 In this video, I want to recast our FIFO axiomatic system into the language of geometry. So you'll remember previously that with our FIFO axioms and axiomatic systems, we introduce these things called fees and we introduce these things called foes. What we're gonna do is actually, instead of calling them fees, we're now gonna call them a point. So interchange the word fee with point and interchange the word foe with line. There was also this relationship between fees and foes that we called belonging. So we'd say that a fee belongs to a foe. Well, now we're gonna call that incidence. So we can say that a point is incident to a line. And so make that substitution there. Fees are points, foes are lines, belonging is incident. And then we changed nothing about the theory of fee and foe. What would that look like when we rebranded as a geometric object? Cause after all points line, point lines and incidents are undefined terms. So we're FIFO and belonging. And so to be a geometry, we just have to have some notion of points lines and incidents, which FIFO, if you rebrand it, does have that. So it turns out FIFO theory is a geometry. And in fact, this is what we call three-point geometry. Where it gets its name is actually from the first axiom of FIFO theory. If we rebrand it using now lines, points and incidents, the first axiom of FIFO theory, which let me remind you, the first axiom stated that there exists exactly three distinct fees in the system. Well, we can rewrite that as there exist exactly three distinct points cause fees are now points. And that's actually where it gets its name. We're calling this three-point geometry because we have the three points given by axiom one. Axiom two of FIFO theory told us that any two distinct fees belong to exactly one foe. Well, if we rebrand that using geometric terms, that becomes the sentence. For each two distinct points, there exists a unique line containing both of them. All right? Again, slightly different phrasing, but it means the exact same thing. We had the two fees, which are now points, and they're incidents to exactly one foe, which is now called a line. Axiom three of FIFO theory was not all fees belong to the same foe. Not all fees belong to the same foe. In lines and points, that actually turns out to be not all points lie on the same line. There's no line that contains all of the points. In other words, the entire set of points is a non-colonial set. All right? Then the last one, axiom four FIFO theorem says that any two distinct foes contain at least one fee that belongs to them. Okay? I've actually really simplified the statement. I've actually written that axiom as all lines intersect because if we took the exact language and just replaced foes with lines and fees with points, we would get that any two distinct lines contain at least one point that belongs to both of them. Now that we've introduced the word intersect, which we didn't have for fees and foes, but if we introduced the word intersect, that axiom can be simplified using the definition to just be all lines intersect. Remember the axiomatic method. We first have undefined terms, then we have definitions, then we have axioms. Axioms come third on the list, and that's because the axioms themselves can use in addition to the undefined terms, they can use the definitions that we've developed. Like in this case, the word intersect simplifies, it simplifies the axiom four here that all lines intersect. Given any two lines, there's at least one point on both of them. Now the fourth step of the axiomatic method is then theorems. Once we have our axioms, we can then start proving theorems from them. Now we proved four theorems for FIFO theory, just based upon the axioms. What we've done now with three-point geometry is we just now have a new interpretation of what phi and foe mean. We're interpreting phi as point and foe as line. And with those interpretations, then all four of the axiom, excuse me, all four of the theorems then become theorems of our three-point geometry. And so they translate immediately and we're gonna look at them in just one moment. Before I do that, I want us to consider this diagram that you see here on the screen. This is actually meant to be a model of three-point geometry and hence is also a model of FIFO theory for which these little circles you see here are supposed to be illustrations of what we mean by points. So we have three circles, three dots. Those represent the three points. These straight segments you see here then represent our lines. And so we have three lines that you see right here. We have the three points that you see right here. Incidents is then indicated that if a line touches the two points, sorry, if a line touches a point, we say it's incidents to that point. So if you focus on this point right here, this line is incident to it and this line is incident to it. Similarly, if we look at this point right here, this line is incident to it, this line is incident to it. And so from this diagram, this honestly, this graph we see on the screen for which we have vertices and edges, we can use a graph to illustrate a finite geometry for which we get the incidence relations directly from it. So this is in fact a model of three-point geometry. I want you to convince yourself of that. We have axiom one, three points, great. For each pair of points, there exists a unique line that contains both of them. So if you look at this pair, there's only one line that contains both of them. If you look at this pair, there's only one line that contains both. If you look at this pair, there's only one line that contains both. So axiom two is satisfied. Axiom three, there are not all points lying the same line. If you look at all three points together, there's no line that harbors all three of them. The set of all points is non-colinear. And then lastly, all lines intersect. If you take these two lines, they intersect at this point. If you take these two lines, they intersect at this point. If you take these two lines, they intersect at this point. And so this is in fact a model of three-point geometry. And therefore it's also a model of FIFO theory. And as we discussed FIFO theory in the previous lecture, we actually discovered that the theory was complete. That is up to isomorphism, there's only one model. And therefore this is the only model of three-point geometry up to isomorphism. Now let's take a look at those four theorems I had mentioned beforehand. I'm gonna zoom out a little bit so we can see the picture of the model and the theorems as well. So look at the first theorem here in three-point geometry. And so I'm gonna start all these theorems with that statement in three-point geometry. So we know that these theorems are not broad theorems for all geometry. It's only for the three-point geometry theory we're studying right now. And three-point geometry, each line, each line has exactly two points that belong to it. You'll notice this line has two points, this line has two points, and this line has two points. It's very easy to see from the model, but if we wanna prove it from the theorem for the theory, excuse me, we need to prove it from the axioms. Cause how do we know there aren't different models than this one? I said the theory is complete, but that's only because of these four theorems. From these four theorems, we can derive that every model is isomorphic to this one. All right, the second theorem of FIFO, which becomes a theorem of three-point geometry states that a line is completely characterized by the points on it. That is, if two lines are distinct, then they do not contain the same points. So if you look at these two lines right here, notice they have different points. If you look at these two lines, notice they have different points. If you look at these two lines, you'll notice they have two different points. So no two lines contain the exact same points. All right, so a line is actually determined, it's characterized by the points it's incidence to. So we actually could have defined lines to be sets of points. There would be no difference in the theory if we had done that. Theorem three in three-point geometry, two distinct lines intersect at exactly one point. So Axiom four told us that all lines intersect, in which we check, they all do, but furthermore, we know that the intersection is unique. You don't get something like the following where you could have two lines that intersect at two different points. We don't have anything like that in the picture. That's easy to see. And then the last one in three-point geometry, there are exactly three distinct lines for which we prove this in general. And that's true for a model because this is the only model of disomorphism. So with the right perception, many axiomatic theorems can be thought of as geometries. As long as we have a notion of line and point, then we have a geometry. And so our FIFO theory is also a geometric theory. And so going forward, we're never gonna talk about fees and foes again. We'll just talk about three-point geometry. And in the next video, we'll introduce another finite geometry that we haven't yet seen so far.