 In this video, we're going to continue developing the theory and notions of order geometry which we defined this in the previous video for lecture 10 in our lecture series here. In the previous video, we provided the four axioms of betweenness. We had our collinearity and symmetrization, axiom extension, trichotomy, and posh's axiom. In which case this lecture we're going to talk about right now really is focused on posh's axiom. So it turns out that posh's axiom is actually named after Moritz posh. And we haven't seen any axiom so far that has been named after a person. I keep on referring to them as Hilbert's axioms, but none of them are named after Hilbert himself. Why did we name it this one posh's axiom? Well, it turns out that our posh's axiom before isn't actually the version that Hilbert himself used. And instead it's named after Moritz posh. I have his name written here on the screen here. And honestly, when you take Hilbert's original axiom before, commonly referred to as the plain separation axiom, in that situation we actually can prove posh's axiom as a theorem. And so in the literature it's often called posh's theorem depending on what your axiomatic framework is. And that's because in 1882 posh recognized that this statement was an assumption made by Euclid, but can't actually be proven from Euclid's poshschlitz himself. So we named posh's theorem after that discovery of posh's there. In which case the version we used for axiom before, which we're calling posh's axiom, we're basically taking posh's theorem as truth. It's true. It's our axiom. And then we're going to prove plain separation as a theorem of betweenness. And mostly this has to do with just style. Basically, I'll make the argument that using posh's axiom, it simplifies some of the theory of betweenness. But it doesn't actually change any of it because it turns out that posh's axiom is logically equivalent to Hilbert's original axiom, which is logically equivalent to this notion of plain separation I want to talk about in this video right here. Because after all, when we introduced betweenness last time, we came up with the idea about being on the same side of a line. What does it mean to be on the same side of a line? Well, remember the definition. If we have a line L and we have two points, call them A and B. If the line segment AB doesn't intersect L, we say they're on the same side of the line. But if the line segment AB does intersect the line, then we say they're on opposite sides. And what's the line segment again? Well, it contains the point A, it contains the point B, and it contains all the points between A and B. So in incidence geometry, we don't have any notion of betweenness. This idea of same side versus opposite side doesn't make sense. By virtue of this, of the definition, being on the same side or being on the opposite side is a betweenness notion. And the fact that we call the term same side versus opposite side, we're already putting the cart in front of the horse, supposing that there's two sides of the line. And one has to be careful. In order geometry, by the end of this video, we'll know there are two sides to a line. That is, a line separates the plane into two pieces. But in non-order geometries, that's not always the case. Like in projective geometry, take the real projective geometry, for example, a line doesn't separate the plane. Both sides of the line are basically the same thing because of wraparound, you can go the other way, you know, you teleport, whoop, come out the other side like this. The two points can be on, so to speak, opposite sides, but the interval doesn't cross. There's some weird things that can happen there. So plane separation is going to provide us something that's logically equivalent to Posh's axiom. But in our lecture series, we took Posh's theorem as the axiom, and then plane separation is a consequence of that. But like I said, they're logically equivalent. So for these three, there's actually three statements to plane separation, but all three of them are basically the same. The proof of one is essentially the proof of the others. And I'll make some more, I'll make that more explicit in just a second. But consider three points A, B, and C, and consider a line L, so that A, B, and C are not points on L. And this is taking place inside of an ordered geometry. So the first one, these are conditionals. If A and B are on the same side as L, and if B and C are on the same side as L, then A and C are on the same side as L. So after improving this one, this relationship of A and B being on the same side of a fixed line L, we'll see is actually a transitive relation. It's much more apparent that it's a reflexive one because honestly, if you're just a single point, I should make mention that by definition here, the segment A to A, this is just the point A. Because it includes the endpoints A to A, and there's nothing between A, so it's just a point. If you're not on the line, then it doesn't intersect. A is on the same side of A. Okay, it's also symmetric, and this comes from the fact that the segment AB is equal to BA. So if AB doesn't intersect L, neither does BA. So being on the same side is symmetric, it's reflexive. The hardest one is going to be it's transitive, and we're going to prove that it's a transitive relation. We'll prove that in just a second. I do want to read the other two statements. So if A and B are on opposite sides of L and B and C are on opposite sides of L, that actually means that A and C are on the same side. And intuitively, you can make sense of this with a picture, right? If