The Continuum Hypothesis in Topos Theory and Algebraic Set Theory (contd.)

0:00 We conclude that there is no more than a few more of these. We can prove that there is an object through a power set. We can prove that there is no one from a power set to an object. And we can prove that there is half of one from an object to its concept. That's just a simple thing. However, what we can't prove is that there is one from a power set to it. So in fact, can't just pull them out and say that an object is a script and that's it. It only holds one object. So for this reason, that's the part I wanted to talk about. So now that we've decided what the cardinality is, we're going to decide what the continuum happens. Included in the continuum hypothesis is the statement that there are no cardinalities between the natural numbers and the continuum. If you feel like you are reading Bayes, it's not clear what you should fill in for R there, what correct generalization happens. But even before we fill in what R is, there's multiple ways of generalizing this, there are no current analogies. 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 the future N, or alpha is bigger than the future Y. These two things are not necessarily as different, but one of them is that we don't have any track body, right? Which, again, without the maximum choice, in fact, we don't have the track body. So, however, what I'm going to use for the continual hypothesis is the second one because it's stronger. And what I'm going to use for the negation of the continuum hypothesis is, in fact, the first one without the negation sign, which is the strong link. Basically, if 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 the statement at the bottom, the existence of something which falls between the natural numbers and this continuum in primality, is the statement we'll use for the negation of the continuum hypothesis. And that's what I'm going to include in the spelling technique system. Okay, so there's also different things we could choose for the continuum. We want the continuum to be an object of the troublesome 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 topocies have natural numbers objects. I'm not going to consider the totals of my axons. 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, the power object, that community. 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 dedicated fields, which basically is just similar to the dedicated fields, it's confined as a sub-object of a power set of natural numbers. In fact, it's a power set of irrational numbers, which of course is isomorphic to natural. But instead of using those five axes to find out, we're going to forget about the fifth axis. The fifth axis basically says that it is decidable whether or not a rational number is less than a 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 classic 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 co-studials are by some logic of the dedicate fields, which lies in some logic of the power set of the natural numbers. So far, I can prove that negation of the Hitchcockian hypothesis is consistent if you take the continuum to be the powers of the financials. That's a really easy generalization of the theory of Hitchcockianism. It's a little bit more of a stretch to get it to work for weak-dedicated 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 on the smallest thing, of course, is that there's a lot of toposies out there for which the cardinality which lies between the natural numbers and the powers of the natural numbers is in fact the definition of the real numbers object, or the quotient of the 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-reference that you want to discover.

5:00 Okay, so now I'm going to talk about how these sort of forces-out interstices and groups work. So this basic structure here is the same structure that is in Cohen's group and in Ruggieri'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 has to be familiar. A plus two could be. 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 you want to prove that in the new model, you have L check less than or equal to, that R should actually be not the equal, let's see, whatever the continuum is, like just a plain R. 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-entering groups, the constructions, the old model and the new model are all focused, right? So, the construction of the new model comes from the old model by two steps. You construct a large object L, you can just take a few of them out as that works, you can construct an internal coset in your blue tables of choice using L, and you can take out tables of internal pre-sheets. Now in this table of 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 ran out of is not yet a domain tool for choice. In order to do that, we've got to booleanize it by taking the category of delegation sheets. And so, like, basically, this group motion from a group of choices to this purpose of delegation sheets on... Anyway, so there's a total of internal sheets on it. We'll create this new model and find which one model can be used in that class. Now, the way that this whole check construction works in this case, it's a composition of the diagonal pre-sheet function, which takes every object in your base to oppose to an internal pre-sheet. And it's a composition that we associate with the 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 the boolean maximum choice in your base totals. So we can't, in fact, use the exact same construction in the case that you use not a boolean totals to the 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 and matrix 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 a 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 is just the internalization of this basic 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 a number of 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 the only one that I've actually completed so far.

10:00 So, in the first case, the four-step B 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, we now 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, in some sense, I will look at it as an approximation of the dedicated real as an only integral. Basically, the stuff on this side will be integral. For your own sake. The topological integral, the topological integral will be definitely in the Dedekind class. Some of the dopamines will be definitely out of it. And the stuff in the middle is indeterminate. And this is just an approximation to the Dedekind class. The Dedekind and the other videos in this interview. So the approximation to a map from L to Dedekind is going to be a map from L to pairs of rationals such that the first rational is lower than the second rational. And the ordering on this post-set is actually an inclusion ordering on open initials, which is given by that formula on the top. Again, this is solved by the internal language of the topos, so this post-set P is not a set periodic post-set, it is an internal post-set being this basic one, which we can stop here. Now, this post-set P will allow us to construct a new category of internal finitions in the EDPL. And the EDPL understood conditions of real and fact. We will have an L check in there, which will be the diagonal pre-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, and we get a congenial number subject. We construct, there's a canonical construction of a map between the L check and the congenial 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 maps from L to the natural numbers to distinguish the elements of L. This has to do with, basically, this makes sure that the post is going to be constructed as big enough 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 I 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. Constructivity and power sets will not work in this case. Power set and natural memory on the subject is not necessarily match for the Speakers rule. However, anything in the form of n to the x will get matched to the Speakers rule. 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 based on this theory, but we need basically another assumption, and we might as well see this. And the reason we can, without a lot of generality, assume that the based on this theory is, we're going to normally, to base topos, we can construct the topos that have double variation sheaths on it. And that would still be a topos, and we haven't. It'll still be non-generative. The first one was non-generative. So there's no reason not to just luminize before we do it. 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 booleanization sheet, we can construct those big L booleanization rules. So, if you take the totals of the total pre-sheets on this, then that will, in fact, negate the continuum hypothesis for the weak-dedicated fields. 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 haven't had time and I have to do it. It's really late. So that's as far as I'll get for generalizing two VH-mesh 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. It's called algebraic set theory. And basically what it is, is it's, as Turco's theory is in some sense an axiomatic generalization of the category of the sets, algebraic set theory is in some sense an axiomatic generalization of the category of classes, so category of classes being sets of thought and class all put together into one category.

