Hi, everybody, welcome. It’s another broadcast from calling power into infinity talking about today entry, the continuum hypothesis. So we’re gonna leave the fourth dimensional mathematics for a little bit and just do a small series on our solution to the continuum hypothesis. A really so that you can understand sort of a little bit more about this number structure and how numbers are actually working. You know what this number is what this pie is. And we can only work those things out once we know about the dent what we call the density of infinity infinity. So just for those who don’t know what infinite densities are, really, you can say it’s it’s looking at, let’s say we have a space, and there’s an infinite number of numbers in that space. And we can say let’s remove a say we take all the the whole numbers for example. And if we remove all of the even numbers, that’s half of the infinite set, isn’t it? Or you think it’s actually not but you know, it appears that way. So, so what you could say is there’s that the you know, how dense then is the all the odd numbers compared to the even and actually when we do in analysis, we say that you know, actually infinity isn’t even number because odd. Zero is a it’s a it’s an even number. And when we start to provide a reiterative square root equation, within reciprocal space, we see that the number reduces towards zero which is it won’t ever cross the zero line. And when it’s when we do that, in whole number space expands into infinity. So in the fourth dimensional view, the reciprocal space is folded into the is awful folds out, shall we say more? Accurately to become the whole number spaces actually no difference between the two spaces. And number one acts like a kind of hinge if you like, and that’s why we don’t have this mathematical squaring on the number one, you know, because it is a boundary condition for that, that that function. So and there’s reasons for that. So with that boundary of number one in place, you know, the infinite number set then manifests itself? Yeah. And so what we’re saying is, is that we’ve, we’ve understood that when we take the we take reciprocal space as a line between zero and one and when we start to divide that line, then we begin to uncover another type of infinite density. And that infinite density is called Aleph 0.5. And so Aleph 0.5 then is really a combination of when we divide that line and we’re just going to count that’s one half and the second half, so we divide the lines duplicants. There’s one half and that’s two half the second half. Yeah, we’re gonna add that up together. So we’ll have a half plus one for example. And then we’re gonna go and do the same for the other lines. And what we find is that we get a density that as we progress, see through the numbers, one ordinals 1234567, counting down, what we find is the total of that line is only progressing by naught point five. So that’s, that’s one of the infinite densities you know, you can look at other infinite densities with the square root function. You know, when when you look at normal numbers that are just progressing 12345678 and so on, we have a density function, whereas the square numbers will increase exponentially as the numbers go through. Because they’re square, that’s another dimension, isn’t it? And and so you could say the square number sequence has less than less infinite density than the whole numbers. Yeah. But when we talk about square roots, what we find is that the square roots progress, exactly double density as far as the numbers concerned that we write at least anywhere here. So let’s say for example, let’s say we take the numbers 1234. And then we take the number square root of two, and then it will jump to the square root of eight after that, so the square root of eight sits between the square root of four. And so when you move up to you know, let’s say the number 16, we get the square root of 32 that sits at that diagonal. So the the relationship of a number to its diagonal is also that he has a number when we wrote the thing but as the actual number of regresses is also progressing at twice the rate as the as the sequential number so we have another infinite density on that side on the square root function, which is once again it’s Aleph 0.5.
And so when we come in and compare all of that, with square numbers, square numbers are all exponentially growing, but the root numbers because of that, square root function is like levelling, living levelling out each square, we find that they just doubling off the square but there’s no you know, compared to the square, so when the root is compared to the square we have a double density infinity as well from the diagonal. So once we know all of that, then we can start to understand how spaces constructed you know from 90 degree 45 degree angles, because that’s what the square root of two is, isn’t it? And all these square roots that are of a certain number, or the square of a line going through the diagonal of a, of a square that says, you know, a square of one is where it starts with the square root of two. When we get to the square root of four, we then we have to have the square root of eight, you see, so we’ve made that big jump because that’s four times two and and so on, and so we can know exactly what the side the side length of a of a square is. Yeah, and it’s diagnosed if you take this you know, big bite. For defining the square root of a square, we can find the side of that square. And when we find want to find the diagonal of that square will use a different kind of square root which is the square root of the diagonal Yeah, and when you compare that to the side length of the square it’s it’s moving Aleph 0.5 so so that’s that’s a little bit about square numbers and that sort of thing and Aleph 0.5 and so that’s where we wanted to start this conversation about the continuum hypothesis. Because what the continuum hypothesis is really saying is that you know, is there a density between the whole numbers which is 123456789 on into infinity, and the reels as they’re defined, which is, you know, whether all of the numbers that could possibly exist, all of the fractions and all of that, and the main problem we have here is about number definition. There’s a few problems. So when we come to the fourth dimension, we see numbers in a completely different definition. here because it’s all change. But that was Aleph 0.5, four today. And so you can try this out at home, you can just, you know, draw a line. And then you can draw another line underneath and divide into two and you can count one half and then you put two, one over two on the first one and two over two and then you get the result, which is 1.5. And that’d be on your first step. So, you’ve got you’ve gone from one at that point, which is the start one one, and then you’ve made your first step. So it’d be number two on that side, and the other one will only increased by one naught point five to one so so then, actually, as you go down to the next step, the other side will increase another nought point five as you add it all up, you know, so you go one or you divide the next line into three, and it’s one over three, two over three, three over three, which is counted like that. And then and then that that equals the infinite density of two, when you add it all up. Meanwhile, we’ve progressed a whole unit on the other side, which is the count as we go down, become on