Cantor's Infinities - Professor Raymond Flood

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Although many people contributed to the study of infinity over the centuries it was Georg Cantor in the nineteenth century who established its modern development. Cantor created modern set theory and established the importance of one-to-one correspondence between sets. For example he showed that the set of all integers can be put into one-to-one correspondence with the set of all fractions and so these two sets have the same infinity. But he also proved the remarkable result that there are infinitely many infinities, all of different sizes.
The transcript and downloadable versions of the lecture are available from the Gresham College website: www.gresham.ac.uk/lectures-and...
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1 июн 2024

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Комментарии : 166

@russelshackleford8169 4 года назад

It's funny to read the comments. People often think they disagree with what Cantor was saying when in reality they are agreeing with him. It's hard topic to discuss.

@gaulindidier5995 4 года назад

Best explanation of Hilbert's Hotel I've seen online.

@peeyushawasthi9843 3 года назад

A great mathematician resting in eternity.

@migfed 9 лет назад

Great lectures and great lecturers as well.

@gregt4202 6 лет назад

Interesting lecture. I had just viewed a talk by Sir Roger Penrose on the possible scale invariance of the universe. I'll have to digest both for some time, but I see that they may relate to each other in ways that I hadn't thought about in the past. Thanks.

@amiralozse1781 9 лет назад

wow, great exercise in logic and great lecture!! 29:01 Create a list of all (real) numbers from 0 to 1, then find out there are still an infinite number of numbers missing ..... first knot in my brain! add all the missing numbers to the first list and .... still there are an infinite number of numbers missing - secont knot or rather a short circuit in my brain!! only solution: ALL numbers between 0 and 1 are not coutable, i.e. cant be put into a list - complete WOW! effect ... and you made my day, thanks!

@Chris-5318 9 лет назад

Amira Lozse If you add all the missing numbers then there will be none missing. If you merely add in another uncountable infinity (but not all) of the missing numbers, then you will still be missing some numbers. The number of missing numbers is less than or equal to the cardinality of the continuum. It could be that only 10 are missed for example.

@antimoniolantanio4597 5 лет назад

i read somewhere that this set is an uncountable infinity. but by the method of construction of this set : first you get the points 0,1 then 1/3 and 2/3, then 1/9, 2/9, 7/9, 8/9... so if you get all this points according to the order of occurence, you have a set : { 0, 1, 1/3, 2/3, 1/9, 2/9, 7/9, 8/9 ...}. by this process all points will appear in this set, and this set can be mapped into the set of natural numbers ( which is a countable set). therefore cantors set is countable too. i cant find my mistake . where does he lies?

@Dyslexic-Artist-Theory-on-Time 6 лет назад

We have an infinity of possibilities and opportunities based on geometry! And this is represented in mathematics!

@mrautistic2580 9 лет назад

Very very interesting...

@user-ec9rc8jj8q 6 лет назад

장학사, 교육감님 오시니까, 우리 열심히 환경미화 하자!

@ghirardellichocolate201 3 года назад

So the gaps between numbers can always be the jumps. 1 to 3, 2 to 4, 3 to 5, 4 to 6...1 to 4, 2 to 1, 3 to 5, 4 to 3.... To count how many ways we can distribute numbers is infinite.

@Kalumbatsch 8 лет назад

At 28:37 I thought "those are not infinitesimals" before I realized that he said "infinite decimals" :D

@markzender4386 8 лет назад

video errata: typo at 31:48 it should say b2 not equal to 5 say 7, etc 34:25 speech typo 'the real numbers are countable'

@euphratesjehan 4 года назад

Correct good looking out

@niveditham1692 4 года назад

That was said so because of example besides it, where u said it in generalised form which is also true for the proof

@niveditham1692 4 года назад

And it was said that it cannot be put into a list that the real numbers are countable

