 Welcome back to our lecture series Math 3130, Modern Geometries for Student Sets of the Intel University. As usual, I'll be your professor today, Dr. Andrew Misseldheim. So up to this point in our lecture series, we've gone through many different axiomatic systems for geometries, continually adding to it over and over and over again. We began with the incidence axioms, followed by the betweenness axioms, followed then by the congruence axioms, and of course, in the mix of that, we did talk about parallel alternative servile as well. And all of this was based upon really David Hilbert's axioms for Euclidean geometry. We just keep on stacking more and more and more axioms on there. Well, the last family of axioms from Hilbert's axioms of planar geometry are in fact the continuity axioms. This is the final piece which completes our notion of Euclidean geometry. Incidents, betweenness, congruence, the Euclidean parallel postulate, and then continuity, which the continuity axiom is sometimes called the completeness axiom, so we can use those terms interchangeably. So the completeness axiom is a property that really is coming from topology, which we've talked about topology before in this lecture series. I mean, we of course have geometry versus topology here. The two in some way are related to each other. You could argue that topology is a geometric notion, but then again, that the two are very spatial oriented, but topology has sort of a different goal in mind than geometry does. But again, the two interplay with each other very much. The completeness axiom really is this property coming from topology. And for the most part, it's really, it's the setting that we're gonna be in this conversation for lecture 22 is really what one would call real analysis because the completeness axiom is what makes the real numbers, the real numbers, as opposed to some other number system. In real analysis, the completeness axiom is often referred to as the least upper bound property, least upper bound property. And the least upper bound property is what sets apart the real numbers compared to some other ordered field like the rationals, which what is the least upper bound property? The least upper bound property says that all non-empty sets of numbers with an upper bound must have a least upper bound called the supremum of the set. The numbers, I should say the real numbers as a set is the unique ordered field, which also satisfies the least upper bound property, which is the difference between the real numbers, say with the rational numbers. The rational numbers is an ordered field. So ordered field, what does that mean? Well, from an algebraic point of view, a field is going to be a set of numbers which have a well-defined notion of addition, subtraction, multiplication, and division. And these four operations behave with the usual properties such as addition and multiplication or associative, multiplicative. They have identities, zero and one. They have inverses, which basically the inversions give you subtraction, division, the distributed property is satisfied in a field. An ordered field would suggest that we can also order the numbers. We can say that these two numbers are greater than or equal to some other number and such. So an ordered field is then a field for which the ordering is compatible with it. And this is then where the least upper bound property enters the field, because the least upper bound property is a condition on that ordering that whenever you have an upper bound, there's always a least upper bound, the supremum for that set. The rational numbers, they are a field that feel is ordered. It has the same ordering that the real numbers have, but you don't have the least upper bound property. There exist sets of rational numbers that have upper bounds that don't have a least upper bound. Well, we'll talk about that in just a second here. And so when it comes to our geometric conversation, we're really interested in the Euclidean plane, which can be given the coordinates R squared. So R squared is then deeply concerned with the properties of the real numbers because if we have this Euclidean plane whose coordinates are real numbers, then what we can say about real numbers means something about the coordinates of our geometry, which then means something about the geometry. So as one studies properties of the real numbers, you learn things about Euclidean geometry. So the completeness axiom is a necessary condition that we need to differentiate between, say a affine congruence geometry like Q2, right? Q2, it's a congruence geometry. It's an affine geometry because it satisfies the Euclidean parallel postulate. So what separates these two geometries is that the real plane, that is the Euclidean plane is complete and the rational plane is not. And we wanna be a little bit more precise about that in this conversation. So the least upper bound property is sometimes referred to as the Dedekin axiom of completeness or continuity because the model of the real numbers that best exemplifies the least upper bound property is the model of Dedekin cuts on the rational numbers. As such, we're gonna recast our continuity axiom for geometry after the least upper bound property and we're gonna use this idea of Dedekin cut because after all, as our goal is geometrically focused, I wanted to explain continuity in a way that'll be most approachable for a