 So I wanna add some more terminology to our discussion of the axiomatic method. And so these are gonna be some adjectives to describe axioms here. So if you take an axiom of a system, right? So this is a statement which is assumed to be true, irrelevant of anything else, right? It's a true statement because it's an axiom. We say that an axiom is independent if it's not a theorem of the other axioms. So let's say like we have these axioms, axiom one, axiom two, axiom three, and axiom four, much like the FIFO theory we've developed here. Let's say that we were to actually prove axiom four, using as a proof we could use axioms one, axiom two, and axiom three. So if we could actually prove axiom four, which is a statement, right? Axioms are statements. If I could prove that statement using only the other three axioms, then that means that our axiom is actually what we call dependent. It's a dependent axiom means it's a theorem of the other axioms. And if you have a dependent axiom, it turns out that you don't actually need it because you can prove it from the other ones. An independent axiom would suggest that this is not possible. The axiom is not, you can't prove it from the other axioms. You need it in a manner of speaking to further the theory. We say that a set of axioms is independent. If each axiom is individually independent, and hence a set of axioms is dependent if one of the axioms is dependent in there. That's to say the set is not independent. Now, we oftentimes have dependent axioms for one or two reasons. One, we might not know the axiom is dependent upon the others because the proof isn't yet found. It could be very difficult, very convoluted, very lengthy. And so because the proof is unknown, we have the axiom, but we might discover later on, oh, we didn't need it. So you could redevelop the theory by scratching it out. The other reason we sometimes take on dependent axioms is that it's for convenience. That I want my theory to be easier. I want the proof to be easier, so I take on this axiom. And as we talk about different geometric axiomatic systems in the future, because there are different ways of developing the axioms for Euclidean geometry as an example, the reason that these different axioms exist has a lot to do with this independence versus dependence. And in our lecture series, we will adopt Hilbert's axioms for Euclidean geometry. The main reason is because they are logically independent axioms. Each and every one of them serves a purpose and you need it to get Euclidean geometry, the removal of any one of them would not give you Euclidean geometry, it would give you a much broader subject. And but how does one actually know whether your axioms are independent or not? Well, to show that an axiom is independent, what you're gonna do is you're gonna construct a model that demonstrates all of the axioms except for the one in question, but then the one in question will actually be negated. So that is to say, if we had four axioms, axioms one, two and three, if you wanna prove that the axiom four is independent, you create a model for which axiom one is satisfied, axiom two is satisfied, axiom three is satisfied, but axiom four is violated. That is the negation of axiom four is a theorem for that model. Because if axiom four was a theorem of axioms one, two and three, that means every model that satisfies axioms one, two and three would have all of the theorems including axiom four. So if you can demonstrate a model that violates axiom four, then axiom four cannot be a theorem of the other ones and therefore it's independent of it. And so constructing a model is how you show that an axiom's independent. And then David Hilbert's great work on Euclidean Geometry, he writes a very long paper that constructs models that satisfy all of the axioms of Euclidean Geometry except for one and thus proving each and every axiom is independent of the other. Now, if you wanna show that a axiom is dependent, you actually provide the proof of that axiom as a statement of the other axioms. And so to prove dependence, you provide a proof. To provide independence, the proof is actually a model. And so that's what I wanna provide here in this video here. I wanna show you that axiom one of the FIFO theory is independent, okay? Remember axiom one told us that there are exactly three fees in the system. So I need to come up with a model of axioms two, three and four that have not three fees. So I could have less than three fees or I could have more than three fees. So I'm actually gonna come up with a model with four fees. So we're gonna call them A, B, C and D and we're gonna define the foes to be sets of, well, sets of fees. And so there's gonna be four foes in this model here. So one of them's A, B, one of them's A, C, one of them's A, D and then one of them is A, A, C, D. So remember axiom one, axiom one was that there are three fees, exactly three fees in the system. Clearly this is negated because we have four fees. What about the other ones? What was axiom two? As a reminder, axiom two states that any two distinct fees belong to exactly one foe. So if