@@Garbageman209 yeah for real. he's not giving any theorems or definitions really, but I understand it way more than if he just gave theorems and definitions.
Yes. Finally i understand diagnoalisation. I always used to think, how DO you know you have a brand new number. What if it's somewhere else on the list. Now i understand. By doing this diagonally, you really do make sure, that you are making this number different to every single other number on the list, no matter how many digits, how long the list goes
I am sorry to say that just as you understood the impossible I proved just the opposite. See the video below : Hello, I designed 2 superb algorithms which incredibly establish 1-to-1 correspondence between positive floats and positive integers. For details please access the RUclips video “Pairing Floats and Integers” at ruclips.net/video/tgVOrCo_5wE/видео.html You may see the step-by-step unfolding of one of the most exciting discovery, which up until now was believed to be mathematically impossible. It is amazing that a single action of digit-reversal made it possible to construct all possible floats using a pair of integers (Whole and Fractional). I predict that my discovery will withstand the combined challenges of the best set-theory experts and will earn its place in mathematics history. Let’s hope that as a result, the understanding, teaching and the relevant literature of this very important subject will considerably improve. Please leave any feedback in the COMMENT section of the video. or email to Tamas Varhegyi at secondcause@gmail.com
ive been struggling to understand this concept as my teacher assigned it for homework without teaching the concept. I learned it thoroughly just after watching this video. Ur enthusiasm inspires me and thanks for teaching it so its easy to understand.!!!!!!
There can be infinitely many rational numbers between 2 naturals numbers as well as in case of real numbers, but the main point is that for each of those rational numbers we can have a *bijection* with some *infinitely countable sets* which is NOT the case for real numbers, that is the main reason for them to be uncountably infinite.. there is no relation between how many real numbers is there between any 2 natural numbers or say integers(which strictly refers to density), if we have a bijection from a set to the other countably infinite set, then that set is countably infinite as well(which is till this date not possible for real number) The readers can give it a try.. :)
@@misterentername8869 Because that's how it is. Natural numbers can be mapped one-to-one with rational numbers (this was proven by Cantor - you can enumerate all fractions by diagonals: 1/1; 2/1, 1/2; 3/1, 2/2, 1/3; 4/1, 3/2, 2/3, 1/4; ...). But natural numbers (or rational numbers) can't be mapped one-to-one with real numbers (for example by Cantor's diagonal proof). A set being dense is property of an order on that set, not of the set per se. For example, the usual order on rational numbers is dense. But because natural numbers can be mapped one-to-one with rational numbers, there exists an unusual order on rational numbers which isn't dense (and conversely, there exists an order on natural numbers - which has nothing to do with their numeric value - which is dense). Those orders are just the order on rational numbers (or on natural numbers) transferred to the other set using the one-to-one mapping. And assuming axiom of choice the same can be done for real numbers. Observe that the usual order on natural numbers is a well-ordering relation (such that every two elements are comparable, and every non-empty set of natural numbers has a minimum). But from axiom of choice it follows that every set can be well-ordered. So while the usual order on reals is dense, there exists a different order on reals which is a well-ordering (and for obvious reasons, a well-ordering can't be dense). (Of course, a well-ordered set doesn't need to be countable.)
@@MikeRosoftJH It's been six months, I actually understood it already (we learned it with Cantor's proof) but you brought along 2 new arguments! I'll look them up but thank you for the explanation, because (from all the platforms I used) you're the first one, so thank you and have a great day bg.
awesome!! i have watched a lot of video which talk about countable and uncountable infinite set but it doesnt work. and now , finally, i has found the best one !!! thank you so much professor
I've had it explained to me in many ways and I still just don't understand how there can be multiple levels of infinity. Maybe different active "progressions". For example Whole numbers vs Even numbers....as you start counting it will appear that the set of "whole numbers" are twice the amount vs the set of "evens". But what does this matter? There is no technical "end" to either set. No matter what point you STOP the sample-size that you have counted "so far" , both samples would represent EXACTLY 0% OF THE WHOLE (0% of infinity) So how can one set be "larger" than the other? I know this is an over-simplified argument against it, but I just can't get past that fact.
It's really an extension of how cardinality (or the number of elements) is defined in case of finite sets. Imagine that you are in a cinema, and see that all seats are occupied: nobody is standing, nobody is sitting on two seats, no seat is empty, and no seat has two people in it. Then - without needing to count them - you can say with certainty that there's the same number of people as seats. Cantor's insight was to use the same definition for infinite sets. So are there more natural numbers than even numbers? In fact no (even though even numbers are a strict subset of natural numbers). The two sets can be mapped one-to-one, and the mapping is n->2*n. Likewise, natural numbers and integers can be mapped one-to-one (so by definition the two sets have the same cardinality), because integers can be enumerated by the following sequence: 0, -1, 1, -2, 2, -3, 3, ... - any integer n appears at a position no greater than 2*|n|+1. But real numbers can't be mapped one-to-one with natural numbers; given any function from natural numbers to real numbers, there is some real number which the function does not cover. So by definition, natural numbers have a strictly lesser cardinality than real numbers. (And again, this uses precisely the same definition as in case of finite sets. The statement that real numbers can't be mapped one-to-one with natural numbers is exactly like saying that you can't put ten guests in nine rooms, such that no two guests share a room.)
@@rob_olmstead What's "weird" about it? I see, the word "weird" is being used increasingly for so many diverse situations. Not talking about you, but clearly notice the decrease in vocabulary of folks over the generations.
@@rob_olmstead Correct. No, I couldn't detect any flaw in your writing. However, I feel a bit sad when folks use fewer words than older generations used to for more "types" of situations. I'm assuming you're not from India. When I point this out to Indians, the most common answer I get is : well, it's slang and slang is commonplace now. And to me, it feels like murder of a language . 😆 I'm nearly 46, how old are you and where from? I'd delete my comments after your reply. Cheers, buddy!
Do you think it would be possible to match up all whole numbers to all real numbers by say starting at 0.000.....001 for the real numbers and on the opposite end; the whole numbers 10000..... So like a 1 with an unending line of 0's. Then the way we count them we could say and 0.000....1 to the real number, and add 100000... to the whole, but when we go from 900000... to 1000000... we shrink it by a tenth. With that we could get to 1 at the same step as the real numbers, right? So we can match up real numbers to whole?
What assumptions do we have in saying that the list is complete? Lets reduce this setup to binary representation where the only elements used are 0 or 1. Does a complete list represent all the possible combinations that we can think of using these digits? For example, lets make a list of all the possible combinations with 3 place values; xxx 000 001 010 011 100 101 110 111 This is not a list of values, it is a list of possible combinations using 2 elements. If we construct a square matrix with any 3 items on this list, then the altered diagonal will not represent any item within the matrix. Spose that we arbitrarily define each item such that there are only 3 values represented by the list. A = 000 = 011 = 101 B = 001 = 010 = 100 C = 110 = 111 Does a complete list only represent all the unique values possible, is it 1to1 and onto (a bijection)? Why does this matter? A: 000 B: 010 C: 111 our altered diagonal is 100 = B, IS contained in our list. A given decimal expansion between 0 and 1, is not necessarily represented by a single, unique decimal expansion. 3/10 = .3000... = .2999... If we keep this in binary: 1/2 = .100... = .0111...; and i can think of an infinite number of duplications. So I ask, what does a "complete list" look like? Do we not allow the list to have duplicate representations of value; or, are we just making a complete list of possible combinations without regard to any value.
Well, sure: you need to be careful when constructing the diagonal number. But that's just a technical detail (the only numbers with two different decimal expansions are those whose decimal expansion ends with infinitely many digits 0, which have an alternate expansion ending with infinitely many digits 9; and all such numbers are rational, and so there's only countably many of these). My favorite way of constructing a diagonal number is by taking the n-th digit of the n-th number in the sequence, and adding 5, wrapping around zero if necessary. (Let a(n) be any infinite sequence of real numbers. Let d(n) be, for all natural numbers, the n-th digit of a(n). Then d'(n)=d(n)+5 mod 10; and the diagonal number is the real number whose decimal expansion is d'(n).) Now it can be seen that not just the decimal expansion of the diagonal number is different from every real number in the sequence, but also that their values differ by at least 3*10^-n. (Alternately, you can just use a mapping which doesn't use the digits 0 and 9 - for example, map digit 5 to digit 6, and all other digits to 5.)
I don't get how the derived decimals are considered "uncountable". They are literally being counted in the video. Also when you are writing the rows and numbering them (i.e. the counting them) every new row is also a number that did not exist previously in the list. How is it we can we even use a comparison operator when they are both infinite? There is some core concept I am missing here.
Construction matters. Imagine that an inclusive list is actually inclusive. Imagine that if you really could write down all the rational numbers, would such a list be square in form, as it must be in order for diagonalization to produce a valid row covering each entry once. Imagine that time is related to distance so that the notion of producing such a creature - apriori - out of a bag and then processing to wax theoretic, wouldn't really be possible 😮
The first number was made by adding one to the diagonal and the next by adding -1. What about he next new numbers? There are only 9 digits you can add to or subtract from to get new numbers. So what do you do to get an infinite number of new numbers?