geometry student. And I think the conversation about Dedekin cuts really will be the best way to approach continuity as opposed to least upper bounds and completeness and other issues, right? I mean, we could talk about every Cauchy sequence is convergent, that's a type of completeness ax and we could use, but that doesn't feel geometric. Dedekin cuts do that. Why keep on saying this word, Dedekin cut, Dedekin cut? What's a Dedekin cut? Well, we're gonna talk about that just right now. So let L be a line and let sigma one, sigma two be subsets of that line L. So the pair, sigma one, sigma two is called a Dedekin cut of the line L if sigma one and sigma two satisfy the following two conditions. Both the sets, sigma one and sigma two, they need to be non-empty and they need to be convex subsets of L. Remember what convex here means. So convex means that if the points P and Q belong to the set, sigma one, that implies that the segment PQ is belonging to sigma one as well. So in other words, if two points belong to the set, everything between the two points also belongs to the set. So sigma one, sigma two are gonna be non-empty convex sets of L such that L is the union of sigma one and sigma two and the intersection of sigma one, sigma two is equal to the empty set. So when you put these last two conditions together, the pair determines a partition of the line. So what is a Dedekin cut? It's a partition of the line into two cells which are convex. So we're basically looking for a convex partition of a line into two pieces. That's what one calls a Dedekin cut, okay? So that's just the definition. What does this have to do with continuity? Well, the Dedekin axiom for continuity for which we, this section, this lecture was called continuity axioms. In this video, we're gonna adopt one single continuity axiom because one of the conveniences of using the Dedekin axiom is that one, it's geometric, but two, it's one axiom that'll give us all of the continuity principles we want to discuss for what we will soon call neutral geometry. So we're actually gonna take a much simpler approach to continuity than what David Hilbert originally provided for us. So the Dedekin axiom says that given any Dedekin cut of a line, so that is we have a partition, a convex partition into two pieces of the line, given any Dedekin cut of the line. And I should mention that's why we call it a Dedekin cut here. It's because we have these two pieces to the line. If you think of it as like, oh, we have a line and the idea is take your scissors and cut, you cut into two pieces. That's the idea. Then convexity is to suggest why there's, it's not a little bit more mixed up than this. So given any Dedekin cut, sigma one, sigma two of the line, there exists a unique point P and an ordering on the line. Remember, all lines have orderings. There's one or the other. And so given this point P, there's an ordering of the line such that sigma one is just the interval negative infinity to P and sigma two is the interval from P to infinity. That's gonna be the case in either situation, but where does P belong? P might belong to sigma one or P might belong to sigma two. So the continuity action tells us that given any Dedekin cut, well, it's really just you cut the line into two intervals, two rays, you could say it that way as well. You cut a line by a Dedekin cut, it forms two rays. And of course the point P belongs to one of them. That's the important thing here. We have this closed ray and an open ray. It doesn't really matter which one it belongs to, but this is the only way you can cut a line using a convex partition, a Dedekin cut. And so that statement itself, it's like, okay, I can kind of see what you're going with. What's the big deal? What does that have to do with continuity, right? We're gonna make, we'll make them more explicit as we explore the examples here. But a geometry that satisfies the Dedekin axiom, we call this a complete geometry. In particular, if you have a complete congruence geometry, remember the vocabulary we have for this lecture, for this lecture series. A congruence geometry is a geometry which satisfies the axioms of incidents between this and congruence. So therefore a complete congruence geometry will satisfy the axioms of incidents between this congruence and continuity. We make no assumption about parallel postulates, whatsoever. A complete congruence geometry is referred to as a neutral geometry. Some people call it an absolute geometry. And the name there is to suggest, we'll take the term neutral or absolute geometry here, where are these names coming from? The thing is, we haven't made any assumptions about parallel postulates yet. So therefore any theorem of neutral geometry will be a theorem of Euclidean geometry, but it'll also be a theorem of hyperbolic geometry and anything else that doesn't depend on the parallel postulates. So the term neutral is to suggest that, oh, we're gonna go Switzerland here, we're not gonna choose a side. I'm not saying whether it's the Euclidean parallel postulate or the hyperbolic parallel postulate or something else. We remain neutral with regard to the parallel alternatives. The term absolute geometry is also used in this situation because anything that's true in absolute geometry will then be true for Euclidean and hyperbolic geometry. So it's something that's always true. Not a big fan of that because there