 Hi, I'm Zor. Welcome to Unizor education. I will continue talking about sets, quantities, cardinalities, infinities. This is a second lecture of mini theorems about all these infinities and cardinalities. The previous lecture I just want to make a very quick reminder. We came up with these formulas. This means that the cardinality of the set of natural numbers is less than or equal to the cardinality of any other infinite set. This means that if we add one element to a set of natural numbers or equivalent to it in the cardinality terms, we will still have exactly the same cardinality of the resulting set. So, one element added to a countable set results in still countable set. Now, this is much stronger actually. It means that if we will take a countable number of times, a countable set, we will still get countable. We were using matrix, countable number across and countable number of them. And finally, this is the canter theorem that the number of other cardinality of the set of all subsets of a set, s, is always greater than cardinality of the original set, s. Whatever the s is, finite, infinite, whatever. The cardinality of set of all subsets is always greater than the cardinality of the original set. Okay. Now, I will continue with certain other theorems. And the first one is cardinality of countable union of a set of all real numbers with itself is the same as a cardinality of just one set of real numbers. So, this basically, it is kind of a similar to this. This means that countable number of times countable set gives the same cardinality as original countable set. Now, I'm going to prove something which symbolically looks like this. So, if we take the set of real numbers and repeated countable number of times, we will still not exceed the cardinality of the real numbers. And, well, the fastest in that, I don't want to say it's 100% rigorous, but it can be made rigorous if we really want to. But let's think about this particular explanation. You know that the real numbers are mapped to straight line using the coordinates 0, 1, 2, et cetera, minus 1, et cetera. Every real number will correspond to the point. So, I can say that the cardinality of the real numbers is the same as cardinality of points on the line. On the other hand, I can always say that the cardinality of the points on the line is the same as cardinality of a unit segment from 0 to 1. Now, this is the problem which I considered in one of the previous lectures. We have built actually, purely geometrically, the correspondence between these points and all these points. Just as a reminder what we did from a segment, we went to a semicircle using, let's say, this type of correspondence. So, we have the same number of points on the segment and as a semicircle. Now, a semicircle has exactly the same number of points as the line, and the way to do it is, from the center, you map it this way. So, every point on the line will correspond to some point on the semicircle and, in turn, the point in the segment. So, again, the cardinality of number of points on the line is exactly the same as cardinality of number of points in the segment, which is exactly the same as cardinality of real numbers, right? But how many segments are in the line? Well, countable number of times. This is our line. We have decided that this is a unit segment and, obviously, this unit segment is repeated countable number of times on the line. So, that's why I'm saying that, on one hand, this line has the same cardinality as the set of all real numbers. On another hand, the same line has a combination of a countable number of times the segment, which, in turn, has exactly the same cardinality as the real numbers. That's why we have this equality. All right. So, this is a geometrical proof of this particular equality between the cardinalities, and I will include it into my set, set of equalities. Next, cardinality of a set of all real numbers is greater than cardinality of set of all natural numbers. Now, first of all, because of this theorem, we know that the cardinality of the set of real numbers is not less than the cardinality of natural numbers, because natural is the smallest. But is it really bigger or is it equal? Maybe there is a way to put all real numbers in one-to-one correspondence to all natural numbers. And by the way, if you remember one of the problems before, consider it was that number of rational numbers, the set of rational numbers has the same cardinality as the set of natural numbers. So, these are equally powerful sets, the rational numbers and natural numbers, although natural seems to be inserted, like part of a subset of rational. But that's not true for real numbers. In case of real numbers, we do have a real inequality here. This is real number and this is straight less. This is strong, not equal. All right, how can I prove it? What we can do actually is the following. Let's consider only numbers from zero to one. Every number can be represented, real number, can be represented as a sequence of, let's say, zeros and ones in a binary system. So, real numbers are zero point one zero zero one one zero zero zero zero zero one zero zero. I mean, these are all real numbers and they are all between zero and one using binary representation. So, every real number can be represented as a string of zeros and ones. Now, this string can be either finite or it can be infinite. In case of infinite numbers, it can be either periodic or aperiodic. If you remember, periodic fractions are representing rational numbers and aperiodic represent irrational numbers and all together are called real numbers. So, basically, again, let's talk about real numbers in terms of sequences of ones and zeros. Now, obviously, there are infinite number of these sequences because there are an infinite number of real numbers. Now, to prove that there is no one-to-one correspondence between this and set of natural numbers, let's assume the opposite and come to a contradiction. So, let's assume that we can really enumerate. So, every real number will have certain number which it corresponds to, natural number, number one, number two, number