an exclusive order in the situation because of the irreflexive property, but we can modify the relationship. So do we want it to be strictly less than or do we want it less than or equal to? And we can do the same thing to the greater than symbol, we can have a greater than or equal to which means X is greater than or equal to Y if X equals Y or X is greater than or equal to Y. And so this gives us a strict order, less than and greater than, but we can also have a partial order, less than or equal to or greater than or equal to. The idea of a partial order and a strict order are very related to each other like we were able to turn, we can turn greater than, integrate and they're equal to. You can also go the other way around by basically forbidding equality in that situation. So if you have one, you have the other. If you have one of these relations, you have all four of them. And so we often talk about all four of them together. Now, the reason I started off with a strict order is because to prove that lines have an ordering to them, it'll be easier to do a strict order because strict order only has two axioms, irreflexive and transitive, transitivity, as opposed to a partial order here. It has three axioms where it's reflexive, anti-symmetric and transitive. So the proof will actually be simpler by using a strict order in the situation. Now, we actually are gonna talk about a totally ordered set or sometimes called a linear ordered set. This is a strict order that we've attached the so-called totality, the totality axiom to it. So given three elements, excuse me, given two elements of the set, we say an order is totally ordered if either x is less than y, x equals y, or x is greater than y. So one of those three relationships happens. We often refer to this as trichotomy as well. It's like, okay, x is less than y or they're equal or x is greater than y. They're always comparable to some degree. And so this is the totality axiom. Not sure why I can't say that in this video. Sorry about that. This axiom is also sometimes, so if you have an ordered set, so a set with a strict order or partial order because you have one hand with the other. A set with an ordering is called a ordered set. It's sometimes referred to as a partial order because some elements might not be comparable. But of course, if you have a totally ordered set, that means every element is comparable by this trichotomy situation here. This is sometimes called a linearly ordered set. Why? It actually comes from the theorem that we're gonna be proving in this video here that all lines are gonna be linearly ordered in an order to geometry. That's why we call it order geometry because lines will always have a total order to it. And in fact, a total order you can think of as a generalization of the ordering's online. So we're gonna see in a little bit. And so I should point out, of course, that there are orders that we've seen before. If you take the real numbers, we then say that a real number a is less than or equal to another real number b if the difference b minus a is non-negative. So notice this is, I'm using the partial order here, the less than or equal to. And so that way you get a non-negative. Because of a equals b, you'll be at b minus a, which is zero, that's non-negative, so that's okay. If you want to use the strict order, a is less than b, then you would swap this to say, oh, the two things are positive. It doesn't really matter which one you do. But the real numbers have a very natural ordering to it, okay? And this is also connected to other subsets of the real numbers, like the rational numbers, the integers, the natural numbers, they all have a total ordering. And when we think of the Euclidean plane, which the Euclidean plane, of course, is R squared, the points have an ordering on lines that is essentially this same ordering right here. And this is what we want to prove just right now, that every order geometry lines have a natural ordering to them. And it's essentially the same as the ordering of the points, because after all, you can think of the plane as like a ordered pair of points, kind of. At least in affine geometry, that's how the coordinates work, but in general, ordered geometries can order their lines hence the name. That then brings us to the main result for this lecture here, that every line has a total ordering. So this is the ordering of a line. Now it's not unique, you have some preferences you can make, right? So it's kind of like thinking of an Euclidean geometry, like where is the origin? You can make that choice, you can pick any point basically to be the reference, like this is X equals zero. So we have sort of that power as well, but really we get only one ordering up to isomorphism of regard to the ordering here. So given any line L, pick two distinct points, A and B on that line. Then there is a unique ordering on the line L such that A is less than B and given any between this relationship so that D is between point C and E, these are all points on the line, we're gonna either have that C is less than D less is less than E or C is greater than D greater than E. So you do have sort of like two choices there, which that's basically what we're trying to say here. So if you've given a line, you give three points on the line, right? C, D, E, do we want