we look at all the possible pairs, A and B, well, there's a set right here, AB, no one else has AB, so that's good. There's AC, you have this set that contains AC, no one else. You have AD, this one contains AD, but no one else does. There's the pair BC, this one contains BC, no one else does. There's the pair BD, this one contains BD and no one else does. And then there's CD, this foe contains CD, but no one else does. So axiom two is actually satisfied, right? Axiom two passes our list. It passes our check, I should say. Axiom three, what was that one? Not all fees belong to the same foe. There is no foe that contains all of the fees. So the first foe doesn't have, it doesn't have C, so it's good there. This one doesn't have B, this one doesn't have B, and then this one doesn't have A. So there's no foe that contains all of the Bs. Excuse me, there's no foe that contains all the fees, so axiom three is satisfied. Axiom four said that 82 distinct foes contain at least one fee that belongs to both of them. So when we look at pairs of foes, the first pair both contains A, that's good. If we look at this one, AB, AD, they both contain A, so that's so far good. If you look at this one, they both contain B, all right? So AB is good. If we look at AC and AD, they both contain A. If you look at AC and then BCD, they both contain C, that's good. Then if you look at AD and BCD, they both contain D. So every pair of foes has at least one that's on both of them. So it satisfies axiom four. So you'll notice that axioms two, axioms three, and axioms two, three, and four are all satisfied, but axiom one is not. Therefore, since we have a model for axioms two through four FIFO theorem, FIFO theory, but axiom one fails, this then shows that axiom one is independent of the other axioms. And I'll leave it as an exercise to the viewer here to prove that axioms two, three, and four are likewise independent. To prove that axiom two is independent, what you would do is you would construct a model where axiom one is true, axiom three is true, axiom four is true, but axiom two is false. That would show that it's independent. If you wanted to show that axiom three was independent, construct a model where axiom one is true, axiom two is true, axiom three is false, and axiom four is true. A similar statement can be done to show that axiom four is independent. But like I said, I'll leave it up as an exercise to the viewer here. To finish up lecture two, I wanna throw a little bit more vocabulary about axiomatic systems. We say that an axiomatic system is inconsistent if a statement and its negation are both theorems of the axioms. We say that an axiomatic system is consistent if it's not inconsistent. So I should mention that in an inconsistent system, all statements are actually both true and false. So we say the system's inconsistent if a statement and its negation is both true and false, right? They're both theorems. So that is a statement is both true and false simultaneously if it's inconsistent. But if one statement is both true and false, so p and not p are theorems, then in fact every statement is both true and false. And it basically follows by a proof by contradiction, right? So it's like take statement a, all right? The proof, the proof is assume not a. Well, then you say p, then you say not p, you get a contradiction. So then by proof by contradiction, you get, oh, a is a statement. So you just proved a, but a was arbitrary. We could have done this exact same proof and prove not a, right? Well, to do that, you would assume not not a, which is just to say you just assume a, then you say, then you do p, then you do not p, you get a contradiction, then you conclude not a. And so now we've proven that a and not a are both theorems of the axioms. So in an inconsistent system, every statement is true and every statement is false, which makes the axiomatics there completely useless. So for this reason, it's important, it's, we don't use inconsistent systems on purpose. The problem is we don't always know if our system is inconsistent. For this reason, it's helpful to determine whether a system's inconsistent or not. Because if your system is inconsistent, everything you prove about it is pointless. It's pointless, you don't care about it. We only care about consistent axiomatic systems. How do you know if a system is consistent? Well, if a system has a model, if there is a model of the theory, then that proves that the system, the axiomatic system is consistent. An inconsistent system can't have a model. Consistent ones have models. So if you can provide me a model, then you know the system's consistent. But this can get a little bit problematic because the model might depend on other logical systems. Like in mathematics, we often use, we often interpret models via set theory, ZFC set theory to be precise. So the consistency is then contingent upon the language for which the model was constructed. So if our model depends on set theory, then our system is only consistent if set theory is consistent. And this is a question I'm not gonna delve into into these lecture series, whether