@ksmyth999 5 лет назад

The last part where power sets are introduced is probably quite confusing for some people. Would it not be better to introduce the concept earlier in a finite setting and introduce the idea of countability within a finite range say M? Then it is easy to show that there would be 2(power)M - (M+1) uncountable elements within range M. You would then need to extend the idea to infinite sets. I have two further observationes/questions. (1) The diagonal argument works on strings not explicitly numbers. Admittedly the strings are defined to be partial sums and therefore have a value. But the diagonal argument does not use this value and is really concerned with combinatorics. Is there not a further step missing from the proof to show that, at least, some of the innumerable strings have unique values. For example if one claimed the natural numbers at the limit of aleph-null are modular then the whole argument collapses. One would need to reference a proof that aleph-null +1 = aleph-null and not 0. I know this is always assumed but how is it proved? (Although I studied maths my career was in software engineering. A computer uses a modular range mainly because it's the simplest and most elegant solution). (2) Cantor's prove emplies there is no solution for n in the equation 2(power)n = aleph-null. Is this not a problem in terms of the completeness of the number system? Surely completeness would have been the main goal in creating the reals.

@countingfloats 5 лет назад

I can't comment on what is better or what is worse. I sort of made it easy on everybody and created what is the second best float to integer pairing. Incredibly enough, I discovered the proof of the Continuum Hypothesis by simply establishing 1-to-1 correspondence between positive decimal floats and positive decimal integers. For the last 125 years such a proof eluded the top set-theory experts of the planet. It was eventually abandoned and classified either as impossible or undecidable. Well, no more. Sit back and enjoy one of the most trivial and elegant proofs in set theory. It is published on RU-vid with the title : “Proving the Continuum Hypothesis”. I would appreciate if you watched and commented on this video. Your input will be given full credit. Tamas Varhegyi

@MuffinsAPlenty 4 года назад

Hi, Kevin. You are correct about point (1). There are details missing about how decimal notation actually represents real numbers. The proofs basically come down to the theory of infinite series, which again, can all be boiled down to the Cauchy completeness of the real number system. In other words, every Cauchy sequence of real numbers converges to a real number. (A Cauchy sequence is a sequence in which the terms get arbitrarily close.) Using this fact, you can prove that: (a) all real numbers have at least one and at most two decimal representations, (b) every real number which has two decimal representations has one with infinitely many trailing 0's and one with infinitely many trailing 9's, and (c) every decimal representation actually represents a real number. Actually, the diagonal argument wasn't Cantor's first proof of the uncountability of the reals. His first proof actually made the reliance on the Cauchy completeness clearer. It went like this: Given any countable list L of real numbers, take the first two elements in the list. Name the smaller one α1 and the larger one β1. Next, look for the next two numbers in L that are in the interval (α1, β1). Take the smaller of the two and name it α2 and the larger of the two and name it β2. Repeat this process. There are three options: Option 1. The process terminates in finitely many steps, in which you cannot find at least two elements in (αn, βn) for some positive integer n. You may be able to find one or maybe none at all. If you can find one element in (αn,βn), call it η. Then (βn+η)/2 is a real number not in L (otherwise you would have been able to find another α and β). If no η exists, then (αn+βn)/2 is a real number not in L, otherwise you could have found an η. If we're not in Option 1, then you get an infinite sequence {αn} which is monotonically increasing and bounded above. Hence, as a corollary of Cauchy completeness, there is a real number α∞, which is the limit of this sequence. Similarly, there is a monotonically decreasing sequence {βn} bounded below, so we get a real number β∞ as the limit. Since every α is less than every β, α∞ ≤ β∞. Option 2. α∞ < β∞. In this case, (α∞+β∞)/2 is a real number not in L. Option 3. α∞ = β∞, which we can denote as η. Then η is not in L. If η were in L, then you would have an infinite sequence of α's before η in L. But since L is a countable, every element of L occurs in a finite position of the list. Thus, η could not actually be in L. No matter what the case, we were able to prove the existence of a real number not in L, so L is uncountable. Q.E.D. I don't know why point (2) you brought up would be a problem.

@user-ec9rc8jj8q 6 лет назад

Cantor, Our mathmatic pioneer! Many followers did not have a fear, because You.We escape the wall ,I love you. cause you give me a chance.Between the Religion and Science, That is future Spaces .your kingdom?Wrong ? or Not?God Bless You my majesty....l love ..belve or not,I betting!