A, I'll just write the letters, if A and B are on opposite sides and B and C are on opposite sides, then A and C are on the same side. That's intuitively what number two is saying, right? The first one, if we were to draw it, it's like, oh, A and B are on the same side, B and C are on the same side. So A and C are on the same side. And intuitively, that's what we're doing. And then the third one, if A and B are on opposite sides, like so, and if B and C are on the same side, then A and C are on opposite sides. So in Euclidean geometry, this seems very straightforward, but we're not proving this for Euclidean geometry. We're proving this for order to geometry, for which we haven't established this yet. What we're going to do is we are going to prove the statement for, we're going to prove statement number one, and I'm going to leave it as an exercise to the viewer to prove two and three, which are very similar to the proof. We're going to see one in just a moment ago. None of these three are really harder than any other. All right, so we're going to prove number one. And so let's go with that assumption. So we have some line, L, like so, and I'll label it. And so by assumption A and B are on the same side. So if we were to draw this picture, we have the point A, we have the point B, the line segment does not intersect L. We also are assuming that B and C are on the same side. So we have C right here. We have A and C like so. They're on the same side. Now let's consider the line segment AC. What's happening there? So we're going to prove this one by contradiction. So by assumption, we have that AB is parallel to L. We have that BC is parallel to L. They don't intersect. So that's what our assumptions are. So by way of contradiction, we're going to suppose that in fact AC intersects. It's not parallel to L. So I'm going to redraw my picture here. So L is going to look a little bit different. Instead, we're saying that AC intersects the line L. So there's some point of intersection going on right here. Now, so we have this point right here. We could call it D or something, but it doesn't actually need a label. This is where we're going to use posh's axiom here. posh's axiom says that if ABC are not on L, which they're not by assumption, and L intersects the line segment AC, then it must intersect AB or intersects BC. posh's axiom requires one of those two things happens. And it's an exclusive or they can't both happen. That's also a part of it, but we don't need that. Well, if L were to intersect AB, something like this, then this would imply that A and B are on opposite sides of L. It's a contradiction to our assumption right here. So that's not the case. But similarly, if the line L were to intersect BC somewhere, that would imply that BC are on opposite sides of L, for which that would contradict our assumption as well. So since all cases lead to contradiction, we have to then negate the statement we just assumed. So A and C don't intersect L. In fact, AC is parallel to L, so they're on the same side. So we get that A and C are on the same side. And so you can see that this argument is basically just playing around with the definition of what it means to be on the same side and then uses posh's axiom. The other two parts, it's the same thing. It's just you're going to use opposite sides in addition to same sides, but it really just comes down to posh's axiom. With the assumptions you get, you're going to have a line entering a triangle that can't exit a triangle. And that's a no-no because of posh's axiom. All right. So with now the plane separation theorem established, there's another definition that we need to introduce when we talk about order to geometry. And this is the idea of a half plane. This is something we use all the time. So imagine P is a point and L is a line that's not on P. So P is not on the line. And this is in an order to geometry. We define the so-called half open plane. And we're going to use the notation H circle L. P, this is not universal notation, but the idea here is this is the line L. This is the point P. H stands for half plane. And the little circle here, this is a topology symbol we're going to use. Circles are often used to denote open sets. And so in the topology, the reason we call it an open half plane is it would be an open set with regard to the topology. But we're not going to do topology, so it doesn't matter too much. But just so you're aware of what the symbol's mean there. So an open half plane is going to be a set of all points on the same side as L as P. Because we just said a moment ago that being on the same side of L is a transitive relation because we're in order to geometry. It's reflexive and symmetric as well. So being on the same side of a line is an equivalence relation. So we're just saying, oh, the open half plane right here is just the equivalence class of P with respect to being on the same side as L. Okay. And then we can define the closed half plane or sometimes it's just called the half plane itself. Closed again is coming from topology here. This would be a closed set. We'll just denote this as HLP. The little circle is now omitted. The closed half plane is just the open half plane union the line. So the boundary, because L separates one side of the line from the other. We include that on the half plane, the closed half plane. The open half plane doesn't include the line. And that's the difference there. Now again, we're using words like open half plane, open closed half plane, half, half, half. You get half of that suggests there's two of them. Are we still