Session: Lextensivity Rig Geometry / Internal Categories / Quantales (& others) — Part 4

0:00 We can prove that there is no epimorphism from an object to a power set. We can prove that there is no monomorphism from a power set to an object. And we can prove that there is anaphylmonomorphism from an object to its power set. That's just a single thing. However, what we can prove is that there is a network from the power set to A. So, in fact, can the theory pull them out such that an object is scripted back to A, but only holds one object? So, for this reason, that's the problem with quantum mechanics. So now that we've decided what the cardinality is, we're going to decide what the continuum of hypothesis is. Included in the continuum of hypothesis is the statement that there are no cardinalities between the natural numbers and the continuum. The continuum here are really vague, because it's not clear what we should fill in or are there. But even before we fill in what are there, there's multiple ways of generalizing this, there are no cardinalities. One way of saying it is that there does not exist some alpha for which n is less than alpha and alpha is less than r, and the other one is saying that for all alpha, alpha is less than n, or alpha is greater than n, or r. These two things are not necessarily equivalent. They're both going to be really neat to have, right, which, again, without the excellent choice in fact, you don't have this type of argument. However, what I'm going to use for the Continuum Hypothesis is the second one because it's stronger, and what I'm going to use for the negation of the Continuum Hypothesis, in fact, the first one without the negation sign, which is strong, like, basically, we can prove that this thing is consistent, we can prove that you can negate the Continuum Hypothesis in a very strong way, and that's the reason that they're stating at the bottom the existence of... Something which falls between the natural numbers and this continuum in primality is the statement I'm going to use for the negation of the continuum hypothesis, and that's what I'm going to prove, so I'll take the consistency. Okay, so there's also different things we could choose for the continuum. We want the continuum to be an object of the torus of the question. We could choose to do a different thing. We could just take the power set of the natural numbers. Oh sorry, I should have said this earlier, but all my total C's have natural numbers objects. I'm not going to consider the totals of my constants. That would just be silly.

2:30 So yes, there's always a natural numbers object, there's always a power set. We could take that, we could power object that, and we could continue on. We could take the Cauchy real numbers object, we could take the Dedecky real numbers object. These are all not necessarily isomorphic objects. However, actually, another one which I'm going to consider is what I call weak Debyekin rules, which basically is just similar to the Debyekin rules. It's defined as a sub-object of a power set of natural numbers. In fact, it's a power set of irrational numbers, which, of course, are isomorphic to the actual. But instead of using those five axioms to find out, we're going to forget about the fifth axiom. The fifth axiom basically says that it is decidable whether or not the rational number is less than the real number. And so we're sort of going to make... Maybe the numbers are a little bit less determined, but it does make them a little bit bigger. And so we've got, in fact, four. And the point is that all these four things, under the assumption of excluding middle, become isomorphic. So, in the classical case, all these things are isomorphic. And the problem is, because we're trying to generalize away from the classical case, we're trying to generalize away from this assumption of this middle, we've got four different candidates. And they're all nicely lined up in the middle. The costal deals are by some logic of the delicate deals, which lies in some logic of the weak delicate deals, which lies in some logic of the powers of the natural numbers. So far, I can prove that negation of the continuum hypothesis is consistent if you take the continuum to be the powers of the natural, so that's a really easy generalization. It's a little bit more of a stretch to get it to work for weak-editing fields. I'm pretty sure you can get it to work for things as small as quotient fields, but I haven't actually done that yet. The reason you want to work with Swastik, of course, is that there's a lot of hypotheses out there for which the cardinality which lies between the natural numbers and the powers of the natural numbers is, in fact, identical to the real numbers object, or the quotient real numbers object. In a lot of cases, these inequalities here are strict. And in some sense, that doesn't actually make the continuum hypothesis. If you have a topos in which you've got a natural numbers object, nothing between it and the real numbers object, and then nothing between the real numbers object and the powers of the natural numbers object, And if you want to say that you think the hypothesis is true, they're not false. So that's not the type of counter-example to the continuum hypothesis that we want to discuss.