In fact, for any infinite sequence of real numbers (function from natural numbers to real numbers) there are uncountably many real numbers which the sequence doesn't cover. This can be seen from the construction: the video gives one specific way of constructing the diagonal number. But it can be seen that for any decimal position there are at least 7 different digits to choose from: for n-th decimal position you can't pick the digit which n-th number has at the same position, and you may want to avoid choosing digits 0 and 9 in order to avoid the problem with numbers which have two different decimal expansions (such as 0.1000... = 0.0999...). This yields uncountably many diagonal numbers (you can re-interpret the choices - 7 for every decimal position - as base-7 representations of real numbers), and none of these appear in the sequence.
The lecture is still great! And I still hate this very piece of math! It was boiling my blood at school, it is was causing me pain in university, and it still hurt me now. Why can't I do pretty much the same for the 'countable' infinity of whole numbers? Except, of course, the diagonal would be heading to the left, ensuring that new number is completely different from 1st number (which is 1), 2nd number (2), 3rd number (3), and so on and so forth.
No, because any natural number is finite in magnitude, and therefore has finitely many digits (and that's by definition: a set is finite, if its number of elements is equal to some natural number: it's an empty set, or it has exactly one element, or it has exactly two elements, or ... and so on). If you apply the diagonal procedure to a sequence of all natural numbers, you get a sequence which has infinitely many non-zero digits, and that doesn't represent any natural number.
This makes sense and you explain it very well, but I can't help but wonder - couldn't you list the real numbers diagonally as you can with the fractions? So you'd start with the following infinite table to begin with: Row 1: 0.1, 0.01, 0.001, 0.0001, 0.00001... Row 2: 0.2, 0.02, 0.002, 0.0002, 0.00002... Row 3: 0.3, 0.03, 0.003, 0.0003, 0.00003... Row 4: 0.4, 0.04, 0.004, 0.0004, 0.00004... ... Row 10: 0.10, 0.010, 0.0010, 0.00010, 0.000010... Row 11: 0.11, 0.011, 0.0011, 0.00011, 0.000011... Row 12: 0.12, 0.012, 0.0012, 0.00012, 0.000012... ... Row 340902: 0.340902, 0.0340902, 0.00340902... Row 340903: 0.340903, 0.0340903, 0.00340903... ... Row 20167398019374678093018390565748765: 0.20167398019374678093018390565748765... (ad infinitum) You could then "list" the numbers diagonally, 1-1. So, in this example the list would be: 1: 0.1 2: 0.01 3: 0.2 4: 0.3 5: 0.02 6: 0.001 7: 0.0001 8: 0.002 9: 0.03 10: 0.4 11: 0.04 12: 0.003 13: 0.0002 14: 0.00001 and so on... It's countable, right? At least in the same sense that the infinite list of fractions can be counted...?
Now take that final list you made, and apply the same argument he made in the video to it. You'll still get a number that, by definition, isn't anywhere in that list. If you try do the same thing with the infinite list of fractions, you'll see there's no way to construct a number that won't be in the infinite list of all possible fractions.
Yerren v. St. Annaland I’m not sure that’s true. Take item 14 on the list, 0.00001. Raise each digit by 1 and you get 0.11112, which would eventually appear on a diagonal list of all possible 0.x. I totally understand the argument, but all it seems to really be proving is that you can always add another number to an infinite list. By definition, an infinite list (or indeed an infinite anything) can never be completed, that’s why it’s infinite. This argument, by contrast, requires the assumption that the list is somehow complete, and yet there is a way to find a number that isn’t on the list. However, if the list was complete, it wouldn’t be infinite. So, of course it will always be possible to find a new number. It’s an interesting video, but ultimately the analogous concept of infinity is not compatible with a digital counting system, hence the problem.
@@will3music So, for example, let's say we list out the positive and negative integers as follows: 0 1 -1 2 -2 ... Can you find any rule to generate a positive or negative integer that isn't on that list? You'll find it's not possible, but it is possible in the case of the Real numbers. Thus, there must be something different about the Reals. (Thats that there's "more" of them). That's really all this video is saying.
Your list of lists will not cover any number with an infinite decimal expansion. It won't cover any irrational number, like pi or the square root of 2. It won't even cover all rational numbers, such as 1/3=0.333... . Because every number on your list is rational, and because there are countably many rational numbers (as proven by Georg Cantor), so is your list countably infinite.
Given all rational number are real and we have sq rt 2. Hence real larger … qed. Problem of ladder is that it seems artificial and given rearrangement of infinity series give you a different number, infinity series laddering …
Is the set of real numbers from (.1,1) countable. Can I match .1 to 1, .2 to 2, ... .11 to 11... Can’t I match any decimal of any length from .1 to .99... to an integer based on the decimal x 10 to the power on n (n being the number of digits in the decimal). Does that not make the set (.1,1) countable? I understand that (0,1) is uncountable, because of the infinite number of .000... that can proceed any small decimal, but what about taking zero out of the set.
Your scheme only covers numbers with finitely many digits after the decimal point. All such numbers are rational, and that there are countably many rational numbers is a well-known result. At what position does a number like 1/3, or √2/2, or π-3 appear? (And what do you mean by "infinite number of .000... that can proceed any small decimal"? There's no such number as 0.000...1. That is, unless you mean a limit of the infinite sequence 0.1, 0.01, 0.0001, ...; but that limit is just plain old 0.) And it's fairly trivial to find a one-to-one mapping between intervals of a different length. So if one is uncountable, the other must be uncountable as well.
Mr. Woo, why does it have to be a diagonal? What if I construct my 'unlistable' number the same way you did, but on the first vertical line (2,7,7,2,0......) instead of on the diagonal? Isn't it the same thing, namely that it won't be on list either?
I mean, what if, as is inevitably the case, you have ten real numbers that all have a different first digit? You wouldn't even have a means of saying what the first digit is.
@@eggynack i'm trying to understand, since math is not my field at all. So my hypothesis is there to be confuted..but in an easy way that i can understand. I'm sure your counter-example is fit, but I still have a hard time
@@biopolis7 Wait, I think I misread you. Do you mean the first digit of our new number is distinct from the first digit of every number, or that the first digit of our new number is distinct from the first digit of the first number, the second digit is distinct from the first digit of the second number, and so on? Either can be shown to not work pretty easily. I think you mean the latter, so I'll just build that. .1 .23222... .111 .1111 .11111 And so on. The first digit of the first number is a 1, so the first digit of our new number is a 3. The first digit of the second number is a 2, so the second digit will be a 3. The first digit of the third number is a 1, so the third digit is a 2. And, y'know, the rest are all 2's. So, the new number is .23222..., which shows up in the second position on our list.
nice explaination, been a hard one to grasp. i still don't agree as it is just rhetoric for something that doesn't make sense. sooner or later assuming infinity you would hit every number you can create on a diagonal line anywhere on the list. like 1,2,3-2,3,4-3,4,5 gives you 2,4,6 which exists in the list assuming it is infinite, just further down. to me it is a strawman argument that doesn't hold and haven't been able to find a way or explanation that can show/teach me why it isn't.
I'm not sure why you think it is a strawman argument. Perhaps if I understand your objection better, I can explain what's happening in the argument. I think that most mathematicians do tend to gloss over a lot of details when presenting the diagonal argument! Usually, glossing over details doesn't cause an issue, but when the result is so counter-intuitive, often people will try to pick apart the argument. So many people will notice "gaps" that aren't really problems with the argument, but rather are just details that have been skipped over in the presentation. The formal definition of "countable" is that there exists a one-to-one matching with the natural numbers. A "list" is just a metaphor for that one-to-one matching, since the number that is matched with 1 can be considered the 1st number on the list, the number that is matched with 2 can be considered the 2nd number on the list, etc. There is something to point out about these lists - even though the list is infinite, each number must appear in a _finite_ position on the list. This is because each natural number is finite in value, even if there are infinitely many of them. So, if the new diagonal number were on the list, it would have to be in some finite position n. (It would have to be matched with a natural number n.) But, by construction, the diagonal number differs from the nth number on the list in the nth decimal place, so the diagonal number cannot be the nth number. This shows the diagonal number cannot be on the list in _any_ finite position, showing it can't be on the list at all. (Since, by what countable means, it has to be in a finite position.) If this doesn't address your concerns, let me know, and I can try to figure out what is bothering you about the argument.
Couldn't you just do the same for the counting numbers if you said that each number has an infinite amount of zeros before the first digit that shows up and you could just add one diagonally moving from right to left? Or start from the "last" and follow the same algorithm you've created backwards? Sure you can't do that because the list is infinite but by the same logic you can't ever get to the end of your irrational number either to make a new number for your list.
I know this is an old comment but I had to reply! Adding an infinite amount of zeros to the left of the digit doesn't change the value of the number. In fact, EVERY number has an infinite amount of zeros in both directions. This is why we learned to "bring down" a zero when doing long division. Same thing. Whether's 1, 01, 001, 0000000001, it's all equal to 1. This isn't so with non-zero values in a decimal. 3.9 does not equal 3.99 does not equal 3.999. All of these are different, and we could continue to add nines for as long as we want and we will have a next number each time, approaching the limit of 4 (until we have an infinite number of 9s, in which case this would be equal to 4, but that's another proof for another day). You can always add new digits to a decimal to create a new number, and you can use Cantor's method in the video above to create a new decimal that will not be on your list even when each real number has an infinite number of place values. It doesn't work with zeros going in the other way. Although, you may be interested to know that there ARE numbers with an infinite amount of digits going off to the left. They're called p-adic numbers, and they're fascinating. I hope this helps!