are other geometries that aren't necessarily absolute geometries like projective geometry, elliptic geometry. These are not absolute geometries. So some statements won't be true for a spherical geometry. And so it's, I don't like the term absolute here because there are other geometries. So I personally prefer the term neutral geometry. Neutral geometry is we have a congruent geometry. We do have continuity in hand now, but we're gonna remain neutral with regard to the parallel alternatives. So, okay, we should talk about, oh, and one other thing I wanna mention here is this idea of complete. In this lecture series, when we defined a ordered geometry in that setting, we said an ordered geometry was an incidence geometry that satisfied the betweenness axioms. Now some authors, when they define ordered geometry actually say that an ordered geometry satisfies the incidence axioms, the betweenness axioms, and the continuity axioms, in which case what we in our lecture series would call a complete ordered geometry. Some people would call that just an ordered geometry. For them, continuity is built into the cake, or cooked into the cake, baked into the cake. That's the word you're looking for. So be aware, that's not gonna be too much of a concern for us, but as you're studying things about completeness, be aware that some geometric terms already have that built into it. When you talk about neutral geometry, continuity is expected, completeness is expected. Some people expect it for ordered geometry we did not in our lecture series. All right, let's get back to some examples of neutral geometry. The most obvious example, of course, is going to be the Euclidean plane, do-do-do. I mean, this is the geometry we're trying to axiomatize using Hilbert's axioms. So every axiom we add is an axiom of Euclidean geometry. We just don't have them all yet. At the moment, we have not accepted a parallel posh that hints neutral geometry. But the Euclidean plane is the poster child of a neutral geometry. But one of the, of course, the most greatest discoveries in mathematics, particularly the branch of geometry was the existence of a hyperbolic geometry. So the real hyperbolic plane is also an example of neutral geometry. The hyperbolic plane is consistent with all of the axioms we've talked about so far, incidents between this congruence continuity. It satisfies all of those conditions, but it doesn't satisfy the Euclidean parallel poshlet, which then proved that the Euclidean parallel poshlet was independent of the neutral axioms. That is Euclid's fifth poshlet could not be proven as a theorem inside of neutral geometry, which many people studied for a long, long time. So, and these two are basically the neutral geometries. Up-tie-somorphism, you basically just have these two neutral geometries, the real plane and the hyperbolic, I should say the Euclidean plane and the hyperbolic plane. But these are not the only examples of complete geometries we could have. For example, in our lecture series, we've talked about the real projective plane, right? One of the models we talked about that is you can take the upper hemisphere of a sphere and you add a wraparound feature on the boundary, the equator there. So, when you hit the equator, you can, whoop, you jump to the other side, like you're in a video game or something. In this geometry, it does in fact satisfy Dedekind's axiom. Whenever you have a line, which in this case, a line is like a semicircle, right? Which again, you have this wraparound feature, whoop, like so, the only way you can cut the line into a convex partition is to be intervals. Which, of course, we have to be a little bit more careful in this setting here because when we talk about the Dedekind cut, the Dedekind cut used some terms about ordering. That is, well, I mean, the Dedekind cut did not require that, the definition. But the continuity axiom was saying that, oh, whatever Dedekind cut you have, you have to cut the line into two intervals. That can be a little bit of a murky issue inside of the real project of plane because the real project of plane does not have a well-defined notion of betweenness. In particular, trichotomy is an issue here. Like if you have three points, right, which one's really between the other? Because you might be like, oh, I went from A to B to C like that. But you could also go from B to C to A like so, or you can go from C to A to B. So in some aspect, all of the points are between each other. Trichotomy doesn't make any sense there. You also do have some issues when it comes to posh's axiom in the real project of plane because it turns out because of this wraparound feature, if I'm on this side of the line and you have some other point on the quote, unquote, other side of the line, sure there's a line segment that connects the two that intersects the line, but there's also the back door which when you hit the boundary, it's like whoosh, you teleport to their side and then you can come over here. And so that second line segment didn't cross the line because this little teleportation part down here is not actually part of the geometry. Do, do, do, do, do, do, do. So in that setting, there's really only one side of a line in the real project of plane. So the notions of betweenness don't exactly work in the real project of plane. This is an issue we'll talk about a little bit later. Therefore, issues