5:00 Okay, so now I'm going to talk about how these sort of forces of existence in groups work. And how, so this basic structure here is the same structure that is in Cohen's group and in Revere's group. First, you take some set theory, either ZFC or movie toposys in place, or elementary toposys, or algebraic set theory. Either of those can substitute for set theory there. You take a model of it. You build some sort of large set or object in this model. Now, by largeness I mean that you have a string of strict polynomial qualities between the natural numbers object and it. You use this large object to build a new model. In the new model, you can in fact associate all the things in the old model to things in the new model, so for every A in the old model, you have an A check in the new model, and that this association preserves and reflects cardinal inequalities. So, A check plus 3 to the B in the new model. A plus two for B. And of course, in the case that I'm going to define my categories, this check's going to be a punter, but of course, in the case that I've seen it, it's going to be a punter. Now, then we want to prove that in the new model, we have L check less than or equal to, that R should actually be not the equal, let's see, whatever the continuum is. In the new model, we will not have a chain of strict carbon quality below this continuum, so we will have negated the continuum. So, how do we make this work? Well, so, in the re-attainment groups, the constructions, the old model and the new model are both of these groups, right? So, the construction of the new model comes from the old model by two steps. First, you start with the total choice, you start with the base of both. You construct a large object L, you can just take a few of them out as that works, you can construct an internal whole set in your blue circles of choice using L, and you can take out total internal pre-sheets. Now in this total internal pre-sheets we will in fact have an object associated with L which will be smaller than the previous one.

7:30 So, in fact, we will have, sort of, in some sense, four stages in the hypothesis work out. However, what we've made up is that it's not yet a diminutive choice. In order to do that, we've got to booleanize it by taking the category of W-ation sheets. And so, like, basically, this group motion from a boolean choice to this topos of W-ation sheets on... Anyways, there's a total of internal sheets on it. We'll create this new model and find which one we're going to be using in that analysis. Now, the way physical check construction works in this case, it's a composition of the diagonal pre-sheet function, which takes every object in your base total to an internal pre-sheet. And it's a composition of that with the associated sheet function, which goes from pre-sheet to sheets. And it turns out that, again, in the case of boolean totals to the choice, this composition will preserve and reflect Carnouan equality. However, the proof that this works depends upon is there any boolean axiom of choice in your base code rules. So we can't, in fact, use the exact same construction in the case that you use not a boolean totals of choice. However, we can use a various type of modified construction, in fact an easier construction, but get rid of the second step, get rid of the boolean axiom step. The diagonal pre-sheet puncture, in fact, always preserves and reflects cardinal qualities. If you want to think about it this way, if you have some category, and you have a category of punctures into it, if you have two diagonal punctures, there will be monomorphism between two diagonal punctures exactly when there is a monomorphism between the two objects. This is just a general idea. The fact that delta preserves the effects of paramotor equality is just an internalization of this basic result of the category theory. So, in fact... Basically, if we start with their arbitrary troubles, we construct this internal pole sequence in the exact same way that your internal used, take the category of internal coefficients, this thing will negate the continuum hypothesis for the power set of the natural numbers object. This will create some sort of object between the natural numbers object and its power sets. However, in some sense this is not negation of the continuum hypothesis that we want. Again, in other cases, a thing that comes between the natural numbers object and its power set is in fact a real numbers object. So we want to see if we can get something to come between the natural numbers object and at least one of the real numbers objects. So I'm going to start with the week that we did this. In fact, this is one of the ones that I've actually completed so far.

10:00 So, in the first case, the four-step P that we used for forcing was a four-step approximation to a map from this big object L to the power set of the natural numbers object. So if we want to, instead of using the powers of natural numbers object of the continuum, now we use the rededicated reals of the continuum, we create a practical set of approximations to maps, from L to the dedicated reals. So basically what is supposed to be approximations of... So I guess, and sometimes I will look at it as an approximation of the dedicated type as an open integral. Basically, the stuff on this side is an open integral. Sorry. For your sake. The topological integrals, the topological integrals will be definitely in the dedicated class. The stuff above the integrals will be definitely out of it. And the stuff in the middle is indeterminate. And this is just an approximation to a dedicated class. To any of the fields in this interview. So the approximation to a map from L to dedicated fields is going to be a map from L to a pair of rationals such that the first rational is lower than the second rational. And the ordering on this post set is actually the main inclusion ordering on both initials, which is given by that one round plot. Again, this is all done in the internal language of the proposed, so this post set P is not a set theoretic post set, it is an internal post set in this basic approach that we can start with. Now, this post set P will allow us to construct a new category of internal predictions, the e-Lithiol. And e-Lithiol understood conditions of real and fact. We will have an L check in there, which will be a diagonal 3-sheet on L. We will have a chain of strict cardinal points below it. However, we also want to make sure that the L check will be below the weak-dedicated real number subject. There's a canonical construction of a map between the L check and the weak-dedicated real number subject. However, under certain conditions, this map here will not in fact be modeled. In order for it to be modeled, we need that there's enough elements, there's enough maps from L to the natural numbers to distinguish the elements of that. This has to do with, basically, this makes sure that the post is going to be constructed as data to create these things in the model.

