 In the previous video, we talked about models and how they are interpretations of undefined terms for an axiomatic system. It's important to point out that when working with models, it's possible to create two different models that really are the same model. You can have different words, different labels, different interpretations, but they're really the same. And in mathematics, this is what we often refer to as isomorphic, which when we look at the etymology here, morph means shape, iso means equal. So these are things with equal or same shapes. And so if two models are isomorphic, so you have one model M, some other model N over here, what we're looking for is some type of one-to-one correspondence between the undefined terms, such that any relationships between the undefined terms is preserved. And that's a little bit ambiguous. We might have to have talked about specific categories to make sense out of this. But in terms of the axiomatic method, isomorphic models are the same model. And so in our previous video, we introduced a model for our FIFO theory. We interpreted fees as people, we interpreted foes as committees, and then belongs to, was then as a person on the committee. So we had Carol, Ted, and Bob with their committees. And I want to, just as a reminder, the four axioms of FIFO theory are here on the screen right now. I want to provide a different model for FIFO theory that's essentially the same model. This time we're going to take our fees just to be three elements, X, Y, and Z, which these are elements that can belong to a set. So we can think of them as sets and from a set theory point of view. The foes are then going to be sets, they're going to be, they're going to be sets of these fees, right? So F is going to be X and Y, the set, the foes G is going to be X and Z, and then H is going to be Y and Z, and then belonging to is just set containment. You'll notice that when we proved theorems about FIFO theory, we eventually proved a theorem that says that it's perfectly okay to replace foes with sets of two fees. We did that. And so that's sort of validating what we're saying right now. But let's check that the axioms are verified, right? There are exactly three fees. Yep, the three of them are right here. Any two fees belong to exactly one foe. So if you take XY, F is the only one that contains it. If you take XZ, G is the only one that contains it. If you take YZ, that pair, H is the only foe that contains it. Not all fees belong to the same foe. There's no foe that contains all three of them, X, Y, and Z. So axiom three is satisfied. And axiom four, any two distinct foes contain at least one fee that belongs to them. If you take F and G, they both have X, that's great. If you take F and H, they both have Y. And if you take G and H, they both have Z. So there's at least one that belongs to them. Now, we don't have to go through and check all of the theorems. We proved four theorems for FIFO theory. We don't have to check them because this interpretation gives us a model. We satisfied all four of the axioms. All of the four theorems and any other theorems we could prove are automatically true for this model because it satisfies the axioms. When you're checking an interpretation to see if it's a model, you only have to check the axioms because the theorems will be gained automatically. Even if there's hundreds and hundreds of theorems, if the four axioms are satisfied, then the model gains all of them. So we checked all of the axioms, one, two, three, and four. So I want to jump here to the bottom. So an isomorphism is going to be this one-to-one correspondence between them. So for us here, we need a correspondence between the fees. We need a correspondence between the foes. So given the first model we did in the previous video and the model we have presently, we're going to identify Bob with X, Ted with Y, and Kara with Z. We're then going to identify entertainment with F, finance with H, and refreshments with G. You'll remember that the entertainment committee had Bob and Ted, the finance committee had Ted and Carol, and the refreshment committee had, what did it have? It had Bob and Carol on it. And so this is a one-to-one correspondence that preserves all the relationships here. So there's this one-to-one correspondence between the fees, we see it. There's this correspondence between the foes. And then the other thing here is the belonging. That's a relation. That relation is it preserved, right? Well, Bob belonged to the entertainment committee, X belongs to F. So that's preserved. Ted belonged to the entertainment committee, Y belongs to the F set. So yeah, that is satisfied. Bob belongs to the refreshment committee, X belongs to G, right? So does Carol. Carol belongs to refreshments, so does Z. That is preserved. If we look at the middle one right here, finance, Ted is on finance, Y is on H. Carol is on finance, Z is on H. All of the relationships between the fees and the foes were satisfied by both of the two models. These two models are isomorphic because we had this correspondence between all of the objects in play. The fees go to the fees and the foes go to the foes. And the belonging relation is preserved. And both of these models are the same model of the same theory. It's isomorphic. In particular, when it comes to fee-fo theory, essentially up to isomorphism, there's only one model. And we'll say a little bit more about that in the next video.