three, etc. So, they enumerate all the real numbers. Now, is it possible? Well, my answer is no because I will present a specific real number which is not part of this sequence, which cannot have a specific number associated with it, natural number associated with it. Now, how can I build this particular ratio, sorry, real number which does not have a corresponding natural number? Well, simply. So, if I have already done this correspondence, let's assume we have done it. How can they come to a contradiction? And here is how. I will find a number which looks like this. Now, I consider the first number and the first position and change it to opposite. Then, from the second number, I take the second position and change it to opposite. From the third number, I'll take the third position, which is one, and change it to opposite. And I will continue. So, I will come up with a sequence of zeros and ones which represent some real numbers because every sequence of zeros and ones represents certain real numbers. But what's interesting is that this particular number cannot be represented among these. It cannot have one of the numbers assigned. Why? Because it's different from the first and different from the second and different from the third. So, it's different from all of them. So, it's not one of those guys. So, we have built a certain number which is not represented in this correspondence. It cannot find the corresponding natural number to it because it's different. It's different from number one. It's different from number two. It's different from number three. It's different from all numbers because that's how we have built it. We just took a corresponding position and changed it to the opposite. What does it prove? Well, we came to a contradiction because, first, we assumed that every number is represented in this correspondence and then we built something which is not represented. So, that's the contradiction. Which means our original assumption is incorrect. Which means there are some real numbers which cannot be put into correspondence with natural numbers. No matter how we try, we can always, I mean, we can change different ways. We can rearrange it, but for every arrangement, whatever we have, we can always have something left which cannot be put into this arrangement, into this correspondence. Okay, so we have proven that the number of the cardinality of the set of all real numbers is greater than the cardinality of all the natural numbers. And how can we put it here? We can put it here this way. And less than the card. Now, this is a strong less than. There is no equal like here. Now, something which is much more interesting. Here's how it looks in formula. Now, what does it mean? Now, if you remember, if this is a set, then two to the power of this set means a set of all subsets. So, if this is natural numbers, so that means that we are considering all different subsets of the set of natural numbers. And what's important is that its cardinality is equal to the cardinality of the set of all real numbers. So, you can actually consider real numbers as well, the first step upwards into infinity, which is a first bigger into infinity, which we found, bigger than the previous one, than the set of natural numbers. It doesn't mean that there is nothing in between. We have built R from N using this technique. That's what's interesting. So, R is bigger than N, but we can have an equivalent of R using a set of all subsets of natural numbers. Now, how can I prove that? All right. I think I'll just use exactly the same, basically, approaches before. All rational numbers can be represented as zeros and ones, as a sequence of zeros and ones. And what is a sequence of zeros and ones? Well, basically, every sequence can be put into one-to-one correspondence with a certain natural number, which is very easy. Now, how can that be done? Well, there are many different ways, but for instance, you can break this sequence into certain groups, and every group basically represents some binary number. And so, the set of all groups is basically a sequence, a subset of natural numbers. So, if you have all the different sequences of ones and zeros, then each sequence basically is, in some way, a subset of all natural numbers. Now, even if it's infinite, obviously, you can do exactly the same thing. You can break it into certain groups, and then each group will correspond to some number. So, there are many different ways to map it into natural numbers, but the fact is that representation as strings of characters is basically sufficient to say that this is countable. I mean, this is the countable. This is part of set of all subsets of a countable set of natural numbers. So, there is a very interesting hypothesis here. Hypothesis is that, yes, we have actually proven that the cardinality of real numbers is greater than the cardinality of natural numbers. We have even built one cardinality from another. Now, the hypothesis which was suggested by the same counter, by the way, is that there is nothing in between. There is no, you cannot build a set which is simultaneously infinite, greater in the cardinality than set of natural numbers, but less in its cardinality than the set of real numbers. He actually was not able to prove or disprove this statement, and later on it was basically decided and partially proven that it's really independent from all other axioms of sets, theories, etc. So, basically, we can just postulate, take it as an axiom that the real numbers represent basically the next infinite set of numbers, set of elements, whatever it is. Alright, what's next? Cardinality of a set of all points in the square of the unit lengths of a side is equal to the cardinality of a set of all points on its side. So, basically, what it says is that if you have a square, then the number of points inside of the square is an equivalent set to cardinality of the points on the segment which is a side of the square. How can that be proven? Well, let's think about this way. If this is a system of coordinates, then each number, each point here can be represented as two numbers, x and y. So, what we are talking