But what does this mean? In other words,. So a division problem is really a multiplication problem with the unknown number on the other side of the equation. But there is no number that works! Since multiplying any number by zero gives you zero, it follows that zero divided by zero could be literally any number at all. Since every rational number can be thought of as a pair of natural numbers, we can draw up the following infinite table to capture every possible rational number:. We only want each fraction to appear once in the table, and at the moment this table captures each fraction over and over again, infinitely-many times.
So we take care to eliminate all the duplicates by crossing out each entry in the table where the fraction can be simplified all the entries where the numerator and denominator share a common factor are removed.
Having done that, we can now zigzag our way through the table to catch every single fraction, like so:. Then all we have to do then is list every fraction in the order we encountered it during the zigzag:.
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We can check that every natural number has a partner the list of fractions never ends and that every rational number has a partner every rational appears exactly once in the table, after we carefully remove the duplicates. But Cantor had one final trick up his sleeve, which he published in his famous paper. There is one more source of numbers we can draw on. Apart from natural numbers and rational numbers, there is one more set of numbers that most people are familiar with—the set of real numbers GLOSSARY real numbers The set of all numbers that live on the number line.
There are definitely more real numbers, in a mathematically precise way, than there are natural numbers.
Cantor began by assuming that you could create the bijection, listing all real numbers in some order, just like we did for the rational numbers, in order to match them with the natural numbers. To create this number, he takes the first decimal place of our first number on our list and adds 1 to it, and uses the result as the first decimal place of his new number. If the first decimal place of our first number was 9, he changes it to a 0.
He then takes the second decimal place of our second number and adds 1 to it, using the result as the second decimal place of his new number. Use this interactive to pair up some natural numbers with the beginning of real numbers. We do this by taking one digit from each real number in a diagonal pattern and incrementing it by one. Is he right?
This is in contrast to the decimal expansions of rational numbers, which must eventually repeat themselves. Different infinite sets can have different cardinalities, and some are larger than others. It may seem esoteric, but the understanding of infinity—and set theory—is vital to understanding the very foundations of mathematics.
Beyond infinity Expert reviewers. Does every infinite set have the same size? Can they all be matched up exactly? Despite intuitions, we can count and compare the size of infinite sets. Rational thinking The first way we can expand our number system is by introducing all the negative numbers, forming the set of integers GLOSSARY integers The set of all whole numbers, both positive and negative. Which leads to one final point: what about dividing zero by zero? Above: The start of an infinite table that lists every possible fraction in its most simplified form.
Then you cut out the middle third. To figure out the dimension of the Cantor set, look at the left segment after the first excision. If we triple the lengths, then that small segment becomes the original Cantor set. But the original Cantor set is just two copies of that smaller piece. Thus, tripling lengths doubles size, so , and. Not really a line, not really a point. With any of these self similar fractals, you can do a similar trick, without too much problem.
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. The Mandelbrot set is defined by looking at each complex number individually, then repeatedly calculating. For example, if , then we make a sequence , , , etc..
Well, the Mandelbrot set itself is, of course, 2-dimensional, since it has area. But what about its boundary? Just from the previous examples, you probably expect the dimension to be somewhere between 1 and 2, which seems reasonable. This turns out to be true.
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Shishikura managed to prove in that the boundary of the Mandelbrot set is two-dimensional. Normally, the boundary is one dimension lower than the main part of a shape. For instance, a square is two dimensional, but its boundary is one dimensional. Fractals can be a bit different. For instance, we had the Koch snowflake. The inside is two dimensional, but the boundary, as we said earlier, is dimensional.
Still less than two. The Mandelbrot set itself is also two-dimensional. Sorry for taking so long on this post. If you believe Banach and Tarski, you can take a sphere, cut it into a handful of pieces, move them around, and put them back together into two complete spheres of the same size. It turns out that making choices is more controversial than it seems it should be. Go ahead, pick one! Like the girl in the red dress, you probably picked a rational number, i. Recall that an irrational number can be thought of as a infinite decimal, that neither repeats nor ends.