12:30 So there's this criteria that I call natural distinguishability. Basically it means if every function from L can match a number that agrees on two elements of L, then L and M are identical. Again, we formulate this in a general language. And so the problem is if L is not natural distinguishable, she may not be modeled. So we basically need to be able to construct a large natural distinguishable object. Constructively, power sets will not work in this case. Power set natural number of subjects is not necessarily natural. However, anything in the form of n to the x will be natural. So we might work on constructing a chain of iterated powers n, n to the n, n to the n, n to the n, etc. However, this iterative power theory will only construct a chain of strict cardinal inequalities, well, not only if the base totals are skewed, but we need basically another assumption, and we might as well see them skewed. And the reason we can, without a lot of generality, assume that the base totals are skewed is, we're going to normally, to base totals, we can construct the totals that have double variation sheets lined, which will be skewed. And that would still be a troplos, and we haven't. It will still be non-generative. The first one was non-generative. So there's no reason not to just luminize before we do anything. So basically the point is we start with the topos. We take this booleanization of double negation sheets. Then, on the other side of the double negation sheet, we can construct those big L booleanization rules. So, if we take the total of the total pre-sheets on this, then that will, in fact, negate the continuous hypothesis for the weak derivative materials. Basically, this is, I mean again, I haven't done this for anything smaller than we've done it for a few years yet. I think it can be done for a few years and a few years. I don't see any conceptual obstacles to do it yet, but I just... I haven't had time, but I have to, you know, it's really late. So that's as far as I'll get for generalizing to the HMS certificates. The next thing I want to do is I want to generalize this entire result to this thing we call algebraic set theory. Now algebraic set theory is relatively recent. I think it was first formulated in the 90s. And basically, what it is, is it's, as Tropos theory is in some sense an axiomatic generalization of the category of the sense, algebraic set theory is in some sense an axiomatic generalization of the category of classes.

15:00 So category classes being sets of proper classes all put together into one category with class functions between them, rather than set functions. So in some sense, a category of classes will have large objects and small objects. Large objects, of course, are part of the proper classes, and the small objects are the sets. And so in order to distinguish between the small and large objects, a category of classes is not just a category, but it's a category with distinguished... Class of small maps. So some of the maps in the category are called small, and they actually have, like, the class of small maps is the size of a bunch of axioms, and those correspond to certain axioms that the class of all maps with small fibers does, in the category of sets and pieces. Peter Lobsang will talk a bit more about categories and classes later. The important reason that I, I mean, that the important thing that algebraics actually encapsulates, topos theory doesn't, is that in any sort of categorological logic situation, the internal language quantifiers you have quantify over the obvious in the category. So in topos theory, you can't quantify over sets because you don't have an object of all sets. However, in a category of classes you do have an object of all sets, it's sort of the class of all sets, it's there as an object, and so you can impact quanti over sets. This allows you to formulate things such as the generalized Newtonian hypothesis in algebraic set theory, in the internal language of algebraic set theory, not using any sort of external quantifiers. The mathematical community has not yet decided upon one axiom of aviation for what a category class is, there's a whole ton of them out there, the one I use is one of the strongest, the reason I use the stronger one is because I want to make sure that, basically that it has all the structure that a typical theory class, that if you take the, sort of, the sets in a category class, if you take all the small objects and look at them as a full-size category, that that thing in fact would be a topos. So basically, in some sense, the axiomatic association that I used is chosen because it corresponds to what was given in this way.