@sebastianlukito6686 6 лет назад

I have a proof of continuum hypothesis, but this commentary section is too short to contain.

@countingfloats 5 лет назад

My proof of the CH does fit in the comments section... Incredibly enough, I discovered the proof of the Continuum Hypothesis by simply establishing 1-to-1 correspondence between positive decimal floats and positive decimal integers. For the last 125 years such a proof eluded the top set-theory experts of the planet. It was eventually abandoned and classified either as impossible or undecidable. Well, no more. Sit back and enjoy one of the most trivial and elegant proofs in set theory. It is published on RU-vid with the title : “Proving the Continuum Hypothesis”. I would appreciate if you watched and commented on this video. Your input will be given full credit. Tamas Varhegyi

@ikaeksen 5 лет назад

If there is just one universe the biggest number is finite.

@mrautistic2580 9 лет назад

I'm finding that the math that I perform calculations with most often can be subdivided into this mathematics combining itself with rate mathematics...it can be thought of as fluid dynamics on steroids.

@euphratesjehan 4 года назад

Regarding Grand Hotel Example 1 Why do the guests already in their rooms have to be inconvenienced by having to move up a room to accommodate 1 new arrival? Why can't the 1 new arrival be placed into the next available room without bumping all the other guests?

@MikeRosoftJH 4 года назад

Because there isn't any available room; the scenario starts with all rooms occupied. There are infinitely many rooms; and infinitely many guests, one in every room.

@user-ec9rc8jj8q 6 лет назад

1. integral: 곱셈의 형태.2. sigma: 덧셈의 형태.

@user-ec9rc8jj8q 6 лет назад

integer, fracture, even, odds, translated undevided number...-error problems-

@jchenergy 8 лет назад

Wrong start Mr Flood! What you express at 1:50 is not necessarily true. The limit of the ratio of two quantities that go sinuktaneously to zero is not always infinitely small...it the denominator goes to zero faster than the numerator , the limit can grow to a big number!

@nmarbletoe8210 7 лет назад

I think he's saying that the two numbers become infinitely small, but then we still have to see what the ratio does. Like you say, that depends.

@markzender4386 8 лет назад

lol what did he say at 26:05

@yousifucv 3 года назад

"two numbers of the form one number over another number to add it on to one number over another number, you get a fraction"

@dramawind 6 лет назад

"ideer"

@user-xc1wl7uu4l 4 года назад

Too complex proof at 44 minutes. Every element of a powerset of an infinite set can be represented as an infinite binary string 1 - we take ith element, 0 - we don't take it. Then just apply the diagonal principle by flipping bits

@MikeRosoftJH 4 года назад

This works, if the original set is countably infinite. But the proof does essentially the same: let f be a function from X to P(X) (set of all subsets of X). Let Y be the set of all elements of X, such that x is not an element of f(x). By construction Y cannot be the same set as f(x) for any x being an element of X, because Y and f(x) differ in at least one element: x is an element of f(x), if and only if x is not an element of Y.

@WarzSchoolchild 7 лет назад

Would an imaginary colony of super-intelligent computers, who only knew "Binary Notation", were aware of other notations, but thought them rather obscure and useless. Would this super-intelligent colony be impressed by "The Diagonal Argument"...???

@MuffinsAPlenty 7 лет назад

How could one possibly judge what would impress (a subjective notion) something else? If you're concerned that Cantor's diagonal argument relies on notation, then you needn't worry - it works regardless of which natural number you choose to be your base. There is a slight issue in binary and ternary (the fact that 1 = 0.111... in binary and the fact that 1 = 0.222... in ternary, which cannot be avoided because there are too few numerals to use - in base four and higher, you can avoid the numeral "0" and the largest single digit numeral to avoid this problem altogether), but one can then adjust this by showing that the collection of binary sequences (or ternary sequences) is uncountable and that the real numbers in (0,1) are in one-to-one correspondence with the collection of binary sequences (or ternary sequences). Of course, even without the binary and ternary patch for the real numbers, Cantor's diagonal argument (as is) still proves the existence of infinite sets of different cardinality in binary and ternary (just use binary or ternary sequences), which was the real shock of Cantor's work anyway. So... notation doesn't really matter for the argument.