So, to pick an irrational number at random, we could just pick digits randomly, one at a time. Choosing one digit, or even a million, is in theory not very hard. There are digits, you pick one. No problem. The axiom of choice says that, for any collection of nonempty sets, you can choose one thing out of each set. For instance, if we were picking an infinite decimal, like before, our collection of sets would be a bunch of copies of the set of digits 0 to 9, one set of them for each of the infinitely many digits we need to pick. The axiom of choice says that we can pick one digit from each set of digits in order to pick an infinite decimal number.
This is the argument of the constructivists. In their view, everything needs to be explicit. A choice only makes sense if you can tell me what you picked, or, at least, a way to make a unique choice. The axiom of choice fails this standard, and so should be avoided. A bit of black magic, indeed. For finite sets, of course, this is not a problem. A set with 42 things in it is bigger than one with 27 things. 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. It turns out that the axiom of choice is equivalent to saying that you can always compare sizes of sets.
But, can you prove that the axiom of choice is not consistent? If you could show the axiom of choice caused inconsistencies, all the accountants in the world would feel more relieved, since then we could throw out the axiom of choice, along with its impossible consequences, like the Banach-Tarski paradox.
But maybe we can do the opposite. The second question about the axiom of choice is whether we can prove it true using only the other axioms. In other words, do we need to assume the axiom of choice at all, or do we get it for free? Here, again, we get an interesting answer. Either way is fine for math. Originally, mathematicians were resistant to the axiom of choice. One well known story is about Tarski of Banach-Tarski fame. He used the axiom of choice to prove a result about the sizes of infinite sets. In response, two editors rejected his paper.
Their argument? Meanwhile, Lebesgue wrote that using one false statement to prove another is of no interest. Nowadays, however, most mathematicians accept the axiom of choice without too many reservations.
Infinity - RationalWiki
Thanks for sticking with me! Learning as much computer science as I could and searching for a job and moving and so on took a lot of time and mental effort, which lead to not many any? New posts should now continue to come out about once a month. More awesome math! The Banach-Tarski paradox says that you can take a ball, cut it up into a handful of pieces, then rearrange them in order to get two balls identical to the original. It certainly seems impossible, though. After all, if all you do is cut up the ball, and move the pieces around no stretching required!
But you can duplicate a ball! The trick is that rotations can create points, seemingly out of nowhere. How can we fill back in that hole?
The obvious thing to do is just infinitesimally stretch the circle to fill in that single-point gap. Can we do it with just a rotation? So, of course, that point should be in the set we will rotate. Of course, moving that point leaves another gap, so we need to also rotate the point one radian clockwise of that. And then we need another point to fill in that gap….
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The trick is that we picked a special angle. Recall that a circle has degrees, or, equivalently, radians. If we keep picking points one radian clockwise of the original gap, we go around the circle once, then twice, then more, but we will never end up back where we started. That is because is an irrational number! Details in this footnote: 4. The original hole is filled by the point that was 1 radian away. That hole at 2 radians is filled by the point at 3 radians… and so on, so a billion radians later, the hole left by the point a billion radians away is filled in by the point a billion and one radians away.
Here's the simple proof that there must be multiple levels of infinity
All thanks to being irrational. Of course, creating a single point is not so impressive. As we talked about in the last post, the key trick is not really about geometry at all. If we take a ball, we can rotate it in different directions, forward F , backward B , right R , and left L. And, we could do multiple rotations in a row, for example, FRB would be backwards rotation, right rotation, then forward rotation. Thus, a series of rotations is represented by a word which is represented by a point on this branched graph.
Of course, we can do any length of words i. Again, words are series of rotations are points on this graph. Where do all these extra points come from? To do that, we need to associate points in the ball with this graph somehow. Fortunately, the basic idea is not too hard. The points on the graph are supposed to represent words which represent series of rotations of a ball. Grab the ball. The north pole, though, is just a point on the surface, and we want to duplicate the entire ball.
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. Though not including the center point at the core itself. Thus, N represents a little line segment. For every other series of rotations, after you rotate, the word i. To guarantee that, we need to pick the angle we rotate carefully. One traditional angle is , but there are infinitely many angles that would work. And those points are spread out all over the ball — it turns out you can get points spread out evenly all over the ball with an arbitrary number of rotations.
Simply pick one of the points we missed on our first go, and start again with that point as the new north pole.
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