putting the cart in front of the horse right here? Well, because, because of the plane separation theorem, we know that being on the same side of a line because as an equivalence relation, so there's equivalence classes, these open half planes, that's just the equivalent. That's an equivalent class for that relation in the closed one just means you throw on the line. But why half? Are there only two half planes? The answer is yes. Because if that wasn't the case, we shouldn't have called them half planes. Let L be any line in an ordered geometry, then that line has exactly two open half planes. So there are two of these things. So for every point in the geometry, you either belong to one side of the line or the other side of the line, or I guess you could be on the line itself. But that's why we call a plane separation. The line separates the plane, the geometry, into two classes, this half plane and the other half plane. Okay. And so it's important that we, as we talked about, we're only going to talk points, talk about points that are not on L, right? So let A be a point that's not on L. And let's talk about P, which is a point on L. Okay. So let's draw a little picture to help us understand what's going on here. The picture's not the proof. The picture just helps us keep track of where we are in the proof. So we have this point P, we have a point A, and this is our line L. The important thing to remember here is that A is not on L, not on L, P is on L, and we're going to go from there. Okay. So since we have these points A and B, we could connect the line between them. By line determination, there exists some line that contains both of them. I didn't quite reach them that time. Let me try this again. That's close enough. Whenever your dot doesn't fit, you just draw a little bit bigger. So it's like, there you go. So consider the line that's determined by A and P. Now this line can be extended by the extension axiom. That is to say, there's a point B that with my diagram would look like this. But mathematical notation, what I'm saying here is by the extension axiom, this is axiom B2 up between this, there exists a point B such that P is between A and B. And so then if we look at the line segment AB, P belongs to that line segment because P is between A and B. And P is on L, therefore the line segment AB intersects the line L at P, in fact. So that tells us that A and B are on opposite sides of the line. After all, that's what it means to be on opposite sides. So this gives us that there's at least two. There's at least two half planes. There's the one that contains A, there's the one that contains B. By construction, A and B are on different half planes. So it's justified on saying there's at least two. But why they're exactly two? Well, that's what we're going to consider now. What if we take some third point C that's not on L? Okay. Now if C were on the same side as A, right? Well then, because this is an equivalence relation, the open half plane that contains C would be the same as the open half plane that contains A. So we would be done at that moment. Alright. So then that means that A and C are not on the same side as L. So we're going to get something like this. Now what does it mean to be on opposite sides? What that means is that the line segment A, C intersects this point. That's what it means. I don't want you to think that this is just becoming like circular reasoning right here. I'm not assuming there are two sides. I'm assuming if you're on the same side, you're here, right? But that's because that's what the definition of this set is. We prove in it by plane separation, it's an equivalence relation. So if C and A are in the same class, their equivalence classes are the same. But now when I draw my picture where I'm saying C is on the opposite side, I'm not saying C belongs to the open half plane that contains B. I'm saying it's on the opposite side. That is, their line segment intersects the line L. So my picture is not the proof here. I didn't draw C over here to say that A, C, and B are on the same side. This actually comes from the plane separation theorem which we did on the previous slide. Part two actually. We prove part one, part two, it's a similar proof. But remember, part two said changing the labels a little bit, if A and B are on opposite sides, and if A and C are on opposite sides, then we get from the plane separation part two that B and C are on the same side. That's when we get in, basically it's posh's axiom because the line enters here and has to exit somewhere. It's going to have to exit here at P, right? So that's since B and C are on the same side, they have the same equivalence relation, excuse me, they have the same equivalence class, thus they're on the same side. So no matter where C was, C is either on the same side as A or it's on the same side as B, which improves, there's only two equivalence classes. And so that brings us to the end of lecture 10 about the betweenness axioms. This is not our last conversation about betweenness. The next several lectures in this lecture series will be about ordered geometry and the betweenness axioms. In particular, in the next video, we're actually going to talk about why is it called ordered geometry? What partial order are we referring to? We'll talk about that next time. It'll be very technical, but very, very important. Stay tuned, of course, subscribe to the channel to get updates about videos like this in the future. And as always, if you have any questions, feel free to leave them in the comments below and I will be glad to answer them.