 Welcome back to our lecture series Math 3130, Modern Geometries for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misildine. This is the first video for lecture two. In lecture one, we introduced the idea of an axiomatic method, what is an axiomatic system? There's four ingredients to an axiomatic system. There's the undefined terms, the definitions, the axioms, and the theorems. We talked a lot about axioms and how they are these true statements because they're true, right? They're true without proof and that the need for axioms and logic is to avoid a logical descent, an infinite logical descent. I should mention that in addition to this logical concern from the last video, the purpose of axioms is to provide a definition to the undefined terms in a way of speaking, because the undefined terms don't have a definition, but the axioms tell you what must be true about those undefined terms. In some essence, it is giving it a definition. The axioms provide a level of rigidity to the undefined terms. That is to say, a reader can apply any interpretation to the undefined terms that they wish so long as all of the axioms associated to this undefined terms are satisfied. That is given the interpretation of the undefined terms, then with that interpretation, the axioms are true. That would then be a valid interpretation. By giving each undefined term in the system a particular meaning, we've created an interpretation of the system. If for a given interpretation of the system, all the axioms are true statements with that interpretation, then we call that interpretation a model. That's what I want to talk about in this video right here. What is a model? What I have in front of us are the four axioms to the fee foe logical system that we introduced at the second video of lecture one. You can see those here. Let me just remind you. We have these undefined terms, fees, and foes for which there's also a relationship between fee and foe called belonging, so fees can belong to foes, and foes can contain fees. The four axioms we had were the following. Axiom one, there exists exactly three fees in the system. Axiom two says that any two distinct fees belong to exactly one foe. Axiom three says that not all fees belong to the same foe, and axiom four tells us that any two distinct foes contain at least one fee that belongs to both. In lecture one, we had proven four theorems about fee foe theory, but we won't worry about those right now. What we want to do right now is give some interpretation to it. Fees and foes at the moment really don't mean anything to us. That's why they're called fees and foes. Intentionally, there's no intuition to what they are, but I'm going to give us some meaning to it. We're going to interpret the word fee as actually people, and we're going to interpret the word foe as committees. Imagine you have people working in the office, and there are committees that are formed to make decisions for the office. Then the relation belongs to, will then mean that the person is a member of that committee. Is that a valid interpretation? Well, it does depend on a little bit more, but that can get us definitely started. If we interpret the four axioms with these meanings of fee foe and belonging, what would they mean? Well, they would turn into the following. Axiom one would say there are exactly three people. Well, if there's three people in the world, there's more than that. But maybe in this particular office, maybe only three people work there. Maybe it's a small business that's very plausible, or maybe we're only talking about three people. So that's what the first axiom would become. The second axiom would become every pair of people belongs to exactly one committee. Axiom three would then be reinterpreted as, not everyone belongs to the same committee, and then axiom four would then be reinterpreted as, every pair of committees has exactly one person on both committees. So if we interpret the fees as exactly as these three people, we have Bob, Ted, and Carol. These are the only three people we're talking about. So these are the only fees that would then satisfy axiom one already. Then we're gonna say that there are three committees that they serve on. There's the entertainment committee, the finance committee, and the refreshments committee, which these two feel a little bit like, entertainment and refreshments, only one committee for finance, right? Clearly party planning is a big deal in this office. And Bob, Ted, and Carol are in charge of these three things. So how does belonging work? So we have our fees, we have our foes. Belonging is the following here. Bob and Ted are both, are the two members of the entertainment committee. Ted and Carol are the two members of the finance committee, and Bob and Carol are the two members of the refreshments committee. So we've now interpreted fees, we've now interpreted foes, and we've now interpreted belonging. Does this belonging fee foe interpretation make us a model? Does it satisfy the four axioms? Well, like I said before, since we're only considering these three people, Bob, Ted, and Carol, then that means that we have exactly three people, right? So that axiom is satisfied. Now, axiom two, every pair of people belongs to exactly one committee. So what are the possible pairings? We have Bob and Ted, that is the entertainment committee. We have Bob and Carol, that's the refreshment committee. We have Ted and Carol, that's the finance committee. So every pair of people does in fact belong to exactly one