it to be ordered like this? So it's less than less than or do we want it to be ordered like this? So it's greater than greater than. You can pick the direction, but once you decide, oh, A is less than B, that enforces the direction of the line so that everyone's bigger on this side, basically choosing who's positive infinity, who's negative infinity. And so based upon your choice, A is less than B, that'll imply that this between this relationship gives you either this ordering or this ordering. You don't get both, but given the two points A, B you choose, you'll then have to, you have to make a choice as A less than B or the other way around. And it doesn't matter which two points you choose, you just have to choose a direction. That's really what you're doing here. And so every line has two total orderings. You know, if you think about it going, if it's like a horizontal line, are we going right to left or left to right? That's what we have to choose. Okay, so in order to do this, we have to construct our relation. So less than or equal to is a relationship. So we have to talk about how it relates between elements of the set. Now we don't have to relate everything. Not everything necessarily has to be comparable. Because of the irreflexive axiom, we actually cannot let an element be related to itself. We won't ever do that. So by construction, the relationship we're gonna make right now will be irreflexive, but we have to argue it's transitive and then it's total. That will take a little bit more effort. So the first thing is, let's consider the two points A and B. We are defining A to be less than B. That's just the choice we made. And this kind of has to be the case because we said we're gonna make a total order that does that. Now, if you take any other point on the line that's not A or B, then we have to decide is P, this point, is it less than A or greater than A? Is it less than B or greater than B? We wanna make that argument. We haven't made that choice because for totality, every point, different points have to be related to each other, not the same point, but different points have to be related to each other. So we're gonna make a choice. So if A is between P and B, we will then say that P is less than A and P is less than B. If P is, if P, Papa here is between Alpha and Bravo, then we'll say that Alpha is less than Papa and Papa is less than Bravo. And likewise, if Bravo sits between Alpha and Papa, then we have that Alpha is less than Papa and Bravo is less than Papa. So I want you to be aware that in this situation, if you have a point P and you're related it to A or B, you'll get these statements because of these three between the statements. Now by trichotomy, one of these three things happens. It has to happen. P is related to A and B since they're all on the same line. So one of those three things happens and so then you get these three statements because of that. And so then looking at this right here, we do have these relationships going on here. So when you have this between this relationship, you're gonna get exactly that. So P is less than A and P is less than B, just like we were saying before, because of the between this relation. A is less than P and P will be less than B because of this one. And then finally A is less than P and B is less than B because of those. So we're building the ordering from the between this relation. At least that's how we're gonna do it for any point as it relates to A and B. So then the last heavy lift here in terms of the definition will prove it satisfies the axioms of this total order in just a moment. But we have to also define, well, if we have two different points, P and Q and these two points are not A and Q, excuse me, they're not A and B. P and Q are different than A and B right here. How are they related? Well, we're gonna say that P is less than Q if one of the following between this relations happen. So if we have P dash Q dash A dash B, then we say P is less than Q. If we have P dash A dash Q dash B, then we say that P is less than Q. And you can kinda see what we're trying to do here with these between this relations, right? Cause A is less than B, so we're forcing P to be less than Q in that situation. So then likewise, number three here, we say that P is less than Q if this happens. A or P Q B, let me say that correctly, P A B Q. We'll say that P is less than Q if we have the between this relation, alpha, papa, Quebec, bravo. We'll say that P is less than Q if we have alpha, papa, bravo, Quebec. And lastly, we'll say that P is less than Q if we have alpha, bravo, papa, Quebec. So if one of those six between this relationships hold, we say that P is less than Q. And so, yeah, so we have those statements there. And I should mention that the way we've defined relationships, we've never defined a relationship where P could be less than P. That was never a possibility. And so by construction, this relationship is irreflexive, just like we said it would be. Now, the totality axiom follows from basically trichotomy and the linearity rules. Like I mentioned before, that any point will have some between this relationship between A and B. Therefore, by construction, A is related to B. P will be related to A and related to B. So there's totality there. But then if we take two other points