This boils down to a variation of the Hilbert’s Hotel paradox, which we talked about way back in our very first post. But let’s rule out stretching — we know that stretching things mathematically can create points. so 1/oo= an infinity … Well, we’ve actually missed almost all the points of the ball! new posts every sometimes So, of course, that point should be in the set we will rotate. That would explain some of it. But there is also another thing you can do. Each side becomes four sides, of one third the length. Therefore and again by definition, the last number that was documented must equal ∞-1 So, there you have it. <– Previous Post: The most controversial axiom of all time How can we fill back in that hole? Yay! A choice only makes sense if you can tell me what you picked, or, at least, a way to make a unique choice. Shishikura managed to prove in 1998 that the boundary of the Mandelbrot set is two-dimensional. So, let N actually represent all of the points below the north pole through the inside of the ball, all the way down to the core. Either way is fine for math. Recall that an irrational number can be thought of as a infinite decimal, that neither repeats nor ends. The step that is the most questionable is the one where we choose points as new north poles. The axiom of choice says that, for any collection of (nonempty) sets, you can choose one thing out of each set. That’s because there are two directions for doubling to affect, so doubling the lengths makes the square times as big. One traditional angle is , but there are infinitely many angles that would work.7 If we pick that angle, each word, or series of rotations, will rotate a new point to the north pole. For instance, the Serpinski Carpet, which is made by taking a square and repeatedly cutting out the middle ninth, increases in size by a factor of 8 whenever the lengths are multiplied by 3, and so has a dimension of , so . For instance, nothing is more obvious than if you have two bags of rice, one has more grains of rice than the other, or maybe the same amount of rice. There are a lot of other theorems that are equivalent to the axiom of choice. Thus, we’ll try to associate each point on the graph, i.e., a word, with points on the sphere that we find via those rotations. But a good intuitive idea is that a fractal is a shape that looks roughly the same, no matter how much you zoom in. In case you’ve forgotten, radians are just another unit for measuring angles, like Fahrenheit and Celsius are different units for measuring temperature. What is the conflict of the story of sinigang? Unfortunately, we’re still missing most of the points in the ball. The obvious thing to do is just infinitesimally stretch the circle to fill in that single-point gap. Again, the axiom of choice simply says you can make a choice, not what that choice is. More than a length, a bit less than an area. Awesome, to be sure, but stressful too. 2 0. zenock. We can associate the word N (for no rotations) with both this north pole and the original one we picked. Can we do it with just a rotation? But if you have to make an infinite number of choices… Well, it’s easy to say that you should make infinitely many choices, but can you really do it? In response, two editors rejected his paper. So, if we take all those points, and rotate them to the right to undo that last rotation, as with the graph, we get all the points in , , , and , exactly like we did for the branched graph! As before, we call the left-last points, rotated right, . But maybe we can do the opposite. The axiom of choice fails this standard, and so should be avoided. (Though not including the center point at the core itself.) So, I applied to permanent jobs this school year. It turns out that covering years of computer science education on my own is time-consuming. It is another way of saying that When did organ music become associated with baseball? Like the girl in the red dress, you probably picked a rational number, i.e. ∞-1 is Bonkers. It’s been… busy. Impossible to progress any further. Here, again, we get an interesting answer. In other words, either you accept the axiom of choice, or else you can’t always compare sizes. Anyway, if you’re curious how you can be bigger than infinity, go back to the very first post, Infinity plus one. In mathematics, infinity plus one has meaning for the hyperreals, and also as the number ω+1 (omega plus one) in the ordinal numbers and surreal numbers. Then, we can do all the rotations, like before, and associate their words with the new points as well. After all, if all you do is cut up the ball, and move the pieces around (no stretching required!) Thus, tripling lengths doubles size, so , and . You can subscribe by email by going to this link! If we had two sets, say A and B, and each thing in A had a corresponding thing in B, then clearly B is at least as big as A. To figure out how to define dimension, let’s look how size grows when we double lengths. Those of you who came from the recent video by 3Blue1Brown may not have realized, but I haven’t posted recently. Admittedly, we’ve glossed over a few important details if you want this to work out perfectly, but I think they can be hidden in a footnote.4. And so on. Of course, moving that point leaves another gap, so we need to also rotate the point one radian clockwise of that. Then, for each of the new, smaller sides, add a new spike. Apparently one cannot blog without cute pictures from wife. So, let’s get back to the Banach-Tarski paradox. then the volume of stuff from the ball shouldn’t change. For instance, a square is two dimensional, but its boundary is one dimensional. With any of these self similar fractals, you can do a similar trick, without too much problem. (It can’t be three dimensional since it’s already confined to a plane.) That, right there, is the heart of the Banach-Tarski paradox. Fractals can be a bit different. For example, for the rotation L, you would rotate left, and that new north pole and the points under it are now “L.”. It doesn’t say how to pick those digits, or what digits you pick, just that you can pick them, somehow. Not only are the sets we’re cutting the ball into hopelessly complicated and delicate, but they assume that matter is infinitely divisible, which is false. Fortunately, there’s an easy way to fix this. The center point, which represents no rotations, we can label N. Thus, a series of rotations is represented by a word which is represented by a point on this branched graph. The most famous of these, we’ve already talked about, the Banach-Tarski paradox. To make one, start from a simple shape, then repeatedly change it in the same way, on smaller and smaller scales, infinitely many times. Why don't libraries smell like bookstores? After all, which directions can you go on the Cantor set? The thing is, when we make that choice, there’s no reason to pick one point over another. A consistent set of axioms is a set of assumptions that can’t prove contradictions. Basically, a line is one dimensional because you can only go in one (pair of) directions — left and right. The problem is that you can come up with complicated sets A and B where it’s not obvious how to line up things in A with things in B. He’s pretty awesome too. Fortunately, like in the circle rotation example earlier, it’s not too hard.