like, well, can you actually talk about the interval from negative p, negative infinity to p, right? Do you need ordering to talk about that? One can refine the version of the Dedekin axiom that we presented on the screen above to include geometries like the real project of plane where, again, betweenness doesn't exactly work. So in particular, I should mention that the real project of plane is not a neutral geometry because we do not have all of the axioms of betweenness. We're missing two of the four. But nonetheless, we can still, we could massage Dedekin axioms to be appropriate in a setting where betweenness is a little bit lacking, such as the real project of plane. So we can get continuity in that situation. We do get completeness in the real project of plane. But I should also mention the real project of plane that notions of congruence of segments and angles, they actually satisfy the six congruence axioms. And so despite these issues of betweenness, incidence and congruence have no issue. And we do, in fact, also have completeness inside of the real project of plane. That's an important thing to note here. So RP squared is not a neutral geometry because it doesn't have trichotomy or posh's axioms, but satisfies everything else, including the continuity axioms. I should say axiom because we only have one in this situation. I should also mention that the sphere also is an example of a geometry, which is complete, right? It satisfies Dedekin axiom. Now with the sphere, you do have some issues that it's not a neutral geometry because as we talked about before, line determination doesn't work in spherical geometry. Note that if you have two antipodal points on the sphere such as the north and south pole, then there are actually multiple straight lines that connect them together. So there's always a line between two points, but for antipodal points, that line is not unique. So line determination doesn't work. Also for similar reasons that we had with the real project of plane, trichotomy doesn't work for the Euclidean sphere. And those are the only axioms we're lacking. The other four axioms of incidence are satisfied on the sphere. The other three, let's see, did I say that correctly? There's four axioms of incidence while line determination doesn't work. The other three work for the sphere. When it comes to betweenness, there's four axioms of betweenness. Trichotomy doesn't work, but the other three axioms of betweenness, they work just fine. That includes plane separation. In the sphere, a line does cut the plane into two sides, one side and the other side. Think of the equator, for example. You have the upper hemisphere and the lower hemisphere. You do have plane separation in that situation. And then when it comes to the axioms of congruence, the sphere satisfies all six axioms of congruence. We can make a notion about segment congruence. We can talk about angle congruence and it'll satisfy the six axioms we have there. So the only axioms of neutral geometry that the sphere doesn't satisfy are line determination and trichotomy. So notice that with both of these elliptic geometries, trichotomy is basically dead on arrival for both of them because in these two geometries, lines are circles. And so you really can't have trichotomy when lines and circles coincide here. But you can trade plane separation for line determination. And so the real project of plane does have that advantage that it is a incidence geometry. But on the other hand, the sphere has the advantage that you have plane separation. But both of these examples, this is the critical thing I'm trying to get to here. Both of these examples satisfy the completeness axiom. They are both complete geometries. And in fact, any manifold that we can come up with is going to be a complete geometry because a manifold, remember, is a locally Euclidean topology. And because it's locally Euclidean, you can inherit the least upper bound property from that local behavior. And so we could list several other manifolds here. These are all gonna be examples of complete geometries. So now we reach a very important part in this video here that we need an example of a geometry that is incomplete. That is not a complete geometry. And the example we're gonna look at is gonna be Q. Q2, I should say. Now, of course, the reason we're gonna go here is that as I've already mentioned, Q2 is an affine congruence geometry, or call it a congruence affine geometry. It satisfies the axioms of incidence between this completeness and it actually satisfies the Euclidean periopostia. The only thing it doesn't have is the continuity axiom. And we're gonna argue in this example right here that it doesn't satisfy the continuity axiom that in fact, Dedekin cuts can be problematic for Q2. So let's first, let's of course make sure we understand what we mean by Q2 in this situation. So the set itself will be the points of the geometry. That's often something we do with the geometry. The set is the set of points and the other things are gonna be implied. So we talk about the set Q2, this is a Cartesian product where points are considered ordered pairs where both coordinates are in fact, rational numbers. So we can have something like one comma two. We can have something like one half comma zero. We can have seven fifteenths comma negative three fourths. Those are all example of rational points. What is a line inside of