committee. There's no pair that belongs to two different committees. How about axiom three? Not everyone belongs to the same committee. Is there a committee that contains everyone? Well, the entertainment committee doesn't have Carol, so nope, the finance committee doesn't have Bob, and the refreshment committee doesn't have Ted. So there's no committee that contains everyone. So axiom three is satisfied. And then axiom four, every pair of committees has at least one person on both committees. So if we take entertainment and finance, you'll notice that Ted is on both of those committees. If you take entertainment and refreshments, you'll notice that Bob is on both of those committees. And then lastly, if you take finance and refreshment, you'll notice that Carol is on both of those committees. So this is in fact a model of the FIFO axioms because all four axioms with this interpretation are then true statements. And so I should mention before we look at the next example that because the theorems of a axiomatic system are logical consequences of the axiom and are thus independent of interpretation, all theorems of an axiomatic system are necessarily true for every model. This is the real strength of the axiomatic method. So let's take this example that we have right here with Bob, Ted, and Carol and the entertainment, finance, and refreshment committees. In lecture one, we prove four theorems about FIFO theory. Those theorems must be true for the interpretation, the model that we see currently on the screen. The first theorem that we proved when interpreted with this language, then would say that every committee has exactly two people that belong to it. And sure enough, every committee has two people on it. In fact, the committees are uniquely determined by the people on it. There's no two committees that have the exact same two people, right? The third theorem we proved was that given two distinct committees, there's exactly one person on both of them. We mentioned something like that already, entertainment and finance. They have, that one has Ted together. Again, we kind of said that one already. That's similar to what we said with axiom four. Axiom four says there's at least one person on both. But in fact, we're saying that with theorem three of FIFO theory that there's only one, there's exactly one person on both committees. We see that. And then the third, the fourth theorem said that there are exactly three committees, which we see right here. All of the theorems of FIFO theory must be true for this model. And so, although these results might seem trivial for this example, this example, this model you see on the screen is just a toy that we can play with. For more complicated axiomatic systems, this sort of strength is a valuable tool for this model theory. For example, any interpretation with peoples and committees with four, if there's four committees, then we know that one of these axioms must have been violated because the theorem was violated. We know for FIFO theory that there can only be three committees. If I had an interpretation with four committees, one of these axioms I know must be violated. This is a very powerful thing. The axioms are assumed to be true for the theory, but in the model, it depends on the interpretation. If the interpretation has valid axioms, then we gain all of the theory. If the interpretation doesn't lead to a model, that means at least one of the axioms was invalid, and therefore the theory doesn't apply. That's why we assume the axioms are true because a later interpretation will then check whether they're true or not. Let's look at another example here. So again, our four axioms of FIFO theory are on the top of this screen. I want you to see them so you can also see the new interpretation. This time we're gonna interpret the fees to be books. We're gonna interpret the foes to be shelves and then belongs to as well the book is on the shelf. So what are interpretations due to the axioms? Well, axiom one would say there are three books. Maybe you only have three books in your library. That's a possibility. Every pair of books belongs to exactly one shelf. Not every book is on the same shelf and every pair of shelves has at least one book that's on both shelves. Now, when you look at the last one, that's kind of like, how can you have a book that's on two different shelves simultaneously? That might break laws of space, time, physics and things like that. So this interpretation is not gonna lead to a valid model, which we should specify what are the books, what are the shelves, which book is on which shelf. But the thing is a book can't belong to two shelves simultaneously. So we already see that this interpretation is not a valid model because it doesn't lead to a truth where axiom four is a true statement. And so again, that's the power of the axiomatic method. It's up to the interpretation. Some interpretations will satisfy the axioms. We have a model that model then inherit all of the theorems of the theory because the theorems are independent of the interpretation they only depend on the truthfulness of the axioms which are taken for granted. It's in the model stage of the interpretation stage we then will scrutinize whether the axioms are true or not. And so I hope these examples really prove to us why the axiomatic method is was important because the axiomatic method when coupled with model theory, that is we come up with these interpretations which give us models or not, then starts to show us the strength behind the theory here.