about, number of all pairs of x and y, this is the number of points inside the square. Now, how can I prove that there is a one-to-one correspondence between these points and, let's say, points which have only one coordinate, let's say, x or y or whatever? Well, here is how. Well, first of all, obviously, the set of all points on the segment is not greater in the cardinality terms than the set of all points in the square. So, I don't want to do this. So, all I have to do is this. I have to prove that there is a one-to-one correspondence between all these pairs of real numbers and set of real numbers. How can I build it? Well, actually, it's easy. Again, using the, let's say, decimal or binary or whatever else representation, I represent x as some kind of an infinite number of zeros and ones, same as y. Now, what I will do is I will put that into the correspondence of the following real number. I will just interlace these two. I will take zero from x, then one from y, one from x, one from y, zero from x, zero from y, etc. So, it goes this way, this way, this way, this way. That's what interlacing means. So, for each pair of real numbers, I built another real number. I mapped it into, now, is it correct mapping? Now, if we have two different elements, two different points, then something would be different here, and that's why it will map into different real numbers. So, the correspondence between the point, which is two real numbers, and one real number, which is built using these interlacing methods, is obvious. Different points will map into different images, different real numbers. So, that's why the number of points inside the square, which is the number of pairs x and y or x and y are real numbers, is from the cardinology standpoint, equivalent to the number of points in a segment, which is just plain real numbers. This is done, mostly. Okay. By the way, how it can be written here, r times r, this is pairs of real number, is equal to cardinology of the all points on the square, which is pairs of real numbers, is exactly the same as cardinology of real numbers. Okay. Now, before I go to the last problem, let me go to a three-dimensional space. Situation is no different. Now, what is a three-dimensional space? Three-dimensional space is a set of all triplets, x, y, and z, right, where each of them is a rational number. Now, how can I map it to rational numbers? Well, basically the same interlacing fashion. I will do the first from x, will go into the first digit in the r. Then the first digits from the y go to the second r, and the first digit from the z goes to the third digit of the r. That's how my first three digits of r are built. Then I go to the second, which will go to number four, number five, number six. Then I go to the third, etc. So, for any finite dimensional space where the point is represented by this finite number of coordinates, whether it's two coordinates or three coordinates for a three-dimensional space, we have exactly the same situation, which means that not only number of points in a square is equivalent from the cardinologist's tenth point to a number of points in the segment, but also number of points in a cube, three-dimensional cube, is also exactly the same as number of points within the segment. All right, and now let's go to one more level of infinity, if you wish. Now, this, by the way, can be written as to the power of n is really close to r. Now, this is a finite number n. My last problem would be to prove r to the power of countable power of r is equal to cardinology of r. Now, what does it mean? Well, it's a, geometrically speaking, it's a countable dimensional space, if you wish. So, we have two-dimensional, we have three-dimensional, we have n-dimensional space. How about countable dimensional space where the number of dimensions is countable? Now, what does it mean? It means that we have not a finite number of elements in this sequence, but countable, etc. So, the whole string of rational numbers, now infinite string, but countable, because I have indices here, represents a point in the countable dimensional space. I hope it doesn't confuse anybody. So, how can I put into the correspondence this to some kind of a real number? What real number corresponds to this infinite, infinite string? Well, let's think about it this way. I'll do this very interesting trick. So, r to the power of n means, now, do you remember this formula? I didn't put it here. I kept it before. 2 to the power of n equals r. This is, yeah, the previous problem, one before it. I actually have proven that the number of elements in the set of all subsets of natural numbers is exactly the same in terms of cardinality as number of elements in the set of all real numbers. So, I'll use it here, and I will use this. It's not really rigorous. It's really like a formality, some kind of formalism, if you wish. I will replace r with 2 to the power of n. Now, you all know that power to the power would be a multiplication of powers, and I will use this. Do you see n times n equals n? Well, obviously, I should have this everywhere. So, not the sets themselves, but their cardinal value. It's not really a true rigorous proof to make it, but it can be made a very rigorous proof, because really what I did, I used whatever logic I had before to prove all these formulas for this. So, basically, from the logical standpoint, I can repeat exactly the same logical transformations as I used before to go through this, but in a very short and concise form, it would look exactly like this, which is quite interesting, because now you see you can use certain formulas which look quite strange if you just take a look at this, but you can still use them to prove certain theorems. Okay. That's the end of this lecture. Don't forget uniseur.com is the site where you can find this and many other interesting lectures, and especially for parents and the teachers, supervisors. Using this site will allow you to very, very precisely control the educational process of your student with the pace which is kind of individual for every student. 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