Q2? Well, we can define lines using the coordinate system that we have since the points are ordered pairs. That is we define lines as linear equations, ax plus by equals C, which the coefficients a, b and c have to be rational numbers, which includes integers of course. a, b, c have to be rational numbers and you can't have that a and b are simultaneously zero. a could be zero, you could have something like that. b could be zero. This would give us vertical and horizontal lines, but other than that, they can't both be zero because you either have zero equals something like two, that's not a line, that's a contradiction or you could have like zero equals zero, that's not a line, that's an identity. We don't wanna consider those possibilities. Now of course, when you have a rational equation, so the coefficients are a, b and c, you could, since there's only three numbers in play here, you could always find the least common denominator of the fractions a, b, c, you could clear the denominators and so every rational linear equation could be turned into an integer linear equation. So if you prefer that, just think of it that way, a, a line in the plane Q2 is just a set of all integer linear equations. Now of course, two different linear equations can actually be representing the same line. Like if I take like x plus y equals two, this is the same thing as three x plus three y equals six. Notice that if I times the first equation by three, you get the other one. Those are considered equivalent equations and we aren't gonna distinguish between equivalent equations. Those represent the same line. So really when we talk about lines in Q2, it's there's an equivalence class there, equivalence relation. A line in Q2 is a linear equation up to equivalence and therefore then a point on Q2 is considered incident to a line when it satisfies the equation. That is a solution to the equation. So something like x plus y equals two, then the point one comma one would be a point on that line. I should say it that way because notice one plus one equals two. You could also do two comma zero, that's on the line. We could take something like three fourths over here and then we have one and a fourth right here, or five fourths if you prefer. We can do lots of points. There's gonna be infinity points on that line. And that gives us the notion of incidents. That notion of incidents will satisfy the four axioms of incidents. We have line determination, point existence, secant C and finally non-colonarity. How do we define betweenness in this setting? Well, betweenness for the plane Q2, we would say that two points or one point is between the other. If you, well, the short answer, there's a couple of ways you could do it. But basically when you have equality of the triangle and equality, that's what betweenness could mean in that situation. So what do we mean? We say that a point, let me offer a linear algebraic perspective here. So if we have two points P and Q, then we could look at their linear combination. So we have T P plus one minus T Q in this situation. Like so, for which I'm gonna require that T be a number on the interval zero to one, like so, for which clearly P and Q are gonna be rational points and T is gonna be a rational number on the interval zero to one. This then we can define to be the segment P to Q, like so, or I should say I shouldn't write as an interval, I should write as a segment P Q, like so. So the points other than P Q themselves, every other point that can be written as a linear combination like this, this is what's often referred to as a convex combination in linear algebra. We could say that's a point between these, this convex combination definition will give us the actions betweenness. We have not, we have co-linearity and symmetrization. We have extension, we have trichotomy, we have Bosch's axioms. Those will be satisfied in that situation. Then what about congruence? What does it mean for, what does it mean for things to be congruent? Now that you have segments defined, we can talk about the length of a segment. So you can talk about the distance between two points, P and Q, that's gonna produce a real number. We can then say that's the length of the segment P Q. And so we'll say that two segments are congruent to each other, if and only if they have the same distance, they have the same length attached to them. We can do similar things. I don't wanna say too much more about this, but we can talk about every angle has a measure and then two angles are congruent if and only if they have the same measure. This is exactly how we define things in R2. What does incidence mean there? What does betweenness mean there? What does congruence mean there? The only problem, I should say the only difference between R2 and Q2 is that when we define points, the coordinates have to be rational numbers. When we define lines, the lines have to have rational coefficients. When we talk about convex combinations for betweenness, the coefficients there have to be rational numbers. With distance, a distance function actually maps into the real number. So if the distance between two points is irrational, that's not a big deal for us. We're okay with that. You can still have a rational segment, even if its length is irrational. Sorry Pythagoras, you can actually do that. So with this conversation here, we've now reached the point where, okay, we've now