[Edited by replacing entirity. (I should have ruminated for another 24 hours before posting.)] (I also commented [with] a formal proof under {CvalbBGhmW4}.) -- Can skip -- Consider the case that the list of all the numbers is a finite list - say all the numbers between 0 (inclusive) and 1 (non-inclusive), to 10 decimal places. . Suppose first that this list is in order. We generate a diagonal number… perhaps by adding 3 to each given digit - getting (offhand) 0.3333333332. . Is this number *not* in our complete list? Of course not - of course it is in the complete list. Suppose, then, that we order the complete list randomly. . Suppose that (arbitrarily) we end up with, say, 0.7529901438. . Is this number in our complete list? . Yes, it is… because the list is complete. There is a trick here - a sleight of hand. . The trick is that… for each 1 digit that we write, there are 10 numbers in the list. . Consider, for instance, a complete list of 4-digit numbers. . Our diagonal system processes 4 digits - one digit per line - thus covering 4 lines in the list. . However, there are 10,000 numbers in the list. . We can indeed guarantee that there is no match for our number, among the first four numbers, but we can be certain that there will be a match for our number among the remaining 9996 numbers (since the list is complete). . Similarly… if we process 12 digits with our diagonal system, we can guarantee that the number we generate will not match any of the first 12 in the list… but it will match one of the remaining (10^12)-12 = 999,999,999,988 numbers. . (Even if we reduce the base to the smallest possible value - 2 - binary… we still have, for instance, 2^12 = 8192 items in our list, against 12 eliminated. . The limit case is (i.e. would be) base 1; in this case, we can cover the whole (complete) list using our diagonal system, but now it is hypothetical. . [I was going to say that this was not a meaningful number system, but perhaps it reduces to just counting. . Either way, anything we write is going to match an item in the list, because there is only 1 numeral that we can write.] ) The argument presented in the video is a “reductio ad absurdum” [proving the opposite of P by showing that P entails something inconsistent]. . The idea in this case is to assume that the list is complete, and then show that it is possible to generate a number that is not in the list. . I have shown that this argument fails, in the case of a finite list. . What about an infinite list? People are funny with the term “infinite” or “infinity”. . Suppose that we have two sequences [a list generated by an equation]. . [I shall write “_” for subscript.] . One is just j_n = n; the other is k_n = 2n. . j and k range from 1 to… they just go on and on, getting bigger and bigger. . We talk about the limit as j (or k) “approaches infinity”. . (Remember that infinity is, not a number, but {the concept of getting bigger and bigger, or having more and more, without limit}.) . It is true that the sequence j (or k) approaches infinity as j (or k) approaches infinity. . The *limits* of these two are the same. . Nonetheless, k_n = 2 * j_n, for any value of n - be it 5 or 10^10,000 or 10^10^10,000. Actually, our list of all the Real numbers is infinitely long in a stronger sense; whereas the list of positive integers has a 1:1 correspondence with an attempt to count them… the list of Real numbers is just a sea of numbers - most of which are infinite in length just by themselves. Nonetheless, the above flaw in the diagonal system still applies; the diagonal system is not apt to generate a number that is guaranteed to not be in the (complete) list, because it has this flaw of having less than a 1:1 correspondence with items in the list. . (Of course, the failure of this particular proof does not prove that its tenet is false. . Conversely, the construct in question is a complete list of all the Real numbers, and we should question any purported rigorous proof that it is not *because the concept is not incoherent*.) -- Can skip -- END [I proved that the diagonal proof is flawed.] Unfortunately, all of the above is a red herring… because we are concerned, not with whether or not the list is *complete*, but whether or not it is *countable*. Of course, to count an infinite list of anything is not *feasible in real life* (even if one lives forever). . However, that is merely an observation about the concept of “infinite”… and the real world. Suppose that we define “countable” to mean that each and every item in the list must have a unique identifier. . We can achieve that by making each number its own identifier (so to speak). . √2 - √2 - check! . √2-1 - √2-1 - check! . 15.397029…[whatever it is] - 15.397029…[whatever it is] - check! . Overall… any number that we can identify is countable… and conversely any number that is not countable is beyond our reach anyway. . Given this… the concept of {a complete, infinite, countable list} is perfectly coherent. . …Given enough time.
Help! I am confused here: can't we use the same argument on natural numbers to prove the set N is uncountable (which obviously is wrong): since the natural numbers can increase indefinitely, we have unlimited number of digits to work with (just like in this case), and therefore we can always construct a new number not in the set. As a matter of fact, after the part of 0., the digits of the real numbers would resemble natural numbers right?
The term countable may not fully explain the concept. I think 'sortable' is a better term. The integers are sortable in the sense that you can start from 0 towards infinity an always be guaranteed you didn't skip a number in the middle. For the real numbers, you cannot guarantee that sorted list... There is always another number in between any other number that you didn't count.
@@ricosrealm you can make all numbers between 0 and 1 in order, it's just a bit weird: 0.0 0.1 0.2 ... 0.9 0.01 0.11 0.21 ... 0.91 0.02 0.12 0.22 ... You basicly write it down like the Natural numbers but mirrored after the decimal point, so you come across EVERY number between 0 and 1. So all real numbers between 0 and 1 are countble but if you do all real numbers you're right because you will have no BEGINNING and no END because you an infinite amount of infinities. (which you can't count)
@@LC19. Apology for my poor English, but it seems like your way of listing all numbers between 0 and 1 doesn’t really list all numbers between 0 and 1. We can observe that every number you listed has a finite number of digits, and irrational numbers (infinite non-repeating decimals) like sqrt(2)/2 cannot be on this list. You might want to argue that since the list goes on forever, an infinite non-repeating decimal can be in the list. But again, because of the fact you construct this list in such way, I can always find out the exact number of digits given the index of a number in the list, which in not infinite. So unfortunately this list cannot actually cover all real numbers between 0 and 1.
As I understand, while arbitratily large, natural numbers are finite numbers by definition so while you can in principle go out arbitrarily far, you cannot go out infinitely far. This is in contrast to a number like 1/3 (repeating 3's in decimal places --> 0.3333333....) where it's "reflection" across the decimal place is ...3333333333333.0 (3's going out to infinity) would not be a natural number. This is why I think you can qualify 0-1 decimals and infinitely large numbers as uncountable, but not natural numbers because, while you can always find a larger natural number, the natural numbers can be listed completely (no contradiction with a complete list), and the others cannot only be listed partially (the idea of a complete list leads to a contradiction).
In a countable infinity you can find a next step. Start from 1 and you can keep going, 2, 3, 4, etc. There are no integers between 1 and 2. so 2 is definitely the "next step". So it's countable, despite it being an infinite list. But with uncountable infinity, like numbers between 0 and 1, there's no "next step", there will always be a smaller number between every 2 numbers you pick. so you can never even list the first few numbers of an uncountable infinity. There's literally an infinite number of numbers in-between any 2 numbers you choose. Making it uncountable.
Grasp no more. CDA is wrong and I have the video below to prove it. Hello, I designed 2 superb algorithms which incredibly establish 1-to-1 correspondence between positive floats and positive integers. For details please access the RUclips video “Pairing Floats and Integers” at ruclips.net/video/tgVOrCo_5wE/видео.html You may see the step-by-step unfolding of one of the most exciting discovery, which up until now was believed to be mathematically impossible. It is amazing that a single action of digit-reversal made it possible to construct all possible floats using a pair of integers (Whole and Fractional). I predict that my discovery will withstand the combined challenges of the best set-theory experts and will earn its place in mathematics history. Let’s hope that as a result, the understanding, teaching and the relevant literature of this very important subject will considerably improve. Please leave any feedback in the COMMENT section of the video. or email to Tamas Varhegyi at secondcause@gmail.com
Sorry this doesn’t make sense to me. If a number goes on forever and we can count it that doesn’t make the infinity countable it makes the number infinitely countable.
I'm not getting why this proves that there are more numbers between 0 and 1 than there are natural numbers. Sure, one may find a decimal number that is not in the list, but at the same time we may just come up with another natural number to index the new item. For example, consider the list below: 1 - 0.02323 2 - 0.06909 3 - 0.06910 5 - 0.07000 6 - 0.07340 Sure a new number that is not in the list may be produced,, such as 0;10000. This number may be added to the list, and to index it we may sum 1 to the counting numbers. 1 - 0.02323 2 - 0.06909 3 - 0.06910 5 - 0.07000 6 - 0.07340 7 - 0.10000 Considering that both natural numbers and the decimal numbers between 0 and 1 could go on infinitely, we may assume that we may keep adding new items to the list, and properly linking it to a new counting number resulting from the sum of 1 to the previous counting number. What am I missing?
yeah man this is what i'm struggling to understand too. Yes, we created a number that isn't in the list, but we can just add it to the list and then +1 the corresponding index. I don't get it, I've read so many comments and explanations but it still doesn't make sense in my head.
When looking at 2 different sets like that I understand that one progresses faster than the other, but if they BOTH go on infinitely then how is it relevant to try and compare or to say one is "bigger" than the other. No matter what your stopping point is on either set, what u have written down in BOTH sets with both equal to 0% of infinity so how can one be larger than the other? I'm just not convinced that you can fit infinity into a contained "set". Wouldn't a "set" imply both a beginning and an end?
racer3d1 racer3d1 it means that one is countable and the other is uncountable.Now what is countable and uncountable?Well,countable always has a next step,like positive integers-infinitely many of them,but 1 is the point to start and the next step is 2,so you can still count.But with real numbers,you don't know where to start and how much to proceed while counting.Its practically impossible.This is what I have understood,but if it's not the correct explanation,then I'm sorry.
