Even though our words represent points that are evenly spread out all of the ball, so that it would look like we’ve covered everything, we’re actually missing “most” of the points in the ball!8. The second question about the axiom of choice is whether we can prove it true using only the other axioms. I’d planned on being a professor since high school, but a few months ago, I decided I was going to change careers. Still less than two. It certainly seems impossible, though. Infinity is not a number. If we pick just the right set of points to rotate, we can! From a physical point of view, this process is, of course, impossible. blog posts. In general, to add 1 to a number means to take the successor of that number. That, right there, is the heart of the Banach-Tarski paradox. Let’s do another example of a fractal, this one somewhere between a point and a line — the Cantor set. ), I know it’s been a while1 since the last post, but we’re not quite dead yet…. Recall that a circle has 360 degrees, or, equivalently, radians. This is the argument of the constructivists. Shishikura managed to prove in 1998 that the boundary of the Mandelbrot set is two-dimensional. The second ball is similar: take all the points in and , rotate forward to get , then put those two sets together, and you have the second ball! Go ahead, pick one! If you can’t tell me what number you picked, did you pick it? See, to become a professor, after you get a PhD, you usually spend 2 or 3 years at one or two universities as a “postdoc,” which is what I am now. But you can duplicate a ball! Infinity is not a real number, so addition is not defined the same way upon it. How do mathematicians deal with the controversy? The boundary of the Koch snowflake (partly shown above) should probably not be two dimensional since it’s just a line that’s been crinkled infinitely many times. The word that represented no rotations, N, we’ll associate with the “north pole” of the ball. They’re all just as good as any other. In fact, without the axiom of choice, you can show it’s sometimes impossible to compare the size of the two sets. Though, on that front, I personally think it’s foolish to believe in anything other than free will. They’re all very special ones that we can write down using a fancy formula, rather than a completely random choice. And then we need another point to fill in that gap…, The trick is that we picked a special angle. Thus, N represents a little line segment. But what about its boundary? Different definitions are useful for different fields of mathematics, but they all agree for simple shapes. Recall that an irrational number can be thought of as a infinite decimal, that neither repeats nor ends. But the original Cantor set is just two copies of that smaller piece. As before, we call the left-last points, rotated right, . Unfortunately, we’re still missing most of the points in the ball. How will understanding of attitudes and predisposition enhance teaching? If we multiplied each length by a factor of 3, the size of a shape of dimension would have to expand by a factor of . As we talked about in the last post, the key trick is not really about geometry at all. For a square, if we double the lengths, the square quadruples in area. but the proof that you can double a sphere does almost nothing questionable. More than a length, a bit less than an area. But, for a fractal, no matter how far you zoom in, it keeps on repeating itself. The set will just represent all the points associated with the north poles. How is it “obvious” that you can make such a choice? If we take a ball, we can rotate it in different directions, forward (F), backward (B), right (R), and left (L). Ahem. Who is the longest reigning WWE Champion of all time? (Though not including the center point at the core itself.) But another big thing is the complete upheaval of my life plans! This boils down to a variation of the Hilbert’s Hotel paradox, which we talked about way back in our very first post. To make the snowflake, start with a triangle, then, add a spike to each side. And so on. Fortunately, the basic idea is not too hard. The Mandelbrot set itself is also two-dimensional. Fortunately, there’s an easy way to fix this. (Subatomic particles are, after all, a particular size, and it’s hard to cut a quark into pieces…). Likewise, 1 / 0 is not really infinity. All thanks to being irrational. For finite sets, of course, this is not a problem. With any of these self similar fractals, you can do a similar trick, without too much problem. I had a bunch of interviews, but I didn’t end up getting hired as a professor. A set with 42 things in it is bigger than one with 27 things. Constructivists reject anything you cannot explicitly construct. 1) It's widely believed that infinity plus one equals infinity, but it's an oversimplification. First, you can’t “disprove” an axiom, since they’re base assumptions. The first is that you can’t actually get your hands on the object(s) the axiom of choice chose. A consistent set of axioms is a set of assumptions that can’t prove contradictions. Many of the most awesome fractals aren’t exactly self similar, like the Mandelbrot set. And again, and again. It’s a really zig-zagging line, like the Koch snowflake, but it’s not exactly self-repeating, so we can’t do the same tricks we did before. We’ve taken the ball and identified its points with the words, which are points on the branched graph. What details make Lochinvar an attractive and romantic figure? Basically, a line is one dimensional because you can only go in one (pair of) directions — left and right. Let’s call the set the set of all the points and line segments in the ball represented by words that start with L, i.e., where the last rotation was to the left. The key observation from last time was that if we take the “words” starting with L, then undo the last rotation by rotating right, we end up with all the words except the ones on the right!