 So in the previous video for lecture 22, we introduced the notion of Dedekin's axiom of continuity, which we then have adapted as our one and only continuity axiom, which when conjoined with the incidence betweenness and congruence axioms, forms what we call a neutral geometry. Now, as Hilbert, David Hilbert, went about to try to axiomize Euclidean geometry, be aware that if you just remove the Euclidean parallel postulate, then you have the axioms for neutral geometry. So Hilbert axiomized neutral geometry when he was axiomatizing Euclidean geometry. He took two axioms of continuity. His first axiom of continuity is the so-called axiom of Archimedes, which we have listed on the screen right now, listed as the Archimedean principle. We will talk more about that in just one second. His second axiom of continuity known as the posture of line completeness. Well, honestly, in that rendition, it was a little bit more complicated than we wanted to take on, which is why we adopted Dedekin's axiom instead. From the geometric point of view, understanding that the material presented in this lecture series is meant for undergraduate mathematics students, I personally believe that the Dedekin axiom has the advantage of being much simpler to approach for the audience, but also one axiom will serve both. We'll talk about in a second how Dedekin's axiom implies the Archimedean principle, because after all, the Dedekin's axiom is trying to capture this notion of completeness of the real line. Hilbert took that as an axiom, but in his language of the axiom, I mean, not German. I mean, we'll say this in English, of course, but when we talk about the posture of line completeness, it was framed in somewhat of the following manner. This is just a quote from Wallace and West Rhodes to Geometry textbook. An extension, so this is the posture of line completeness, an extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follow from axioms one through three and five is impossible. Wow. Sorry, that's really an impregnable axiom in my mind. It's like, what does that even mean? So by that in contrast, you could perhaps see why the Dedekin's axiom that we presented in the previous lecture was much more approachable than Hilbert's original axioms of congruence. But as Hilbert presented it, he took two axioms of congruence, basically that the real line is complete and he took upon the Archimedean principle. For which in this video, I want to talk about three sort of weaker axioms of continuity. That is the weaker than the Dedekin axiom we adopted in the previous lecture video but I also want to make mention that these three principles which we're going to call continuity principles because we won't adapt them as axioms of neutral geometry because all three of these are theorems of neutral geometry. That is Dedekin's axiom combined with the other axioms of neutral geometry can be used to prove these. So these are all theorems of neutral geometry but if you didn't accept these axioms, that is if you didn't accept Dedekin's axioms, you could still take the axioms of congruence geometry and equip one of these principles to it and you would have some notion of continuity then added to that geometry. It wouldn't be full-blown neutral geometry but you would have a weaker form. And I'll comment about that in a second because for example we know that our, the Euclidean plane, our squared is a neutral geometry. We have, we also talked about how the rational plane is not a neutral geometry because there are some problems with continuity. I'm going to use these two examples to help tease out the difference of these weaker continuity axioms. Alright, so I mentioned it a couple of times. Let's refer to the Archimedean principle. I like, what is it? So the Archimedean principle, we'll state this as a theorem of neutral geometry but you could adopt this as a weaker axiom of continuity. The Archimedean principle says the following. Given a segment AB and a ray R who emanates from the vertex C, for each point D that's on the ray but is distinct from the vertex itself, there exists an integer N and a point E on the ray such that the segment CE is congruent to the segment N times AB. And we have one of two things, either the points E and D coincide with each other or D is between C and E. This is a situation where drawing a picture really helps, helps clarify what's happening here. So we have our line segment and we have a ray. The ray is the longer one. I just put an arrowhead on over here. This ray is our ray R. It has as its vertex the point C. We have this line segment A and B over here. The Archimedean principle says the following. If we take any point on the ray, which we'll call that point D, what we can do is we can take some number of copies of the segment AB and eventually we will surpass the point D. So if we take one copy of AB, we get something like this, A and B. Then we take some other copy of AB. We can copy it over here. So this is another A. This is another B. We can take another copy of AB. Let's say something like this. So we take another copy of AB, something like this. So with my diagram, we have the segment 3 AB. And so this very final point, we call that point E. And the idea is the segment CE is congruent to the segment 3 times AB. And where did D lie in this situation? D then sits between C and E in this construction. Now, it's possible that D and E could actually be the same point. By quink-a-dink, you could have that E lands on D. That's a possibility. But most likely E and D are not the same point in which case D is improperly in this interval. So the idea is if you take any ray, you could copy it. Eventually, it'll surpass any other ray. And so this is the Archimedean principle. This is a very important principle in real analysis. And as such, it's an important consequence on, of course, the neutral geometry here. Now, I should mention that, well, if we have the Archimedean principle, does it make sense to talk about a non-Archimedean geometry, a geometry that doesn't satisfy this condition? That would be made possible if you had a geometry where intervals potentially have infinite length or perhaps infinitesimal length, all right? If you had an interval, for whatever reason, if you have a geometry where it's consistent to give segments length and say there's some segments that are infinite length, but there are some segments with finite length. Well, if you have a segment of finite length, there's no amount of copying that segment that'll ever make it be infinite length. And therefore, a finite interval can never cover an infinite interval. You know, assuming your geometry allowed for such things. It also works if you had some type of infinitesimal length. For example, with the hyperreal numbers, that's something you might consider. If you have a segment whose length is an infinitesimal, it doesn't matter if you take any finite amount of an infinitesimal length, you'll never reach a finite length. And so a non-Archimedean geometry would be something of that nature. Now, of course, the real plane, that is the Euclidean