established that Q2 is a congruence geometry. And also I should mention that Q2 satisfies the Euclidean Parallel Apostolate, because if you have a rational line and a rational point off of that line, there will exist one rational line that passes through it. Because in particular these lines and points are subsets of the Euclidean plane. So you can't have more than one parallel line because that would be more than one parallel line in the Euclidean plane. But then the other thing is to remember is in the Euclidean plane, two lines are parallel to each other, if and only if they have the same slope. And for a rational line, the slope has to be rational. And so therefore there will be a rational line over here with that slope. So Euclidean Parallel Apostolate is good. Everything'sunky dory. So now we're in a situation where I can actually then give us a dedicum cut that cannot be turned into an interval. And this will show us that the continuity axiom fails in this situation. And what better line to use than the x-axis? Because after all, we've proven previously that in a congruence geometry, all lines are congruent to each other. So continuity axioms, we take our line, which is the x-axis. Then we have to come up with a dedicum cut, which is gonna be a convex partition of the line into two pieces. Our sigma one is gonna be the following. We take all points on the x-axis. So it's x-coordinate could be potentially anything. It's y-coordinate, of course, will be zero. But we take all points on the x-axis so that the x-coordinate when you square it is greater than or equal to two. And then the other set sigma two will be all those points on the x-axis such that when you square the x-coordinate you get something less than two. And so the first thing to note here is that every point on the line on the x-axis belongs to the set sigma one or to the set sigma two. Because if you take your x-coordinate it squares either greater or equal to two or it's less than two because we have a total order on real numbers and hence the total order on ration numbers. Every point will fall into one of those two sets. It should also be said that the intersection of these two sets is the empty set. There is no point, there's no number whose square is both greater or equal to two and less than two. Those options are mutually exclusive. So this is clearly a partition of the line. Why is it convex? Well, we have to, we have to then remind ourselves what's between this here, right? So let's suppose we have two points on the set sigma one A and C which I'm really, since this is just the x-axis here I'm really just focusing on the numbers x here. You can sort of ignore the y-intercept for right now excuse me, the y-coordinate because it's clearly zero. So we have, we essentially have two numbers A, C on the set sigma one. So this means that A squared is greater than equal to two and C squared is greater than equal to two. So without the lots of generality let's assume that A is larger than C. So A is greater than equal to C and then let B be between the numbers A and C. So B is some number that some rational number that sits in between the rational numbers A and C in this situation. So then there exists some rational number T such that B is equal to TA plus one minus TC like we see here. And so play around with that. If I square B squared that means I'm gonna take TA plus one minus TC and I'm gonna take that quantity and I'm gonna squared, okay? Since, since A is larger than C I can actually replace A with C in this situation for which then notice that TC plus one minus TC that adds together just to be CC of C squared that's greater than two, greater than equal to two they should say. And so therefore B squared is also greater than equal to two. So this tells us that B belongs to sigma one and as B was an arbitrary point between A and C we then get that sigma one is convex. A similar argument will show that sigma two is convex I will let the viewer of the video prove that if you want to, it's the same argument. Thus, sigma one sigma two is by definition a dead, a Dedekin cut of the line but this is the important but this is what we're building up to. There is no rational number P such that P belongs to sigma one or sigma two that makes this into a interval of rational numbers. And the issue kind of boils down to this the square root of two is an irrational number. This is something we can prove that we're not going to in this video but it's a very simple elementary proof that the square root of two is an irrational number it's not a rational number. So in the real number system, if you took this Dedekin cut the least upper bound property within produce for you oh P is equal to the square root of two and that's the thing that separates it and that situation for the real numbers sigma one would be equal to the interval the square root of two to infinity and then sigma two would be the interval negative infinity to the square root of two. This is of course, this is of course in the Euclidean plane but the rational plane does not have this point the square root of two. And so in that situation there is no boundary point that can be obtained. And so therefore this illustrates that q two fails the continuity axiom. So it is in fact a congruence affine geometry that it doesn't satisfy continuity. So the continuity axiom is independent of the other axioms of neutral geometry.