besides A and B, why are P and Q related to each other? Well, because if you look by trichotomy, there are forced relationships between points, right? So we could say something like, okay, so we know that P-A-B holds, right? Okay, so let's assume we're in that situation, which I think was the first one we had listed right here. Yes, absolutely. So let's say this is the relationship between P, A and B, which by trichotomy, one of these three things happens. So we have this relationship. Well, how does Q fit into all of this? Well, you could have something like, oh, A-B-Q, there is some trichotomy relationship between A and B. Now, if you put these two things together by linearity rules, you're gonna end up with relationship three right here. So, and then that would tell us that P is less than Q. You could have instead that Q is between A and B, which then by linearity rules, you're gonna have to have the relationship that P-A-Q-B happens, like so. So again, linearity replies. And then the other trichotomy relationship is that Q-A-B. Now, when we talk about the linearity rules, there was sort of a problem here. It's like, oh, that's not enough information to say what happens. Do we have a P-Q-A-B, or do we have Q-P-A-B, right? That's the possibility. And so the first one implies P implies Q, a lesson Q, excuse me. The other one then says that Q is less than P, so which one is it? Well, then, okay, so if we have these relationships, we also have to have a relationship between P, Q, and A. So by trichotomy, do we have this, or this, or something equivalent to that, like Q-A-P? Did I list all the possibilities? Nope, that's just the first one backwards. So we'll do this one. I don't think we did that one yet either. Yeah, so by trichotomy, one of these three relationships happens. Again, we're assuming these ones right here. So this one right here is incompatible with those. That's not possible. You get a contradiction, so you can remove that one. This one would imply that P is less than Q because that gives us condition one. This one would actually tell us that Q is less than P because we'd get kind of Q, this one, but it's Q and P. Like, so this is what I mean by using linearity rules and using trichotomy. You always are gonna get a relationship that looks something like this. So again, the P, Q, A, and B could be scrambled around, but as these are four points on a line, there's gonna be some foray between this relationship of those points, and there's only gonna be one of them, unless you reverse the direction, but that's beside the point. We're always gonna make it so that A is less than B. So with that fixed, A is less than B, then there will be some relationship forced by trichotomy and linearity rules so that P is less than Q or Q is less than P. So that gives us totality. So we basically, by construction, got that it's irreflexive and we have the total axiom. Transitivity is the hard one, but that's what we're gonna work for right now. So we're gonna break it up into some cases. So let's suppose that, well, okay, by transitivity, of course, we're assuming, right? We're gonna assume that P is less than Q. We're gonna assume that Q is less than R, and then what we want is that P is less than R. That's what we have to prove right here. And so we're gonna make some assumptions because the ordering kind of depends about it's relative to these points A and B. So what if A and B are two of the three points in consideration right now? So something like, what if this was A and this was B, then this would be B as well. And so we know, obviously that A is less than B, that's always the case here, but what if B is less than R, okay? So coming up to this original picture here, so if we know A is less than B, if B is less than R, we're sitting in this situation right here because the only possibility here where B is less than R would be this one. So if B is less than R, that implies that B is between A and P, which of course implies that A is less than R in that situation. There's no other situation where B would be less than R unless A is less than R as well. So that's exactly what we're looking for. A, which is P right now is less than R. So that's the first possibility. There's three possibilities we're considering of course. There's also the possibility that Q is A and B is R, right? Now in that same situation here, if you look at these three possibilities, there's only one that allows for P to be less than A. And so if P is less than A, that then forces in this situation that P is less than B. So that one also satisfies transitivity. So the last one that's a little bit harder is what if A is P and what if R is B? Now you might think there's some other possibilities, like why couldn't we do something like B is less than A? Well, I could stop you right there. By assumption A is less than B, we don't get the other way around. So that's a contradictory assumption there. So but we could have that P is A, Q is unspecified and R is B, something like this. So if A is less than Q, that's gonna give us these possibilities, excuse me, these possibilities like so. And if Q is less than B, that would give us this one, this one. So wherever they overlap, it's this one right here. But honestly, that consideration isn't even necessary. I want you to be aware here because A is