I get what your saying, it just hurts the brain to think about it lol. I'm still not fully sold on a "countable" infinity, its like the ultimate oxymoron. for example: if the argument is there are infinite real numbers between lets say 1 and 2 and therefore, real numbers cant be in a countable set bc you couldn't even get past "2" , then that IMO is proof that its equally as impossible as counting regular infinity. If 1-to-2 has more numbers then you can count, then regular infinity itself is not "countable" either and therefore IMO can't be contained in a "set". Of course this is just my opinion, I only made it to pre-calculus in college so I'm sure most people debating on here are beyond my understanding/education of it but I just can't seem to get past putting infinity in a countable "set"
@@lifeofphyraprun7601 I get what you are saying with the "don't know where to start" but couldn't technically the same be said for any set of infinite numbers? For example if u were going to start counting integers, isn't a starting point just arbitrary? There is no rule that says you have to count things in any particular "order" , your starting point in this case could literally be ANY integer from negative infinity to positive infinity..... so how would that be any different from saying we can't find a starting point from 0 to 1 bc how many zeros can we inject before the .1 0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 or a quintillion more zeros perhaps? How is this any different, than saying with integers we are going to create a starting point at some number that is so big that we can't even comprehend it? There are never-ending "starting points" even in integers..... I guess I just can't follow the difference between that and real numbers
I don't know why, but this bugs me. I'm sure I'm bathing in ignorance, but....(before I start, be warned I'm not sure how to unpack this and I'm no master with words... so bare with me) I think the biggest issue I have with this is that I believe the idea of infinity is misrepresented. Infinity is uncountable. I understand as a thought experiment you could theoretically count an infinite number if you had infinite time, but I don't think that's actually true. When there is an infinite of any size, it means it is impossible to get to the end. So, this uncountable infinity doesn't make sense to me because you claim this number doesn't exist in the first set, but that number could never be established, because it could never encompass the entire list. We tend to want to think of infinities like we do normal numbers and that is where I think we tend to get this wrong. I don't believe infinites have a size or amount, instead I think its more like they have boundaries, but the contents are all a shapeless void. It is never possible in any thought experiment to count an infinity (I have a feeling someone will say "well you could count if ..." and I would say.... no you can't... that's why it's infinite, if you could it would be finite) and pairing two infinities together doesn't work (counting to infinity for an infinite amount of time) . It's like north of north. I know that this is rambling, so I apologize. I'm at work and I wanted to vomit this out before I lost it. Whoever gets here... Thank you for your time and reading this far. If anyone wants to educate me I'm all ears and appreciative of the information.
But how is this proofable? At some point there must be a countable infinite match with a uncountable infinte, or else its not infinitely to begin with.
There's no such thing as uncountable infinities. It's mathematicians not realizing that infinity is only a limit concept. Infinite sets don't actually exist. What exists are potentially infinite sets, meaning that the sets have no limits or bounds to how big they can become. Both the set of natural numbers and real numbers are exactly the same. They have no limits. Saying that one's "no limit" is bigger than the other's is an absurdity. I can actually map every decimal point to every natural number. For example, .1 --> 1, .3 --> 3, .001 --> 100. For every decimal in the real numbers, there exists a corresponding natural number. Real numbers that have an infinite number of decimal places like Pi are never reached because they have no limits to how precise they can become. Though, I can say the same about natural numbers that have an infinite number of digits. Neither will ever be reached! I'm not going to say that what I am saying here is definitive because there are a lot of smarter people than me who actually buy into this set theory nonsense. However, I think I'm probably right considering how dumb it sounds. Experts have been wrong on many things in the past so I think the safe bet is that they're wrong here.
Infinities have their own classification and we shouldn't conflate one infinity with another as if they are the same infinity. "Countable infinity" is a contradiction in terms which violates the concept of infinity. If something has a measurable value aka value limit then it is not an infinity. It is a relative measurable value set.
What exactly is your objection? Just the terminology? A set is countably infinite, if it can be mapped one-to-one with natural numbers; in other words, if its elements can be enumerated by an infinite sequence, such that any element appears at a finite position. For example, integers are countable, because they can be enumerated by the following sequence: 0, -1, 1, -2, 2, -3, 3, ... - any integer n appears at a position no greater than 2*|n|+1. The same can be proven for rational numbers. But real numbers are uncountable - given any function from natural numbers to real numbers, there exists a real number (and, by extension of the proof, uncountably many real numbers) which the function doesn't cover.
"forever" and "still missing" makes no sense. I am still searching for a mathematician that can deal with infinity. Apparently none can. They just can't, don't ask me why. (tip: adding something to something that "goes forever" makes no difference. Actually, might as well think you can't: it's like to put something in a moving thing. You need it to be static. Does this analogy helps?)
Just because you don't understand doesn't mean no one else does LOL. Mathematicians and physicists are working with infinities all the time, there's even infinities that "expand faster" than other infinities (n^x vs n^n). Just because you shake your head and refuse to follow the logic doesnt mean its not true!
Real is equal to natural numbers, did his numbers do all the sqrts(2.36987..), sqrts(3.874512...)no.no.no. All the numbers have different names , N,Q,R same thing all reals.
This is so unintuitive😭😭 Real numbers are made of only 9 digits. Surely you can go through all possible combinations of digits in each decimal point (say 'bit') until you get all numbers. No? What if I say: 0.0 0.1 0.2 and so on... 0.9 Now I can move on to the next 'bit' 0.01 Notice how "ten" (flipped) moves the zero to the front because, for decimals, significant zeros are in front. Now we count (flipped): 0.11 "eleven" 0.21 "twelve" 0.31 "thirteen" 0.41 0.51... 0.81 0.91 0.02 "twenty" and continue counting 'flipped': 0.12 "twenty one" 0.22 "twenty two" and so on You'll eventually get to: 0.99 ninety nine (note: it's written backwards) And then one hundred flipped: 0.001 0.101 0.201 "hundred and two" I will eventually exhaust all numbers, no? I started with one bit and exhausted all possibilities and moved to two then three, making sure I add zero in front because adding it at the end creates redundancies (0.1= 0.10= 0.100). I will eventually exhaust all possible combinations with three bits then 4 and so on to infinity. What real number won't be on the list???? I am only using 9-digits after all. Surely, if I keep adding bits, I'll exhaust all possible combinations and thus all real numbers between 0 and 1, no? And if you think about it, natural numbers are made of only nine digits and so are real numbers. How can it be that there are combinations of numbers after the decimal that do not exist (in a different form) before the decimal, just flipped? Maybe I'm missing something 😂😂 but I've been struggling to get this diagonalisation thing. Like... The reason it's not on the list is because you haven't gotten there yet😅. Please help me finally understand this conundrum🙈.
You have to keep in mind that the number of digits is also infinite, which means the length of the diagonal will also be infinite. Therefore, there are also going to be an infinite number of numbers that you are missing.
@@crystalexia- That's because your list is finite. With an infinite list, all numbers are there. If I move down every decimal point exhausting all the possible combinations of digits for 5 decimal points, then, 6 and so on I will eventually run through all numbers and even get to the number that is supposedly not on the list.
@@crystalexia- but couldn't you do the same thing for natural numbers?? ..36363861528636 ..27363836363648 ..47363736363737 ..37363736363736 All infinite numbers use diagonalisation to find a number that is not on the list. The problem with this listing method is that it is Random and has no pattern so to say diagonalisation will find a number that is _not on the list_ doesn't make sense. Why did you start by listing infinite numbers? Why don't you start with finite numbers like 0.1 or 0.12? You don't do that with natural numbers but you do with real numbers between 0 and 1. The whole thing just seems like unnecessary confusion to me.
this guy is great. what an awesome professor.
You mean school teacher ;p
Finally, an explanation of this concept that makes sense. Thank you Eddie Woo, you are a genius.
8:24
“Six”
Mood 😂
I learnt more from this video than 4 months in my semester.
and he seems to be a high school teacher, better than most university professors
@@Garbageman209 yeah for real. he's not giving any theorems or definitions really, but I understand it way more than if he just gave theorems and definitions.
Then u are doing the wrong thing
Probably shit school u go to
Yes. Finally i understand diagnoalisation. I always used to think, how DO you know you have a brand new number. What if it's somewhere else on the list. Now i understand. By doing this diagonally, you really do make sure, that you are making this number different to every single other number on the list, no matter how many digits, how long the list goes
Same
This is absolutely amazing! I finally understand it! God bless!
I am sorry to say that just as you understood the impossible I proved just the opposite. See the video below :
Hello,
I designed 2 superb algorithms which incredibly establish 1-to-1
correspondence between positive floats and positive integers.
For details please access the RUclips video “Pairing Floats and Integers” at
ruclips.net/video/tgVOrCo_5wE/видео.html
You may see the step-by-step unfolding of one of the most exciting discovery,
which up until now was believed to be mathematically impossible.
It is amazing that a single action of digit-reversal made it possible to
construct all possible floats using a pair of integers (Whole and Fractional).
I predict that my discovery will withstand the combined challenges
of the best set-theory experts and will earn its place in mathematics
history.
Let’s hope that as a result, the understanding, teaching and the relevant
literature of this very important subject will considerably improve.