the lines. So there we are, what we’ve got there is we have an ordinal number on the left what we call it yet, and a an ordinal ordinal number on going through the right so just sort of clarifying some of that stuff, because it’s nice to have a little bit more clarity on how we’re doing, we’re counting down we’re counting across and by counting across and counting down. We’re creating these this this, this infinite density, that’s how it’s working. That’s why it’s working is because in our in our system, ordinal numbers are different from Cardinals Cardinals of thing, and ordinals are US counting the thing, and ordinal numbers don’t have a number zero, they only have a number one whereas Cardinals have a zero and a minus have actually have a plus or minus zero. But it looks like zero from the centre of the card not because zero looks like zero, doesn’t it? Yeah. And so, but zero is at the centre of those cardinals reduced, there is another type of zero, which is zero. Neutral, which is outside all the number sets as it acts as a boundary. Then the similar the number one acts as a boundary as well. So although the number one is the start of the one ordinals at the same time the Cardinals themselves start at zero, and it’s that space between them zero and one, which creates all of those reciprocal numbers. And that is kind of why the Russell paradox appears that is because it says you know, once you how can it be we can have an infinite number of numbers between zero and one. And we can move to number two and having the same infinite amount of numbers, but we can also find the reciprocals like one over XYZ See, so And the answer to that is is that everything else is is you’re making calculations in the whole number space once you’ve crossed over the one and the one does have double density and printed infinity which accounts for the Russell paradox. So there we are. lf point naught naught point five resolves the Russell paradox, and it’s a simple process of folding. And you can find out more about that on our website. When you can see that there’s many ways we can fold a piece of or even just draw on a on a square. How and show you how all that works. It’s a very simple solution anyone can understand. So we have a solution to the continuum hypothesis in the fourth dimension, which was hyper cubic solution. And what we’ve done is we’ve we’ve done downgraded that, so that we you can see in the in the lower dimensions, such a second dimension in mathematics, which is the square and in doing so, when we were just showing you how the line runs and so that we can show you how that paradox can be overcome. There is a far more interesting solution in the fourth dimension. But we have to get rid of the number I in all that business in order to observe it because we have a an extra set of numbers that aren’t actually in the current mathematic at the moment that currently mathematics only employs
1/4 of our number set. Maybe you could say in fourth dimension if we flattened it into two dimensional space, it could be said that we we’ve got only half of the numbers and that’s really the truth you know, which is another reason why to the eyes of the pie only makes half a circle you know doesn’t make a whole circle UTI two pie made with minus one. And that you know, that that’s you know, that’s why because we’re kind of missing half the numbers. But it’s harder, obviously, if you’re adjusted to the number I to sort of think about how that might be but as it just sort of idea, you know, the number line itself needs to turn into a cross first into a number plane before it can be elevated into the third dimension. And so that that isn’t actually number is not as the axis it’s an X axis. And there’s nothing special about it in that sense, and it creates interesting patterns because if you keep going around the number one all the time yet you’re going to go around in circles and that’s why you have this thing called the unit circle in in the complex plane. But actually, that’s kind of like, you could say it’s like an aeroplane spinning propeller, you know, and that’s, you know, and actually only one propeller at that. So, yes, while you’re going around in circles, you know, from the fourth dimension, you can see what the square root of minus one is. And actually we have, you know, the square root of minus one into two places. We’ve got two of those, and we’ve got two square root plus ones as well. And so those are just the sum of the square root numbers that are missing. And then most of the negative square root numbers are missing from the numbers at the moment, you’d have negative squaring at the moment and standard mathematics. The concept is zero squared zero divide is not there. So you can see actually, once we, once we start, you know, all these axioms of mathematics, they exist in fourth dimension. And that’s based on the idea that where does mathematics come from? The division of the infinite has to be divide first can’t be any other mathematical function, if you think about it, and so because it’s based actually on the idea of where does mathematics come from, it actually is founded in logic, whereas the current mathematical system that we’re using is kind of like being pieced together from just sort of like trying to explore our environment and, and saying, Oh, look, here’s a stone, you know, put that stone there’s two stones, and that’s the kind of mentality that we’ve been trying to solve the universe with. And it doesn’t work and it doesn’t work because the atoms are fourth dimensional structure. But once we get into fourth dimension, we can collapse on the Schrodinger equations, do all that stuff, and quantum gravity just becomes a piece of cake basically. So all of that stuff you know, is the sorts of things that you know, sounds pretty incredible, pretty amazing. And it is, and that’s why fourth dimensional mathematics, we think, is a revolution to planet earth or mathematical structures. And we’re happy to be sort of like spreading spreading the knowledge around fourth dimensional maths, and I’m happy to sort of like be building the community around that. So if you find this interesting stuff interesting. And we can sort of look more into the continuum hypothesis. Yep. And how we’ve resolved the numbers. And then we come up with the conclusion that actually we don’t need the number. The number is not correct. Interpretation of numerical space. And that’s why we can’t we can’t solve the continuum hypothesis. Whereas actually in fourth dimension, it just is very easy to solve. So as easy as folding a piece of paper, so look, there’s something on infinite densities there. And it’s important to know about those densities, particularly when it comes to rotational squaring, and the number three, and how triangles rotate and blah, blah, blah, unless while some of our algorithms for the primes as well also involve the square root three square root five and all the other stuff, you know, because we understand the infinite densities of number of three rotation. So that’s it for today. Thank you very much for tuning in. We’ll have a lot more on all sorts of numbers, stuff and all that business on our website, so you can go and check them. And if you want to stay tuned, we’ll be having lots more podcasts, about the nature of infinite maths and how it all works. Thank you very much for tuning in today and have a very great day. Wherever you are. Bye bye.