 In the previous lecture, we introduced the five axioms of phanogeometry, which were listed right here, axioms one through five. We'll look at them in a moment ago. And so we proved that phanogeometry is consistent. We also proved that phanogeometry as an axiomatic system is complete, that there's only one model up to isomorphism. And that model basically looks like the following, which I'll just draw a small replica of it really quickly. You know, you basically have these, like the Harry Potter deathly hollows is one of the worst drawings ever done, sorry, but three elder ones there. So this is our phanogeometry. Check a look at this previous video if you need to see a better drawing of that. What I want to prove in this is that the theory is not just consistent. That's easy because we have a model. It's complete because we proved this is the only model. But I also want to show that the axioms we have for phanogeometry are independent. That is, if I removed any one of these axioms, I could construct a model which satisfies the negation of the opposite one. That is, none of these axioms is a theorem of the others. So let's take a look at that. What if we take away axiom one? Axiom one says there exists at least one line. The contradiction of that, the negation of that would be there exists no lines. So what if there's no lines? And heck, let's say there's no points. What if we take the empty geometry? There's nothing there. There's no points. There's no lines. The nice thing about having no points and no lines is that the incidence relation, which also is empty, you can't have any incidence between points and lines because there aren't any. The four axioms are going to be vacuously true. Notice that axiom two, there exists exactly three points on every line. If there's no line, then there's no lines that don't have exactly three points. Vacuously true. Axiom three, not all points lie on the same line. Notice there is no line that contains all of the points because there's no lines. It's again, vacuously true. Now for each pair of distinct points, there exists a unique line that contains it. There does not exist any pairs of points that don't have a unique line. So again, vacuously true. And then all lines intersect. Why is that true? Well, do you have any lines that are parallel? Nope, because there are no lines. So all of these axioms are vacuously true for the empty geometry. So this proves to us that axiom one is in fact independent of axioms two through five for phano geometry because using axioms two, three and four, two, three, four and five, we cannot prove the existence of a line because the empty geometry would then be a counter example. So the existence of a line is an independent axiom. What if we move on and take out axiom two? So there exists exactly three points on every line. So what that means is that there exists some line that doesn't have three points, but the other ones are true. I want you to convince yourself that three point geometry is a model that satisfies axioms, phano axiom one, three, four and five, but negates phano axiom two because there does exist a line. There's three of them. Not all points lie on the same line because given your points here, there's only three of them. There's no line that has them all. Between any two points, there's exactly one line and all lines intersect. So axioms one, three, four and five for phano geometry is satisfied, but axiom two fails. Why? Because in this situation, all lines contain two points. In particular, not all lines have three points. So axiom two is then independent of the other phano axioms. What about axiom three? So axiom three says not all points lie on the same line. So it's kind of natural on how this model is going to have to be. So let's take a single line that has three points on it. There's no other points in the geometry. Is there a line? Yes, there is because that's all there is, honestly. Every line has exactly three points. Yes, there's only one line and it has exactly three points. Every pair of points has a unique line that contains it. Yep. This pair has a line between it. This pair has a line between it. This pair has a line between it and it's the same line. There's only one line total that does everything, right? And then axiom five, all lines intersect. That is vacuously true in this situation because there are no parallel lines. So all lines intersect. There's no pair of lines that aren't intersecting because there's no pair of lines at all. You can get, you have to be careful with these, these vacuous statements, right? It's vacuously true. It's true because the hypothesis of the conditionals never satisfied. And therefore it doesn't matter what happens to the conclusion. So axiom five is satisfied, but clearly axiom three is violated because there is a line that contains all the points. Now I want to make mention to you that as we're proving these axioms are independent, I'm providing you models that negate one of the axioms but satisfies all the other ones. Okay? These models are not necessarily unique. There could be other models that satisfy four of the axioms and negates the others. But we just need one model to show that the axiom is independent. So this model right here shows that axiom three is independent because we constructed a model of phano axioms one, two, four, and five, but negates axiom three. All right? What about if we negate axiom four? Negating axiom four would mean that, because axiom four says that given any pair of lines, any pair of points, excuse me, there's a unique line between them. We wanted to negate that, which means that there's at least one pair of points which doesn't have a unique line between them. That could be because there's no lines between them. That because there could be the multiple lines between them. Look at our model here. So in this situation, we have a line with three points on it, and we have some point that's off of a line, off of that line, and there's no other lines. Is there a line? Yes, there's exactly one. Does every line have three points? There's only one line and it has three points on it. Not all points lie on the same line. There's only one line, but it doesn't contain this point, so axiom three is satisfied. And then axiom five here, all lines intersect. Yeah, that's true, because there's only one line. All lines intersect there. And then why is axiom four violated here? Well, because there's no line between these two points. That would be a violation of that. I could also do something like this. This would be an alternative picture. This would also satisfy axiom one. There's now three lines. We need some more points on there to satisfy axiom two, so we could do something like that. Or you could just do something like this. This is a line. These three points. So that would do it, because in this situation, the lines intersect. Not all points are on the same line. Same thing. All lines have three points and there exists a line. So there could be multiple models you come up with that then show the independence. This shows that axiom four is independent. Now the last one, why is axiom five independent? Can we construct a geometry which satisfies axiom, phano axiom one, two, three, and four, but then violates axiom five? That is, there are parallel lines. And we're going to do that, but we're actually going to do that in the next lecture. Lecture five is about what we call young geometry. And our young geometry is a finite geometry that's going to do exactly this. We are going to assume axiom one, axiom two, axiom three, axiom four, phano geometry. So the first four axioms of young geometry are identical to the first four axioms of phano geometry. But we're then going to, we're going to replace axiom five with a different axiom that actually, it doesn't just allow for parallel lines. It actually guarantees some existence of parallel lines. And therefore, because there will be parallel lines, young geometry will then show that axiom five is independent. But we'll wait for the next video to do exactly that. Thank you for watching lecture four here about phano geometry. I hope you learned something. If you did, give these videos a like, please. If you want to see more videos like about young geometry or other geometric videos, any mathematical videos, in fact, subscribe to the channel so you always get information about new videos that are being posted. And as always, if you have any questions, please, please post them in the comments below and I will answer them at my soonest convenience. See you next time, everyone. Bye. Thank you.