plane, is Archimedean, because it's a neutral geometry. I should mention, of course, that an Archimedean geometry, what we're defining here is going to be a geometry that satisfies the Archimedean principle. Remember, as we defined previously, a complete geometry, it's going to be a geometry that satisfies Dedekind's axiom. We'll say an Archimedean geometry is a geometry which satisfies the Archimedean principle, which you could take as a continuity axiom. Of course, neutral geometries are Archimedean, so the Euclidean plane, hyperbolic plane, these are Archimedean geometries. But I should mention that the rational plane is likewise an Archimedean geometry. We saw in the previous video that Q2 does not satisfy Dedekind's axiom of continuity, but it does satisfy the Archimedean principle of continuity. So while Q2 does not have the full-blown continuity that the Euclidean plane does, Q2 does have some notions of continuity. In particular, the continuity principle is in play there. So the second one that we have listed on the screen is what we call the circular continuity principle. Again, this is a weaker form of the Dedekind axiom. As in, the Dedekind axiom implies this. If you have Dedekind's axiom, you have the Archimedean principle. If you have Dedekind's axiom, you have the circular continuity principle, which therefore for neutral geometry, every neutral geometry has the circular continuity principle, which states if each of two circles has one point inside of the other and one point outside of the other, then the two circles intersect at two distinct places. So if we have two circles, you have one illustrated here in yellow, another one illustrated here in blue, and so notice that with respect to the yellow circle, there is an interior point here, which is out of the blue circle. There is an exterior point of the yellow circle that's inside the blue circle, but then it goes the other way around. This blue point is actually an interior point to the blue circle, but not in the yellow circle. And then this point is exterior to the blue circle, but interior to the yellow circle. So this situation satisfies the assumptions in play here about interior and exterior points. So then the circular continuity principle then guarantees that there will be two points of intersection between the circles, just as is outlined here on the screen. Now of course in the real plane, which is a neutral geometry, we always have this. These two circles are going to intersect. On the other hand, if you look at the rational plane, this property is not satisfied. You can have two rational circles, which one point is inside the other exterior interior points there, but there's no points of intersection. That is to say, we can have these two rational circles because what does it mean to have a rational circle? You're taking all these points which are equidistant from the center for some fixed distance right here, your radius, but to be a rational circle, your coordinates have to be two rational numbers, two fractions of integers. It could be that, oh, we have a rational circle in blue, we have a rational circle in yellow, but the points of intersection would actually be irrational points, which are not part of the rational plane, but are still part of the Euclidean plane. And so this is the problem with the rational plane with regard to continuity that is. Sure, it satisfies the Archimedean principle, but it doesn't satisfy circular continuity because circles might not intersect each other, even though they're overlapping circles. In a neutral geometry, overlapping circles always intersect, but that's not always the case. And so this shows you that the Archimedean principle is too weak a continuity principle to imply other things like circular continuity, because the rational plane is Archimedean, but it's not circular continuous, which is evidence that it's not a neutral geometry. All right, but Dedekind's axioms can prove this. We're not going to provide the proof in this video, but using Dedekind cuts and such, you can argue that these circles must have two points of intersection if they're overlapping. And then finally, I want to mention a simpler version of circular continuity. You have the elementary continuity principle, which will be a theorem of neutral geometry because it turns out that the Dedekind's axiom can be used to imply circular continuity. And then if you have circular continuity, that itself is alone to prove elementary continuity. Therefore, Dedekind, of course, implies both of them, and it implies all three of them on the screen right now. So this is a theorem of neutral geometry. What is circular continuity here? If one endpoint of a segment is inside of a circle and the other endpoint is outside of the circle, same circle there, then the segment must then intersect the circle. So we have our circle right here. We have a line segment that might be doing something like this. So one endpoint of the segment is inside the circle. The other one is outside of the circle. Then the elementary continuity principle then provides that there must be a point of intersection between the segment and the circle. Now with the rational plane, you have the sort of the same problem that we saw before with circular continuity. You can have a circle which consists of only rational points. That is, you only take the points in the plane with rational coordinates and those points are equidistant from the center, which necessarily is a rational number. That would then give you a rational circle. You can take a rational segment which, okay, this endpoint is a rational coordinate, this endpoint is a rational coordinate, and you take all the points in between those which are rational points. Well, it could be that the point of intersection that you would expect is actually irrational and therefore doesn't belong to the rational plane. So the rational plane is in fact our comedian, but it does not have circular continuity, nor does it have the weaker form of elementary continuity. But of course in the real plane, aka the Euclidean plane, you have circular continuity, which then implies elementary continuity. The hyperbolic plane has this as well. So these are all theorems that we can take for granted in a neutral geometry, but you could take these three principles as weaker versions of continuity. Dedekin, like I said, the Dedekin axiom here, it implies the Archimedean principle. It implies circular continuity. Circular continuity implies elementary continuity, thus the Dedekin axiom takes care of all of those. And so there do exist geometries which are Archimedean, but they don't have these other principles like Q2. There do exist non-Archimedean geometries that violate that one, but maybe they could satisfy circular continuity, right? Of course there can be geometries which satisfy elementary continuity, but don't satisfy circular continuity. So these are all weaker forms of continuity axioms, but with our coverage of neutral geometry, because we have the Dedekin axiom of continuity, we have all three of these principles as well.