15:00 So in some sense, a category of classes will have large objects and small objects, and large objects will correspond with proper classes, and small objects with sets. And so in order to distinguish between small and large objects, a category of classes is not just a category, but it's a category with distinguished characteristics. Class of small maps. So some of the maps in the catalog 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 that represents. Peter Lobsang will talk a bit more about categories and classes later. The important thing that algebraics actually encapsulates, topos theory doesn't, is that in any sort of categorical logic situation, the internal language quantifiers you have quantify over the obvious in the category. So the topos theory 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 theory has not yet decided upon one axiomatic addition 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 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 situation that I used is chosen because of the correspondence that 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 from the old one before, we need to get the shape and pre-sheet constructions. In fact, given a category of classes, there is an obvious definition of an integral pre-sheet. The dogma definition of a category can be categorized into internal pre-sheets, and an obvious definition of a 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 theories, but I think there are one or two out there. They both actually prove the opposite of this. The first one, 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 Vandervoort paper, unified of course by a secretary. Sadly, they didn't prove anything in there. They just stated the result and said the proofs didn't follow. The proofs have not yet followed, but they will soon. And the actual other thing, so the first paper that I will be ending on, is one of the things that I was talking about, and that one, in fact, does have proofs in it, which I'm happy about. And it's actually kind of different. Again, because there's different academic positions out there, that was also, more like that, it was also for four different academic positions, and I should be able to talk about that a little bit more later. But basically, what I'm doing here is an application of their results. So, in order to create a new moral level of one, we're in fact going to cheat a little bit. I'm going to use the fact that we've already proved that we can use this construction for a couple of instances. Because you take a category class, and you take a school-style category of small objects, that can be a neutral case. And in fact, it will in fact, the internal language will be the same, basically. If you have a statement which can be stated in the internal language of a topos, of a topos of small objects, it will be true, given the topos of small objects, exactly when it is true in the whole category of classes. So, in that, yeah, in terms of how I have to build a small object, it would just be a restriction of internal language in a category of classes.

20:00 I mean, again, under the academic condition of music. You can use a meter out of this in a condition that doesn't always hold. And so, the two, in fact, so if you want to prove that an education that you can do in my classes holds, what we need to do is just prove that an education that I can do in my classes holds in the totals of small objects, and that's enough. So, we start with an arbitrary category of classes. We rule the answers. We construct the category classes of delegation sheets. If the totals of small objects in this e naught naught, in that total of small objects, we do the element p, we construct the object element, we construct the thing p as we did before. And in our previous theorem, if we take the internal equation from that, we'll satisfy the deviation between that hypothesis. The fifth statement is an argument which is not that hard to prove, which is that internal pre-sheets on the small objects are the small objects in the internal pre-sheets. So these two constructions communicate. So, in fact, the small objects in this categorical class of internal pre-sheets satisfy the equation of the Hilbert hypothesis and we're done. So this allows us to piggyback results from topos theory into algebraic step theory, provided that we use the axiomatic division in general class, which of course, for some people might be good for the future. And I guess that's it from the time of questions. When you constructed your co-set of finite approximations to a mathematical realist, did you have a reason for using total functions of LQ cross Q rather than part of the functions of the finite domain, which is the model we are getting? So the reason I do that, total functions of LQ cross Q, is because the point is that... Okay, we want to approximate, so we could do, we could do functions from L to the reals with phi and domain, and that would do something, in fact that would be important, so what I did was instead of approximating a function into the reals by taking a partial function with phi and domain, I instead approximated it by making, instead of taking it as a single real, I'm going to take it as an integral, and I think it's what we all keep doing.

22:30 Each approximation does the refinement of its nature, say, this L and L and this L and this L and this L and this L and this L and this L and this L and this L and this 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, Not much, actually. The reason, I mean, the reason is because we don't really know, so the point is, of course, we know the negation of the generalization between the hypothesis consistent because the negation between the hypothesis is consistent. As far as I know, I haven't seen any, like, basically, to prove the generalized consistency of the hypothesis is consistent, you have to do some sort of constructible-status-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 going along with this. That's if you're going to do one direction, you shouldn't get a direction check.