Please leave any feedback in the COMMENT section of the video.
or email to Tamas Varhegyi at secondcause@gmail.com
last point
n(Z) < n(R)
that point saved my life . THANKS sir
From Pakistan
I love his energy and passion! Mathematics is so complex, that it must be true.
Another showstopper, Eddie. Thank you
ive been struggling to understand this concept as my teacher assigned it for homework without teaching the concept. I learned it thoroughly just after watching this video. Ur enthusiasm inspires me and thanks for teaching it so its easy to understand.!!!!!!
There can be infinitely many rational numbers between 2 naturals numbers as well as in case of real numbers, but the main point is that for each of those rational numbers we can have a *bijection* with some *infinitely countable sets* which is NOT the case for real numbers, that is the main reason for them to be uncountably infinite.. there is no relation between how many real numbers is there between any 2 natural numbers or say integers(which strictly refers to density), if we have a bijection from a set to the other countably infinite set, then that set is countably infinite as well(which is till this date not possible for real number)
The readers can give it a try.. :)
why can we have a bijection between rational and natural numbers but not real numbers?
@@misterentername8869 Because that's how it is. Natural numbers can be mapped one-to-one with rational numbers (this was proven by Cantor - you can enumerate all fractions by diagonals: 1/1; 2/1, 1/2; 3/1, 2/2, 1/3; 4/1, 3/2, 2/3, 1/4; ...). But natural numbers (or rational numbers) can't be mapped one-to-one with real numbers (for example by Cantor's diagonal proof).
A set being dense is property of an order on that set, not of the set per se. For example, the usual order on rational numbers is dense. But because natural numbers can be mapped one-to-one with rational numbers, there exists an unusual order on rational numbers which isn't dense (and conversely, there exists an order on natural numbers - which has nothing to do with their numeric value - which is dense). Those orders are just the order on rational numbers (or on natural numbers) transferred to the other set using the one-to-one mapping.
And assuming axiom of choice the same can be done for real numbers. Observe that the usual order on natural numbers is a well-ordering relation (such that every two elements are comparable, and every non-empty set of natural numbers has a minimum). But from axiom of choice it follows that every set can be well-ordered. So while the usual order on reals is dense, there exists a different order on reals which is a well-ordering (and for obvious reasons, a well-ordering can't be dense). (Of course, a well-ordered set doesn't need to be countable.)
@@MikeRosoftJH It's been six months, I actually understood it already (we learned it with Cantor's proof) but you brought along 2 new arguments! I'll look them up but thank you for the explanation, because (from all the platforms I used) you're the first one, so thank you and have a great day bg.
awesome!! i have watched a lot of video which talk about countable and uncountable infinite set but it doesnt work. and now , finally, i has found the best one !!! thank you so much professor
Thank you very much, you are an excellent teacher!
Excellent! Such clarity of thought!
mind-blowing again.. after 7 years ...still mind blowing
Another clear lecture
I've had it explained to me in many ways and I still just don't understand how there can be multiple levels of infinity. Maybe different active "progressions". For example Whole numbers vs Even numbers....as you start counting it will appear that the set of "whole numbers" are twice the amount vs the set of "evens". But what does this matter? There is no technical "end" to either set. No matter what point you STOP the sample-size that you have counted "so far" , both samples would represent EXACTLY 0% OF THE WHOLE (0% of infinity) So how can one set be "larger" than the other? I know this is an over-simplified argument against it, but I just can't get past that fact.
It's really an extension of how cardinality (or the number of elements) is defined in case of finite sets. Imagine that you are in a cinema, and see that all seats are occupied: nobody is standing, nobody is sitting on two seats, no seat is empty, and no seat has two people in it. Then - without needing to count them - you can say with certainty that there's the same number of people as seats. Cantor's insight was to use the same definition for infinite sets. So are there more natural numbers than even numbers? In fact no (even though even numbers are a strict subset of natural numbers). The two sets can be mapped one-to-one, and the mapping is n->2*n. Likewise, natural numbers and integers can be mapped one-to-one (so by definition the two sets have the same cardinality), because integers can be enumerated by the following sequence: 0, -1, 1, -2, 2, -3, 3, ... - any integer n appears at a position no greater than 2*|n|+1. But real numbers can't be mapped one-to-one with natural numbers; given any function from natural numbers to real numbers, there is some real number which the function does not cover. So by definition, natural numbers have a strictly lesser cardinality than real numbers. (And again, this uses precisely the same definition as in case of finite sets. The statement that real numbers can't be mapped one-to-one with natural numbers is exactly like saying that you can't put ten guests in nine rooms, such that no two guests share a room.)
@@MikeRosoftJH weird to see a 2 weeks reply to a 2 years comment.
Nevertheless, you nailed it. Thank you so much and congratst.
@@rob_olmstead What's "weird" about it? I see, the word "weird" is being used increasingly for so many diverse situations. Not talking about you, but clearly notice the decrease in vocabulary of folks over the generations.
@@ranjittyagi9354 better phrasing would be "unusual", wouldn't it? Furthermore, I'm not entirely fluent.
@@rob_olmstead Correct. No, I couldn't detect any flaw in your writing. However, I feel a bit sad when folks use fewer words than older generations used to for more "types" of situations. I'm assuming you're not from India. When I point this out to Indians, the most common answer I get is : well, it's slang and slang is commonplace now. And to me, it feels like murder of a language . 😆 I'm nearly 46, how old are you and where from? I'd delete my comments after your reply. Cheers, buddy!
idk why 5 people disliked this. super helpful for understanding the core concepts
Nobody disliked
@@LC19. youtube played me. I posted this a year ago before they got rid of the dislike count.
Eddie is a great teacher... god bless
First time I can watch an educational video twice without feeling bored.
Do you think it would be possible to match up all whole numbers to all real numbers by say starting at 0.000.....001 for the real numbers and on the opposite end; the whole numbers 10000..... So like a 1 with an unending line of 0's. Then the way we count them we could say and 0.000....1 to the real number, and add 100000... to the whole, but when we go from 900000... to 1000000... we shrink it by a tenth. With that we could get to 1 at the same step as the real numbers, right? So we can match up real numbers to whole?
Give that man a trophy !
I love him
Beautiful
Great video!Fairly informative!And as always,I love your exuberance and the way you interact with the students.
Sir you are wonderful!
Please continue uploading many lectures
5:14 really puts it in perspective, you can have two sets of infinite lists and not one will repeat
amazing
Love you passion for teaching! :)
damn he is great
Amazingly intuitive explanation!
this was awesome
thanks for this, ive been confused and behind on lectures for like 2 weeks now because i couldnt understand this concept
What assumptions do we have in saying that the list is complete? Lets reduce this setup to binary representation where the only elements used are 0 or 1. Does a complete list represent all the possible combinations that we can think of using these digits? For example, lets make a list of all the possible combinations with 3 place values; xxx
000
001
010
011
100
101
110
111
This is not a list of values, it is a list of possible combinations using 2 elements. If we construct a square matrix with any 3 items on this list, then the altered diagonal will not represent any item within the matrix. Spose that we arbitrarily define each item such that there are only 3 values represented by the list.
A = 000 = 011 = 101
B = 001 = 010 = 100
C = 110 = 111
Does a complete list only represent all the unique values possible, is it 1to1 and onto (a bijection)? Why does this matter?
A: 000
B: 010
C: 111
our altered diagonal is 100 = B, IS contained in our list.
A given decimal expansion between 0 and 1, is not necessarily represented by a single, unique decimal expansion.
3/10 = .3000... = .2999...
If we keep this in binary: 1/2 = .100... = .0111...; and i can think of an infinite number of duplications.
So I ask, what does a "complete list" look like? Do we not allow the list to have duplicate representations of value; or, are we just making a complete list of possible combinations without regard to any value.
I have the same question with U.
Well, sure: you need to be careful when constructing the diagonal number. But that's just a technical detail (the only numbers with two different decimal expansions are those whose decimal expansion ends with infinitely many digits 0, which have an alternate expansion ending with infinitely many digits 9; and all such numbers are rational, and so there's only countably many of these). My favorite way of constructing a diagonal number is by taking the n-th digit of the n-th number in the sequence, and adding 5, wrapping around zero if necessary. (Let a(n) be any infinite sequence of real numbers. Let d(n) be, for all natural numbers, the n-th digit of a(n). Then d'(n)=d(n)+5 mod 10; and the diagonal number is the real number whose decimal expansion is d'(n).) Now it can be seen that not just the decimal expansion of the diagonal number is different from every real number in the sequence, but also that their values differ by at least 3*10^-n. (Alternately, you can just use a mapping which doesn't use the digits 0 and 9 - for example, map digit 5 to digit 6, and all other digits to 5.)
I don't get how the derived decimals are considered "uncountable". They are literally being counted in the video. Also when you are writing the rows and numbering them (i.e. the counting them) every new row is also a number that did not exist previously in the list. How is it we can we even use a comparison operator when they are both infinite? There is some core concept I am missing here.
Construction matters. Imagine that an inclusive list is actually inclusive. Imagine that if you really could write down all the rational numbers, would such a list be square in form, as it must be in order for diagonalization to produce a valid row covering each entry once. Imagine that time is related to distance so that the notion of producing such a creature - apriori - out of a bag and then processing to wax theoretic, wouldn't really be possible 😮
nicu
Great explanation! very cool.
The first number was made by adding one to the diagonal and the next by adding -1. What about he next new numbers? There are only 9 digits you can add to or subtract from to get new numbers. So what do you do to get an infinite number of new numbers?
