 If you've watched the first two videos for lecture 22 in our lecture series, you'll notice that I'm skipping a lot of the details. We've reached this last family of axioms for Euclidean geometry. And without including the parallel posture that we have, we've talked about all the axioms then of neutral geometry. And I keep on emphasizing that the continuity axiom, it's important, but it really is a calculus principle. That is we need it for like topology. We need it for real analysis and things like that. It's related to geometry, so we can't just avoid the issue entirely. But with regard to proofs, I'm proving nothing. In video one, we proved nothing. I did provide a counter example. That is I provided a geometry, which was that geometry that was a congruent geometry that didn't satisfy the continuity axiom, aka Q2. But, you know, we didn't prove a principle. I just provided a counter example that, oh, this is that and not that, right? I mean, R2 was an example, of course, of a neutral geometry. Q2 was a congruent geometry that's not a neutral geometry. But other than that, right? You know, other than like examples and counter examples, we've proven nothing. And I'm not gonna break that streak as we end lecture 22. It is not my goal to prove anything about continuity. But I do want the viewers of this lecture series. I want my students to be aware of continuity and to use continuity to prove other things. And the reason I've been avoiding the proofs is really that the proofs of continuity belong not to this course on modern geometries, but instead it belongs to a course in topology or more appropriately, of course, in real analysis. That's really where continuity should be vetted out, okay? But as we end this discussion of continuity, we then want to look at the implications of continuity on, of course, geometry, particularly we can't talk about neutral geometry without some appreciation of continuity. So in this video, I want to highlight what are then the consequences of continuity or better yet, what are the benefits of continuity? What do we gain from continuity? And we get the notion of measure. We've been avoiding that until this moment. We have a notion of segment measure and we have a notion of angle measure. Let's talk about segment measure first. So when we have continuity, fix some segment, which we'll call that segment OI. So O and I are points in a neutral geometry. With that segment fixed, we will call that the unit segment. And we're gonna define a function, a function that attaches to every segment in the geometry, a real number, which that real number will be denoted AB without the line. With the line, you're talking about the segment. Without the line, you're talking about the measure of the segment. And we will attach to every segment a number, a real number called the measure of that segment. And this function exists, the measure function, and it'll satisfy the following principles. Now, first of all, the measure of the unit segment is one, which is why we call it the unit segment. This is then our baseline of how long is a length in this geometry. And I should be aware that there should be multiple, there's multiple measures you can attach to a neutral geometry. And it depends on what that fixed distance is. What does one mean with the length of a segment? Is it one foot, one centimeter, one light year, one kilometer, one astronomical unit? These are all measurements of distance and different choices will lead to different measures. And we're not gonna worry about unit conversion right now, but it's like, okay, I chose a unit and that unit length is then one, okay? Then what we also benefit from measure is that if two segments have the same measure, this happens if and only if those segments were congruent to each other. So the measure function is a congruence preserving function. Different measure, different segments that are congruent to each other will have the same measure. And also if two segments have different measures, they were different congruences, different congruent classes. So the measure function interacts very well with congruence. Your measures are equal if you are congruent segments. The measure function also behaves well with betweenness. If the point B is between the points A and C, this happens only if the sum of the two measures, AB plus BC is equal to AC. So the measure function, as these are real numbers, is compatible with betweenness, all right? It also behaves well with the inequality we've seen. It makes sense to say that a real number is less than or equal to another real number. Well, the measure AB will be less than the measure CD only when the segment AB is less than the segment with CD, which when it comes to segments here, this had to do with translation. Can I translate a copy of AB onto CD so that it's interior to the segment CD? That's what it meant to be less than. Over here, as these are real numbers, AB is less than CD if it's a smaller number. These two notions coincide with each other in an if and only a statement, okay? We potentially could have introduced these objects earlier. The main reason that continuity is in play here, it mostly comes to do with property E. Like I said, we could have done something like this with the real number, excuse me, with the rational plane. With rational numbers, we could develop a function that behaves like this for a fixed interval, okay? Of course, there are some issues about the way we usually put measure on the rational numbers where you can have a rational interval, but maybe has an irrational length. Pythagoras had some issues with that, but we won't go into that any further. So while you potentially could capture a function that has the first four properties, the last property is really where you must have continuity. There's no way of avoiding it. So for each positive real number X, there exists a point A and B such that A, B realizes that number. So there exist points in your geometry that can realize any real number as their measure. You can't do that without something equivalent to Dedekind's axiom. And that's where the continuity axiom comes into play here. Now by analog, we can do the exact same thing for angle measure. I will go through this one a little bit quickly, a little bit quicker in this situation, but given any angle A, B, C, we can attach to that angle. It's measure, which is commonly denoted as M, A, B, C. And this number right here will be a number between zero and 180. So when we describe measure in this class of an angle, we're gonna actually use degree measure, but you could do this with radians or even gradians or some other angle measurement unit. Just for simplicity, we'll just stick with degrees. We don't have to do any trigonometry in this lecture series. So the actual unit doesn't matter. So we'll just stick with degrees as people are very comfortable with that. So the angle measure is a function that connects to every angle, some type of angle measurement. Now when it comes to angle measurement, we don't actually have a fixed angle in mind. We don't have a reference unit. That's actually because we've already done that along the way. Our angle measure function will give an angle, a measurement of zero degrees, if and only if it's a null angle, which a null angle means the ray B, A and B, C actually coincide with each other. So null angles have zero degree measure. If an angle has a measurement of 180 degrees, that happens if and only if ABC is a flat angle. That is to say ABC is a half plane. That is to say B is between A and C in that situation. For the null angle, I should mention that it's a null angle if and only if C belongs to the right BA, like so. So null angles have zero measure, flat angles have 180 degree measure and then right angles, right? With the angle measure function, this function will give an angle, a measurement of 90 degrees, if and only if it's a right angle. So those are all nice properties. And so these three properties sort of capture the idea of a unit interval that we solve for segment measure. These are our baselines. We rank from zero to 180 degrees. Right angles are also noteworthy here. But just like segment measure, two degrees have the same measure if and only if they're congruent as degrees. Like a segment measure here, that a ray is between two other rays if and only if there's this additive statement here. The ray, BC sits between BA and BD, only if the sum of the two measures of the angles, ABC plus CDB adds up to ABD. So that betweenness is captured there. This measure function, it preserves congruence. It preserves betweenness. Interiors of angles are preserved there. The inequality is also true. That one angle measure is less than another angle measure if and only if the angle measures less than itself. Other, excuse me, measures less than the other one. ABC is less than DEF with regard to their angle measure. And again, we might be able to construct functions without full blown continuity, but the last property is where the continuity axiom is necessary. Given any number in the interval zero to 180, there exist points, ABC, such that the measure of that angle will equal the real number X. And so continuity is necessary for that. We can realize any segment measure. We can realize any angle measure because of Dedekind's axioms. And for us going forward in the series, this is what continuity means. It means the angle measure theorem. It means the segment angle theorem. We really aren't gonna use, we're not gonna use Dedekind cuts in this lecture series, but we can use the measure functions. The measure of a segment, the measure of an angle, and we use these properties. So really, these are the properties we're gonna use going forward. And so that's gonna end our lecture about the continuity axiom in some weaker forms as well. Of course, in the next couple of lectures in this series, we'll look at consequences of continuity, which will mostly be consequences of segment measure and angle measure. But be aware, that's where we're gonna go forward here. So if you learned a little bit about the continuity axiom, Dedekind cuts, like these videos, subscribe to the channel to see more videos like this in the future. And of course, if you have any questions, post them in the comments below, and I'll be glad to answer them.