17:30 So, okay. So what do we need to do? Well, in order to do the consistency approach before, in order to construct a new model for normal learning before, we need to get shape and pre-sheet constructions. In fact, given a category of classes, there is an obvious definition of an internal pre-sheet. There's a categorization of the category of internal pre-sheets, and an obvious definition of the category of double negation sheets. So, what we need in fact is that this category of pre-sheets, the internal pre-sheets are still the category of classes, and that category of double negation sheets are still the category of classes. So you need these two theorems, but I think we'll run them two up here. They both actually improve laws of physics. So the free sheet one comes first, in some sense. It was proven first. The second one is just a special case of basically categories of class being closed and taking sheet considerations. The first statement of that result I saw in more like a vendor workbook. Unified, of course, writes that. Sadly, they didn't prove anything in there. They just say the results and said the proofs didn't follow. The proofs have not yet followed, but they will soon. And the actual other thing, um, so the, um, so the first paper that I will be ending on Wednesday morning is one of the few Wednesdays we've been talking about, and that one, in fact, does have proofs in it, which I'm happy about. Um, and it's actually kind of different. Again, because there's different academic positions out there, that results in the fact that there's also four different academic positions that I should be able to talk about a little bit more later. But basically, what I'm doing here is an application of the results. So, in order to create a new moral level of normal, we're in fact going to cheat a little bit, and we're going to use the fact that we've already proved that we can do this construction for cohomology. Because we take a category of classes, and we take this full stop category of small objects, that can be a neutral place, and in fact... It will, in fact, the internal language will be the same. If you have a statement which can be stated in the internal language of the totals, the totals of small objects, it will be true in the totals of small objects exactly when it is true in the whole category of classes. So, in fact, the internal language of the totals of small objects will just be a description of the internal language in the category of classes.

20:00 I mean, again, under the academic condition of music. Music recreates in a condition that doesn't always hold. And so do two, in fact. So if we want to prove that the negation of the continuum hypothesis holds, what we need to do is just prove that the negation of the continuum hypothesis holds in the totals of small objects, and that's enough. So, we start with a monetary category hypothesis. We booleanize it, so we can start the category classes and delegation sheets. The totals of small objects in this E00 In that total control of small objects, we do the L and the P construct of the object L and the P construct of the thing P as we did before. And in our previous theory, if we take the internal pre-sheet from that, what's outside of the creation of the theorem hypothesis, the very same thing is in the theorem which is not that hard to prove, which is that the internal pre-sheet used on the small objects are the small objects in the internal pre-sheet used. So these two constructions can be used. So, in fact, the small objects in this category class of internal pre-sheets satisfy the condition of the human hypothesis, and we're done. So this allows us to piggyback results from Topol's theory into algebraic set theory, provided that we use the axiomatic division in general, which, of course, for some people might be good for the future. And I guess that's it. Any final questions? When you reconstructed your process of finite approximations to a mathematical realism, did you have a reason for using total functions of LQ cross Q, rather than partial functions of the finite domain, which is not what we are getting at this point? So the reason I do that, total functions of LQ cross Q, is because the point is that... Okay, if you want to approximate, so we could do, we could do functions without reveals with phi and domain, and that would do something, I checked that in the book, so what I did was instead of approximating a function with reveals by taking a partial function with phi and domain, I instead approximated it by making, instead of taking a single real and taking an integral, and I think it makes a lot more people.

22:30 He is building a class on topos for the theory of injection and that would be the science of the post-doc of homo-proclination of the epithelium in general. You know, each approximation just needs a finite number to make it, say, this element will map something apart, this element will keep it all below that. Well, sometimes also a map into... Take into account that you're not giving yourself confidence that you can make it at once. If you first have a problem, then you need to know you have lots of maps to help you with the mapping. I'll see, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, I'll, Not much, actually. The reason, I mean, the reason is because I, we don't really know, so the point is, of course, we know the variation of the generalization between the hypothesis is consistent because the variation between the hypothesis is consistent. As far as I know, I haven't seen any, like, basically, to prove the generalization between hypothesis is consistent, you have to do some sort of constructible sets type construction. I haven't seen one, and it doesn't mean it doesn't exist, but I haven't seen anything like something that I could, should be more than obvious, but it's the other way around. That's if you're going to do one direction, you shouldn't do the other direction.