In fact, for any infinite sequence of real numbers (function from natural numbers to real numbers) there are uncountably many real numbers which the sequence doesn't cover. This can be seen from the construction: the video gives one specific way of constructing the diagonal number. But it can be seen that for any decimal position there are at least 7 different digits to choose from: for n-th decimal position you can't pick the digit which n-th number has at the same position, and you may want to avoid choosing digits 0 and 9 in order to avoid the problem with numbers which have two different decimal expansions (such as 0.1000... = 0.0999...). This yields uncountably many diagonal numbers (you can re-interpret the choices - 7 for every decimal position - as base-7 representations of real numbers), and none of these appear in the sequence.
I wish you were my professor
good stuff proof
why does he have a whiteboard on a whiteboard?
My man here asking the real question.
recursive whiteboard
The lecture is still great!
And I still hate this very piece of math! It was boiling my blood at school, it is was causing me pain in university, and it still hurt me now. Why can't I do pretty much the same for the 'countable' infinity of whole numbers? Except, of course, the diagonal would be heading to the left, ensuring that new number is completely different from 1st number (which is 1), 2nd number (2), 3rd number (3), and so on and so forth.
Brilliant
Can do the same diagonalisation on the integers?
No, because any natural number is finite in magnitude, and therefore has finitely many digits (and that's by definition: a set is finite, if its number of elements is equal to some natural number: it's an empty set, or it has exactly one element, or it has exactly two elements, or ... and so on). If you apply the diagonal procedure to a sequence of all natural numbers, you get a sequence which has infinitely many non-zero digits, and that doesn't represent any natural number.
this video was like watching an awesome magic trick!!!!
This RUclips channel is basically all of Numberphile in a school, taught by a single teacher
This makes sense and you explain it very well, but I can't help but wonder - couldn't you list the real numbers diagonally as you can with the fractions?
So you'd start with the following infinite table to begin with:
Row 1: 0.1, 0.01, 0.001, 0.0001, 0.00001...
Row 2: 0.2, 0.02, 0.002, 0.0002, 0.00002...
Row 3: 0.3, 0.03, 0.003, 0.0003, 0.00003...
Row 4: 0.4, 0.04, 0.004, 0.0004, 0.00004...
...
Row 10: 0.10, 0.010, 0.0010, 0.00010, 0.000010...
Row 11: 0.11, 0.011, 0.0011, 0.00011, 0.000011...
Row 12: 0.12, 0.012, 0.0012, 0.00012, 0.000012...
...
Row 340902: 0.340902, 0.0340902, 0.00340902...
Row 340903: 0.340903, 0.0340903, 0.00340903...
...
Row 20167398019374678093018390565748765: 0.20167398019374678093018390565748765...
(ad infinitum)
You could then "list" the numbers diagonally, 1-1. So, in this example the list would be:
1: 0.1
2: 0.01
3: 0.2
4: 0.3
5: 0.02
6: 0.001
7: 0.0001
8: 0.002
9: 0.03
10: 0.4
11: 0.04
12: 0.003
13: 0.0002
14: 0.00001
and so on...
It's countable, right? At least in the same sense that the infinite list of fractions can be counted...?
Now take that final list you made, and apply the same argument he made in the video to it. You'll still get a number that, by definition, isn't anywhere in that list.
If you try do the same thing with the infinite list of fractions, you'll see there's no way to construct a number that won't be in the infinite list of all possible fractions.
Yerren v. St. Annaland I’m not sure that’s true. Take item 14 on the list, 0.00001. Raise each digit by 1 and you get 0.11112, which would eventually appear on a diagonal list of all possible 0.x.
I totally understand the argument, but all it seems to really be proving is that you can always add another number to an infinite list. By definition, an infinite list (or indeed an infinite anything) can never be completed, that’s why it’s infinite.
This argument, by contrast, requires the assumption that the list is somehow complete, and yet there is a way to find a number that isn’t on the list. However, if the list was complete, it wouldn’t be infinite. So, of course it will always be possible to find a new number.
It’s an interesting video, but ultimately the analogous concept of infinity is not compatible with a digital counting system, hence the problem.
@@will3music
So, for example, let's say we list out the positive and negative integers as follows:
0
1
-1
2
-2
...
Can you find any rule to generate a positive or negative integer that isn't on that list?
You'll find it's not possible, but it is possible in the case of the Real numbers. Thus, there must be something different about the Reals. (Thats that there's "more" of them). That's really all this video is saying.
Your list of lists will not cover any number with an infinite decimal expansion. It won't cover any irrational number, like pi or the square root of 2. It won't even cover all rational numbers, such as 1/3=0.333... . Because every number on your list is rational, and because there are countably many rational numbers (as proven by Georg Cantor), so is your list countably infinite.
What about 1?
This is beyond My understanding
Given all rational number are real and we have sq rt 2. Hence real larger … qed.
Problem of ladder is that it seems artificial and given rearrangement of infinity series give you a different number, infinity series laddering …
fantastic video!! Thank you so much for your explanation! very helpful ;)
Is the set of real numbers from (.1,1) countable. Can I match .1 to 1, .2 to 2, ... .11 to 11... Can’t I match any decimal of any length from .1 to .99... to an integer based on the decimal x 10 to the power on n (n being the number of digits in the decimal). Does that not make the set (.1,1) countable? I understand that (0,1) is uncountable, because of the infinite number of .000... that can proceed any small decimal, but what about taking zero out of the set.
Your scheme only covers numbers with finitely many digits after the decimal point. All such numbers are rational, and that there are countably many rational numbers is a well-known result. At what position does a number like 1/3, or √2/2, or π-3 appear? (And what do you mean by "infinite number of .000... that can proceed any small decimal"? There's no such number as 0.000...1. That is, unless you mean a limit of the infinite sequence 0.1, 0.01, 0.0001, ...; but that limit is just plain old 0.)
And it's fairly trivial to find a one-to-one mapping between intervals of a different length. So if one is uncountable, the other must be uncountable as well.
thanksssssssssssss a lot great explanation my brain can finally understand infinity
🤘
Sooo where can one use this piece of math? Surely somewhere in informatics, can someone make an example?
7:11 please turn on captions.It says Khan Academy for can't count it.🤣😂
(｡☬０☬｡)
🤣😂🤣
Mr. Woo, why does it have to be a diagonal? What if I construct my 'unlistable' number the same way you did, but on the first vertical line (2,7,7,2,0......) instead of on the diagonal? Isn't it the same thing, namely that it won't be on list either?
I mean, what if, as is inevitably the case, you have ten real numbers that all have a different first digit? You wouldn't even have a means of saying what the first digit is.
@@eggynack i'm trying to understand, since math is not my field at all. So my hypothesis is there to be confuted..but in an easy way that i can understand. I'm sure your counter-example is fit, but I still have a hard time
@@biopolis7 Wait, I think I misread you. Do you mean the first digit of our new number is distinct from the first digit of every number, or that the first digit of our new number is distinct from the first digit of the first number, the second digit is distinct from the first digit of the second number, and so on? Either can be shown to not work pretty easily. I think you mean the latter, so I'll just build that.
.1
.23222...
.111
.1111
.11111
And so on. The first digit of the first number is a 1, so the first digit of our new number is a 3. The first digit of the second number is a 2, so the second digit will be a 3. The first digit of the third number is a 1, so the third digit is a 2. And, y'know, the rest are all 2's. So, the new number is .23222..., which shows up in the second position on our list.
This was awesome Thank you!!
Thank you! You made this so clear :) Im reading analysis this semester :)
nice explaination, been a hard one to grasp. i still don't agree as it is just rhetoric for something that doesn't make sense. sooner or later assuming infinity you would hit every number you can create on a diagonal line anywhere on the list.
like 1,2,3-2,3,4-3,4,5 gives you 2,4,6 which exists in the list assuming it is infinite, just further down.
to me it is a strawman argument that doesn't hold and haven't been able to find a way or explanation that can show/teach me why it isn't.
I'm not sure why you think it is a strawman argument. Perhaps if I understand your objection better, I can explain what's happening in the argument. I think that most mathematicians do tend to gloss over a lot of details when presenting the diagonal argument! Usually, glossing over details doesn't cause an issue, but when the result is so counter-intuitive, often people will try to pick apart the argument. So many people will notice "gaps" that aren't really problems with the argument, but rather are just details that have been skipped over in the presentation.
The formal definition of "countable" is that there exists a one-to-one matching with the natural numbers. A "list" is just a metaphor for that one-to-one matching, since the number that is matched with 1 can be considered the 1st number on the list, the number that is matched with 2 can be considered the 2nd number on the list, etc.
There is something to point out about these lists - even though the list is infinite, each number must appear in a _finite_ position on the list. This is because each natural number is finite in value, even if there are infinitely many of them. So, if the new diagonal number were on the list, it would have to be in some finite position n. (It would have to be matched with a natural number n.) But, by construction, the diagonal number differs from the nth number on the list in the nth decimal place, so the diagonal number cannot be the nth number. This shows the diagonal number cannot be on the list in _any_ finite position, showing it can't be on the list at all. (Since, by what countable means, it has to be in a finite position.)
If this doesn't address your concerns, let me know, and I can try to figure out what is bothering you about the argument.
Couldn't you just do the same for the counting numbers if you said that each number has an infinite amount of zeros before the first digit that shows up and you could just add one diagonally moving from right to left? Or start from the "last" and follow the same algorithm you've created backwards? Sure you can't do that because the list is infinite but by the same logic you can't ever get to the end of your irrational number either to make a new number for your list.