ZFC set theory is consistent or not. Again, that's a conversation for another one. So oftentimes we have conditional consistency, but if we produce a model, then we know the system is consistent and not inconsistent. So models are very important for that reason alone. But then another important thing to talk about is completeness. We said that an axiomatic system is complete if every statement or its negation is a theorem of the axioms or not. So you either get A or not A. So we don't want both A and not A, but we would love it if you get A or not A in an exclusive sense, right? Every statement is either true or false. The problem is when your axiomatics get too complicated, then we get incomplete statements. That is to say that you can get new statements that are then independent of the previous ones. So in a nutshell, we don't expect models to be complete. I mean, like the FIFO theory we've developed, so far actually is a complete model. Those four axioms we have about FIFOs determine all of the, those axioms determine every model. That is to say in a complete system, in a complete system, uptie some morphism, there is only one model. And for FIFO theory, there's only one model. You have the fees, which are X, Y, and Z, and you have the foes, which are gonna be X, Y. You're gonna get X, Z, and you're gonna get Y, Z. And then belonging is just set containment. Uptie some morphism, every model is this model right here. It's a complete theory. But most theories and mathematics are incomplete. That is to say there are gonna be some statements which are not true or false. That's because you could take on that, you could take on that statement or its negation as a new axiom and thus get a new theory. Like take group theory, for example. Group theory is incomplete. Group theory, given a binding operation, you assume the operation is associative, you assume the operation has an identity, you assume the operation has an inverse. Those are the three axioms of group theory. It's incomplete because we can't determine is the operation commutative or not because there exist groups, which are the model of group theory, which are non-commutative, and there exist groups which are commutative. And so group theory is incomplete because I can't say everything, I can't say anything about whether it's commutative or not. And that's not necessarily a problem. It's like, oh, okay, I'll just add a new axiom. I'll have a theorem just for commutative groups that we call abelian groups, or I'll have a different theorem for non-abelian groups maybe. And so this incompleteness is not such a big deal as some people sometimes fret about, right? Like the Gödel incompleteness theorem says that any axiomatic system that can perform basic arithmetic, basically you can embed the natural numbers into, is incomplete. And so the axioms of geometry, such as Euclidean geometry, will actually be incomplete because you can do arithmetic with Euclidean geometry. And so this incompleteness statement means that there will be some theorems we can't prove because they're not actually theorems, right? They're independent of the axioms. Again, we're not gonna worry so much about that because honestly these incompleteness theorem, these incompleteness proofs usually involve like silly self-referential statements that really are only studied by logicians. But like if you take the group theory example like is the operation by the commutative or not, that's a relevant statement. Not all groups are commutative, but we actually accept that because we know, okay, this statement's independent and we can then refine our axioms by adding new ones if we need to. So I do want you to be aware of the idea of completeness and incompleteness, but I don't want you to fret too much about it. Certainly if you're spending your whole life trying to prove a theorem that's independent of your axiomatic system because it's incomplete, that's sort of a big deal. People ask questions like this with, for example, the let's take the, well, let's take the Riemann hypothesis, for example, it's a pretty big open question still in mathematics. Is it independent of the mathematics we're trying to prove it in, right? Is it both true and false? Not in an inconsistent saying that just with different models. Is there one model of mathematics through which the Riemann hypothesis is satisfied and another one where it's not? That would be very interesting to know, right? And honestly, I would probably consider that a solution to the millennial problem there because if you take the continuum hypothesis from set theory, it has been proven that the continuum hypothesis is independent of ZFC set theory. There exist models of ZFC set theory, which satisfy the continuum hypothesis, but there are others which then negate the Riemann, the continuum hypothesis. So then we can develop set theory using one or the other. The question that kind of comes down to that one is that as mathematicians, what does our set theory do? Does it satisfy continuum hypothesis or not? And this is a question that set theorists will continue to debate into the future.