by definition less than B. So the transitivity happens in a sort of a trivial manner. So I do want you to consider this very important but simple case that if two of the points coincide with P, Q or R, so if two of those points are A and B, then the way we defined the ordering, it's gonna be transitive in that situation. All right, so then we move on to the next situation. What if the overlap between the sets P, Q and R with this points A and B is only one, okay? What happens in that situation? So we could consider a few possibilities, right? So for example, what if the point R was actually the point A? So we wanna argue that P, if P is less than Q and Q is less than A, we want that P is less than A. How are we gonna get that? Well, let's focus on Q for a moment because we know that A is always less than B. That's by assumption. And we've already proven transitivity on points where both A and B are involved right here. So this implies that Q is gonna be less than B in this situation. Now, since Q is less than A and Q is less than B, then by how we define the relationship that only happens because A is between Q and B. So that's an important thing to pull off right here. We have that A is between Q and B. But we have also the assumption that P is less than Q, okay? So what type of between this relationship can we have between P, Q and A? We could have P, Q and A is like that. So Q is between A and B. By the linearity rules, if this happens and this happens, that implies that P dash Q dash A dash B happens, okay? We can't have any other one because if we try to do that, that's gonna be in violation of this assumption. We can't have something like this either because again, that's in violation of this statement right here. We could try to do something like this, but this is just what we said earlier backward. So the only thing that's consistent with our assumptions about the ordering and how ordering is defined using between this, it has to be that P dash Q dash A happens. So then by linearity rules, we have P dash Q dash A dash B, which let's see is that on the screen right now, it's not scoot up a little bit. That's our very first one right here. And so this implies that P is less than Q. We already have that. So I mean, this is the only possibility that's consistent with our assumptions. Remember, we're trying to prove right now that P is less than A, for which of course, since we have this assumption, then we can look at this, we can imply from this that P dash A dash B holds, which then implies that P is less than A. It also implies that P is less than B, but we don't even need that one. We're trying to prove for transitivity to this one right here. Another possibility, of course, would be something like the following. Let's suppose that B, let's suppose that B is greater than Q, and then Q is also greater than some other point P. What's happening there? Well, by trichotomy, we have to consider what's going on here. We're gonna have that A dash Q dash B holds, or we're gonna have that Q dash A dash B holds. Trichotomy does allow for a third possibility, but since we know A is less than B, that other one doesn't happen, okay? We also have to relate what's going on right here. The P is less than Q. How does P relate to A? How does P relate to Q? As you try to work through all the details, there's gonna be only one of these statements that's consistent with your assumptions. And again, I'm not gonna go through all the details of this. The statements are very similar to what we've talked about already, and I'm gonna leave it as an exercise to the viewer to provide all those extra details. But I want you to be aware that there's six possibilities total, because basically A could be P or B could be P. That's a possibility. A could be Q or B could be Q, or we have A is R or B is R. And we talked a little about those already, but there's these six possibilities. And if you run through your assumptions and how those assumptions of the relation correspond to between this relations, then there's only gonna be, in each and every case, there's only one of these four area between this relationships that hold. And then from that, you can ascertain all of the statements you need. Okay? And so then the last possibility, again, I know this is a very technical theorem here, and I'm skipping a lot of the steps for the sake of time. The last possibility is what if none of the points P, P, Q or R coincide with A and B? So we have this idea that we're assuming P is less than Q and Q is less than R. So we then have to say, we wanna prove that P is less than R. Well, even though A and B are not related, we wanna fit them in there somehow, right? So by the assumptions here, we wanna deduce that P is less than R. How do A and B play around with these points, P, Q and R? It could be, right? It could be that we have P less than Q less than A less than B. And then likewise, we have Q less than R less than A less than B. If that happens using linearity rules, we can, well, I guess it's just for the definition of things, not the linearity rules. We would get that P is less than A less than B, and R is less than A than less than B. Then by linearity rules, we could then infer that P is less than R, which is less than A, which is less than B. In which case that then tells us that P