I know this is an old comment but I had to reply!
Adding an infinite amount of zeros to the left of the digit doesn't change the value of the number. In fact, EVERY number has an infinite amount of zeros in both directions. This is why we learned to "bring down" a zero when doing long division. Same thing. Whether's 1, 01, 001, 0000000001, it's all equal to 1.
This isn't so with non-zero values in a decimal. 3.9 does not equal 3.99 does not equal 3.999. All of these are different, and we could continue to add nines for as long as we want and we will have a next number each time, approaching the limit of 4 (until we have an infinite number of 9s, in which case this would be equal to 4, but that's another proof for another day).
You can always add new digits to a decimal to create a new number, and you can use Cantor's method in the video above to create a new decimal that will not be on your list even when each real number has an infinite number of place values. It doesn't work with zeros going in the other way. Although, you may be interested to know that there ARE numbers with an infinite amount of digits going off to the left. They're called p-adic numbers, and they're fascinating.
I hope this helps!
this is so cool
you are my best asmr
Hey Eddie, love your vids, you should wear more fitting shirts though!! This shirt is like sooo big for you
...
Thank you, was struggling with this. :)
[Edited by replacing entirity. (I should have ruminated for another 24 hours before posting.)]
(I also commented [with] a formal proof under {CvalbBGhmW4}.)
-- Can skip --
Consider the case that the list of all the numbers is a finite list - say all the numbers between 0 (inclusive) and 1 (non-inclusive), to 10 decimal places. . Suppose first that this list is in order.
We generate a diagonal number… perhaps by adding 3 to each given digit - getting (offhand) 0.3333333332. . Is this number *not* in our complete list? Of course not - of course it is in the complete list.
Suppose, then, that we order the complete list randomly. . Suppose that (arbitrarily) we end up with, say, 0.7529901438. . Is this number in our complete list? . Yes, it is… because the list is complete.
There is a trick here - a sleight of hand. . The trick is that… for each 1 digit that we write, there are 10 numbers in the list. . Consider, for instance, a complete list of 4-digit numbers. . Our diagonal system processes 4 digits - one digit per line - thus covering 4 lines in the list. . However, there are 10,000 numbers in the list. . We can indeed guarantee that there is no match for our number, among the first four numbers, but we can be certain that there will be a match for our number among the remaining 9996 numbers (since the list is complete). . Similarly… if we process 12 digits with our diagonal system, we can guarantee that the number we generate will not match any of the first 12 in the list… but it will match one of the remaining (10^12)-12 = 999,999,999,988 numbers. . (Even if we reduce the base to the smallest possible value - 2 - binary… we still have, for instance, 2^12 = 8192 items in our list, against 12 eliminated. . The limit case is (i.e. would be) base 1; in this case, we can cover the whole (complete) list using our diagonal system, but now it is hypothetical. . [I was going to say that this was not a meaningful number system, but perhaps it reduces to just counting. . Either way, anything we write is going to match an item in the list, because there is only 1 numeral that we can write.] )
The argument presented in the video is a “reductio ad absurdum” [proving the opposite of P by showing that P entails something inconsistent]. . The idea in this case is to assume that the list is complete, and then show that it is possible to generate a number that is not in the list. . I have shown that this argument fails, in the case of a finite list. . What about an infinite list?
People are funny with the term “infinite” or “infinity”. . Suppose that we have two sequences [a list generated by an equation]. . [I shall write “_” for subscript.] . One is just j_n = n; the other is k_n = 2n. . j and k range from 1 to… they just go on and on, getting bigger and bigger. . We talk about the limit as j (or k) “approaches infinity”. . (Remember that infinity is, not a number, but {the concept of getting bigger and bigger, or having more and more, without limit}.) . It is true that the sequence j (or k) approaches infinity as j (or k) approaches infinity. . The *limits* of these two are the same. . Nonetheless, k_n = 2 * j_n, for any value of n - be it 5 or 10^10,000 or 10^10^10,000.
Actually, our list of all the Real numbers is infinitely long in a stronger sense; whereas the list of positive integers has a 1:1 correspondence with an attempt to count them… the list of Real numbers is just a sea of numbers - most of which are infinite in length just by themselves.
Nonetheless, the above flaw in the diagonal system still applies; the diagonal system is not apt to generate a number that is guaranteed to not be in the (complete) list, because it has this flaw of having less than a 1:1 correspondence with items in the list. . (Of course, the failure of this particular proof does not prove that its tenet is false. . Conversely, the construct in question is a complete list of all the Real numbers, and we should question any purported rigorous proof that it is not *because the concept is not incoherent*.)
-- Can skip -- END
[I proved that the diagonal proof is flawed.]
Unfortunately, all of the above is a red herring… because we are concerned, not with whether or not the list is *complete*, but whether or not it is *countable*.
Of course, to count an infinite list of anything is not *feasible in real life* (even if one lives forever). . However, that is merely an observation about the concept of “infinite”… and the real world.
Suppose that we define “countable” to mean that each and every item in the list must have a unique identifier. . We can achieve that by making each number its own identifier (so to speak). . √2 - √2 - check! . √2-1 - √2-1 - check! . 15.397029…[whatever it is] - 15.397029…[whatever it is] - check! . Overall… any number that we can identify is countable… and conversely any number that is not countable is beyond our reach anyway. . Given this… the concept of {a complete, infinite, countable list} is perfectly coherent. . …Given enough time.
nicely explained!
This is awesome. Why are you not my math teacher?
Help! I am confused here: can't we use the same argument on natural numbers to prove the set N is uncountable (which obviously is wrong): since the natural numbers can increase indefinitely, we have unlimited number of digits to work with (just like in this case), and therefore we can always construct a new number not in the set. As a matter of fact, after the part of 0., the digits of the real numbers would resemble natural numbers right?
The term countable may not fully explain the concept. I think 'sortable' is a better term. The integers are sortable in the sense that you can start from 0 towards infinity an always be guaranteed you didn't skip a number in the middle. For the real numbers, you cannot guarantee that sorted list... There is always another number in between any other number that you didn't count.
@@ricosrealm you can make all numbers between 0 and 1 in order, it's just a bit weird:
0.0
0.1
0.2
...
0.9
0.01
0.11
0.21
...
0.91
0.02
0.12
0.22
...
You basicly write it down like the Natural numbers but mirrored after the decimal point, so you come across EVERY number between 0 and 1.
So all real numbers between 0 and 1 are countble but if you do all real numbers you're right because you will have no BEGINNING and no END because you an infinite amount of infinities. (which you can't count)
@@LC19. Apology for my poor English, but it seems like your way of listing all numbers between 0 and 1 doesn’t really list all numbers between 0 and 1. We can observe that every number you listed has a finite number of digits, and irrational numbers (infinite non-repeating decimals) like sqrt(2)/2 cannot be on this list.
You might want to argue that since the list goes on forever, an infinite non-repeating decimal can be in the list. But again, because of the fact you construct this list in such way, I can always find out the exact number of digits given the index of a number in the list, which in not infinite. So unfortunately this list cannot actually cover all real numbers between 0 and 1.
As I understand, while arbitratily large, natural numbers are finite numbers by definition so while you can in principle go out arbitrarily far, you cannot go out infinitely far. This is in contrast to a number like 1/3 (repeating 3's in decimal places --> 0.3333333....) where it's "reflection" across the decimal place is ...3333333333333.0 (3's going out to infinity) would not be a natural number. This is why I think you can qualify 0-1 decimals and infinitely large numbers as uncountable, but not natural numbers because, while you can always find a larger natural number, the natural numbers can be listed completely (no contradiction with a complete list), and the others cannot only be listed partially (the idea of a complete list leads to a contradiction).
Thank you for a "rational" discussion of the subject. I'm still not grasping the concept of a countable infinity. If a list goes on forever...
In a countable infinity you can find a next step.
Start from 1 and you can keep going, 2, 3, 4, etc. There are no integers between 1 and 2. so 2 is definitely the "next step". So it's countable, despite it being an infinite list.
But with uncountable infinity, like numbers between 0 and 1, there's no "next step", there will always be a smaller number between every 2 numbers you pick. so you can never even list the first few numbers of an uncountable infinity. There's literally an infinite number of numbers in-between any 2 numbers you choose. Making it uncountable.
Grasp no more. CDA is wrong and I have the video below to prove it.
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I predict that my discovery will withstand the combined challenges
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Please leave any feedback in the COMMENT section of the video.
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Aria A thank you,now I think I understand it.
Exam in 3 hours, lessggo bois
Why are all these illustrative mathematicians ignoring that real numbers also encompass negative numbers? 🤔
Cantors a G
lol subscribed
I means if he only has 9 numbers after the decimal you can’t go on forever without finding the number he wrote.
Sorry this doesn’t make sense to me. If a number goes on forever and we can count it that doesn’t make the infinity countable it makes the number infinitely countable.
Can’t you do the same thing for the counting numbers, just going to the left
Don't wanna like to mess up the 420 likes!
wizardry
If this is a college class why don't these guys know the difference between a racional number and a real number? Smh
I'm not getting why this proves that there are more numbers between 0 and 1 than there are natural numbers. Sure, one may find a decimal number that is not in the list, but at the same time we may just come up with another natural number to index the new item.
For example, consider the list below:
1 - 0.02323
2 - 0.06909
3 - 0.06910
5 - 0.07000
6 - 0.07340
Sure a new number that is not in the list may be produced,, such as 0;10000. This number may be added to the list, and to index it we may sum 1 to the counting numbers.
1 - 0.02323
2 - 0.06909
3 - 0.06910
5 - 0.07000
6 - 0.07340
7 - 0.10000
Considering that both natural numbers and the decimal numbers between 0 and 1 could go on infinitely, we may assume that we may keep adding new items to the list, and properly linking it to a new counting number resulting from the sum of 1 to the previous counting number.
What am I missing?
yeah man this is what i'm struggling to understand too. Yes, we created a number that isn't in the list, but we can just add it to the list and then +1 the corresponding index. I don't get it, I've read so many comments and explanations but it still doesn't make sense in my head.
@@inx1819 I've seen other explanations but I still can't tell what I am missing here. Hopefully some day I will manage to understand it.
When looking at 2 different sets like that I understand that one progresses faster than the other, but if they BOTH go on infinitely then how is it relevant to try and compare or to say one is "bigger" than the other. No matter what your stopping point is on either set, what u have written down in BOTH sets with both equal to 0% of infinity so how can one be larger than the other? I'm just not convinced that you can fit infinity into a contained "set". Wouldn't a "set" imply both a beginning and an end?
racer3d1 racer3d1 it means that one is countable and the other is uncountable.Now what is countable and uncountable?Well,countable always has a next step,like positive integers-infinitely many of them,but 1 is the point to start and the next step is 2,so you can still count.But with real numbers,you don't know where to start and how much to proceed while counting.Its practically impossible.This is what I have understood,but if it's not the correct explanation,then I'm sorry.
I get what your saying, it just hurts the brain to think about it lol. I'm still not fully sold on a "countable" infinity, its like the ultimate oxymoron. for example: if the argument is there are infinite real numbers between lets say 1 and 2 and therefore, real numbers cant be in a countable set bc you couldn't even get past "2" , then that IMO is proof that its equally as impossible as counting regular infinity. If 1-to-2 has more numbers then you can count, then regular infinity itself is not "countable" either and therefore IMO can't be contained in a "set". Of course this is just my opinion, I only made it to pre-calculus in college so I'm sure most people debating on here are beyond my understanding/education of it but I just can't seem to get past putting infinity in a countable "set"
@@lifeofphyraprun7601 I get what you are saying with the "don't know where to start" but couldn't technically the same be said for any set of infinite numbers? For example if u were going to start counting integers, isn't a starting point just arbitrary? There is no rule that says you have to count things in any particular "order" , your starting point in this case could literally be ANY integer from negative infinity to positive infinity..... so how would that be any different from saying we can't find a starting point from 0 to 1 bc how many zeros can we inject before the .1 0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 or a quintillion more zeros perhaps? How is this any different, than saying with integers we are going to create a starting point at some number that is so big that we can't even comprehend it? There are never-ending "starting points" even in integers..... I guess I just can't follow the difference between that and real numbers
Woo....i have a serious problem with you! 😑😒
I don't know why, but this bugs me. I'm sure I'm bathing in ignorance, but....(before I start, be warned I'm not sure how to unpack this and I'm no master with words... so bare with me) I think the biggest issue I have with this is that I believe the idea of infinity is misrepresented.
Infinity is uncountable. I understand as a thought experiment you could theoretically count an infinite number if you had infinite time, but I don't think that's actually true. When there is an infinite of any size, it means it is impossible to get to the end.
So, this uncountable infinity doesn't make sense to me because you claim this number doesn't exist in the first set, but that number could never be established, because it could never encompass the entire list.
We tend to want to think of infinities like we do normal numbers and that is where I think we tend to get this wrong. I don't believe infinites have a size or amount, instead I think its more like they have boundaries, but the contents are all a shapeless void. It is never possible in any thought experiment to count an infinity (I have a feeling someone will say "well you could count if ..." and I would say.... no you can't... that's why it's infinite, if you could it would be finite) and pairing two infinities together doesn't work (counting to infinity for an infinite amount of time) . It's like north of north.
I know that this is rambling, so I apologize. I'm at work and I wanted to vomit this out before I lost it.
Whoever gets here... Thank you for your time and reading this far. If anyone wants to educate me I'm all ears and appreciative of the information.
But how is this proofable? At some point there must be a countable infinite match with a uncountable infinte, or else its not infinitely to begin with.
There's no such thing as uncountable infinities. It's mathematicians not realizing that infinity is only a limit concept. Infinite sets don't actually exist. What exists are potentially infinite sets, meaning that the sets have no limits or bounds to how big they can become. Both the set of natural numbers and real numbers are exactly the same. They have no limits. Saying that one's "no limit" is bigger than the other's is an absurdity. I can actually map every decimal point to every natural number. For example, .1 --> 1, .3 --> 3, .001 --> 100. For every decimal in the real numbers, there exists a corresponding natural number. Real numbers that have an infinite number of decimal places like Pi are never reached because they have no limits to how precise they can become. Though, I can say the same about natural numbers that have an infinite number of digits. Neither will ever be reached!
I'm not going to say that what I am saying here is definitive because there are a lot of smarter people than me who actually buy into this set theory nonsense. However, I think I'm probably right considering how dumb it sounds. Experts have been wrong on many things in the past so I think the safe bet is that they're wrong here.
Infinities have their own classification and we shouldn't conflate one infinity with another as if they are the same infinity.
"Countable infinity" is a contradiction in terms which violates the concept of infinity. If something has a measurable value aka value limit then it is not an infinity. It is a relative measurable value set.
What exactly is your objection? Just the terminology? A set is countably infinite, if it can be mapped one-to-one with natural numbers; in other words, if its elements can be enumerated by an infinite sequence, such that any element appears at a finite position. For example, integers are countable, because they can be enumerated by the following sequence: 0, -1, 1, -2, 2, -3, 3, ... - any integer n appears at a position no greater than 2*|n|+1. The same can be proven for rational numbers. But real numbers are uncountable - given any function from natural numbers to real numbers, there exists a real number (and, by extension of the proof, uncountably many real numbers) which the function doesn't cover.
"forever" and "still missing" makes no sense.
I am still searching for a mathematician that can deal with infinity. Apparently none can. They just can't, don't ask me why.
(tip: adding something to something that "goes forever" makes no difference. Actually, might as well think you can't: it's like to put something in a moving thing. You need it to be static. Does this analogy helps?)
Just because you don't understand doesn't mean no one else does LOL. Mathematicians and physicists are working with infinities all the time, there's even infinities that "expand faster" than other infinities (n^x vs n^n). Just because you shake your head and refuse to follow the logic doesnt mean its not true!
In a topic such as mathematics, if no one agrees with you, you should re-evaluate your view.
Real is equal to natural numbers, did his numbers do all the sqrts(2.36987..), sqrts(3.874512...)no.no.no. All the numbers have different names , N,Q,R same thing all reals.
This is so unintuitive😭😭
Real numbers are made of only 9 digits.
Surely you can go through all possible combinations of digits in each decimal point (say 'bit') until you get all numbers. No?
What if I say:
0.0
0.1
0.2 and so on...
0.9
Now I can move on to the next 'bit'
0.01 Notice how "ten" (flipped) moves the zero to the front because, for decimals, significant zeros are in front.
Now we count (flipped):
0.11 "eleven"
0.21 "twelve"
0.31 "thirteen"
0.41
0.51...
0.81
0.91
0.02 "twenty" and continue counting 'flipped':
0.12 "twenty one"
0.22 "twenty two" and so on
You'll eventually get to:
0.99 ninety nine (note: it's written backwards)
And then one hundred flipped:
0.001
0.101
0.201 "hundred and two"
I will eventually exhaust all numbers, no?
I started with one bit and exhausted all possibilities and moved to two then three, making sure I add zero in front because adding it at the end creates redundancies (0.1= 0.10= 0.100). I will eventually exhaust all possible combinations with three bits then 4 and so on to infinity.
What real number won't be on the list???? I am only using 9-digits after all. Surely, if I keep adding bits, I'll exhaust all possible combinations and thus all real numbers between 0 and 1, no?
And if you think about it, natural numbers are made of only nine digits and so are real numbers. How can it be that there are combinations of numbers after the decimal that do not exist (in a different form) before the decimal, just flipped?
Maybe I'm missing something 😂😂 but I've been struggling to get this diagonalisation thing. Like... The reason it's not on the list is because you haven't gotten there yet😅.
Please help me finally understand this conundrum🙈.
You have to keep in mind that the number of digits is also infinite, which means the length of the diagonal will also be infinite. Therefore, there are also going to be an infinite number of numbers that you are missing.
@@crystalexia-
That's because your list is finite. With an infinite list, all numbers are there. If I move down every decimal point exhausting all the possible combinations of digits for 5 decimal points, then, 6 and so on I will eventually run through all numbers and even get to the number that is supposedly not on the list.
@@crystalexia- but couldn't you do the same thing for natural numbers??
..36363861528636
..27363836363648
..47363736363737
..37363736363736
All infinite numbers use diagonalisation to find a number that is not on the list.
The problem with this listing method is that it is Random and has no pattern so to say diagonalisation will find a number that is _not on the list_ doesn't make sense. Why did you start by listing infinite numbers? Why don't you start with finite numbers like 0.1 or 0.12?
You don't do that with natural numbers but you do with real numbers between 0 and 1. The whole thing just seems like unnecessary confusion to me.