is less than R. So again, play around with trichotomy, in this case, particularly linearity rules, we can then get something like that. Because after all, if P is less than Q, right? Coming back up to the definition here, if P is less than Q, that happens in this situation, or it happens in this situation, it could happen in this situation, this situation, this one and this one. So if P is less than Q, all of these things happen. Not all of them simultaneous. I should say one of them happens, right? That's what I'm saying. So which of those ones happen? But we also assume that Q is less than R. And so when you go through all the possibilities, there's a lot of possibilities, and there's gonna be, there's 10 possibilities here, because essentially what we're looking at is you're having basically something like this. You have P, Q and R, like so. Where do you squeeze in the A's and B's? You can squeeze them in, you can try to squeeze A right here, B right here, that's one. You could try to put B right here, that's two. You try to put B right here, that's three. You try to put B right there, that's four. You could put A right here, then B could go here, here or here. You could try to put A right here, and then B could be here and here, or you could just put them both over here. And so essentially there's 10 possibilities that you have to consider. And so while I'm not gonna drive all of them right now, one of the possibilities is you assume this relation and this relation, which these are relations that are compatible with the assumptions. Cause after all, like I said, if P is less than Q, then one of these six things is true. And if Q is less than R, then one of these six things is true where you of course replace P, Q with QR instead. But not all of those ones will be compatible with each other because how they interact with A and B. And so there's only gonna be 10 that work out. And so that's what I'm trying to say there. Those 10 possibilities have to be considered and each and every one of them will lead to a transitive relation. So this might be the most unsettling of all the proofs we do in this course, but it's really technical. There's a lot of cases, but they're all very simple. This proof is a tedious proof. It's not necessarily a difficult proof. It's all that matters is keeping track of things. So in particular, every line has a total order attached to it and we will utilize that in the future. It's a very powerful result because there's a lot we gotta check here. And again, we didn't check everything, but everything we didn't check is similar and does pan out. I've checked them all, even if I didn't include it all in this video. So every line has an ordering. And so that leads us to a definition that we're gonna end lecture 11 with because lines have orderings, right? We can talk about what it means to be less than or equal to something. We then define intervals and this is gonna match up with what you would see in calculus. We can talk about the closed interval from A to B. The closed interval for A to B is gonna be all points C on the line such that A is less than or equal to C and C is less than or equal to B. In particular, includes the points A and B themselves. We have the open interval A to B, which will take all lines C in, all lines C on, excuse me, all points C on the line such that A is less than C and C is less than B. All right, so if you look at those two just for a moment, I want you to be aware that the open interval from A to B is the collection of all points between A and B. And it doesn't even matter what order you do. You can have one way or the other, left order or right order, it doesn't matter. This interval, the open interval will get you all the points between A and B. The closed interval right here, this is just the line segment AB that we defined previously, but now we define it using orders instead of betweenness. But the two things are related to each other. We also have half open, half closed, sometimes called clopin intervals. So there's the close on the left interval, there's the close on the right interval, which if you're closed on the left, that means you take all points C on the line such that A is less than equal to C and C is less than B. And then if you're closed on the right, you take all points C such that A is less than C but C is less than equal to B. And so with these definitions, we will talk about intervals in the future. For my students, you'll check out a few homework questions about intervals, but we can define them using orders and so many things about betweenness become fundamentally easier when you use order and set your, when you use order to prove things. So I encourage anyone, if you're trying to prove something about an order geometry, trying to prove something about betweenness, use the ordering if appropriate. It's a very powerful tool. Don't let all of the technical details we discussed in this video go to waste, use them. If you learned anything about ordering here, in this video, in this lecture 11, please like these videos, subscribe to see more videos like this in the future as always. If you have any questions, which is very natural, especially when I skip lots of steps. If you have any questions, please post them in the comments below and I'll be glad to answer them when I can.