@awhodothey 6 лет назад

Alastair Carnegie No his proofs are irrational. Not to say they are wrong, but they are not based on the same logic as arithmetic, and at times represent wholly arbitrary preferences.

@MrKmanthie 5 лет назад

Great Moose Detective sounds like a typical reactionary response for someone who has no clue about what transfinite mathematics & set theory are all about.

@ksmyth999 5 лет назад

Yes. In fact it is much more logical to use the binary system in the argument. Because you really need the power set theorem to understand the whole argument. The power set theorem is based on combinatorics. If you have n objects it is easy to show you can create 2(power)n (including the null case) combinations of these objects. If you use a number base greater than 2 then you have a set of sets with base(power)n combinations. An unnecessary complication.

@markzender4386 8 лет назад

I would like to critique cantor's theory if I may. I believe much of the confusion in the interpretation of Cantor's mathematical results lies in speaking of size with respect to infinite sets. The notion of size only makes sense with finite sets. One of the results of cantor's theory is that one can have smaller and greater infinities, i.e. you can order eternities from smaller eternities to bigger eternities. Surely this is silly. Infinity is not finite, period. If a set is endless, then all you can say about its size is that its infinite or unbounded, or that it lacks any size because it blows up. We are accustomed of thinking of size with respect to such things as finite area or finite volume for good reason. I don't think we should abandon this notion of size. Size is necessarily tied to finiteness, and is a finite concept word, and not useful when referring to infinite sets. It makes no difference if we define size using a 1-1 correspondence criterion. We know that for any given n, the set of even numbers less than or equal to n has 'fewer' terms than the set of natural numbers, since the natural numbers include both even and odd numbers. It is also known, since Galileo, that we can match up the even and natural numbers in such a way that the missing (odd number) gaps collapse, by dividing each even number by two. And so what? That doesn't mean that the two sets are the same size. On the contrary. Both sets lack size , since having size implies we can count the two sets. All we can say is that both sets are endless, and it makes no sense to talk about size. Though we can make the observation that infinite sets have some interesting 'collapsing' or interweaving properties. That being said, cantor's results do seem valid for sequential *lists*, because a list can be defined as a function from the positive integers to the list's values, e.g f(1), f(2), f(3) , f(4) , and so on. Cantor showed that the real numbers cannot be listed sequentially, not even with an 'implied list' as we have with even positive integers 2,4,6,... This can be made more precise in the language of functions. There is no function from the natural numbers to the decimals in (0,1) such that it eventually hits every decimal. There is a function from the natural numbers to the even integers that covers all the even integers. f(n) = 2*n This implies that the even numbers can be listed : 2,4,6,8 ... Therefore cantor's result that the real numbers are uncountable is equivalent to saying there is no sequence f(1),f(2),f(3) ... which includes all real numbers. To put another way, there is no way to collapse the natural numbers so that it matches up with the infinite decimals and fills up the gaps, analogous to the way we collapsed the even integer gaps on to the natural numbers. I understand mathematicians are free to define concepts as they please. I have no problem defining cardinality using one to one correspondance, and thus imposing an equivalence relation on sets which can then be ordered. Fine. But don't tell me two infinite sets have the same *size*, where size means I can count the members of the two sets and they have the same number. That leads to confusion and paradox. Two infinite sets are both endless and thus lack size. Cardinality and size are not the same thing when applied to infinite sets. There is even evidence to suggest that georg cantor may have not intended to imply two denumerable infinite sets are of the same size in the normal sense of the word. He used the word power (in german Machtigeit) to denote that two sets have a 1 -1 correspondance. Power is a different word entirely than size. There are other objections one can make. Cantor assumed that an infinite set exists in the first place. This assumption does not bother me perse. In conclusion, it is misleading to say that the even numbers E and the natural numbers N have the same size , or even worse to say that they have the same number of elements. The expression "same number "implies that the elements of the two sets can both be counted 1,2,3,4 ... n, and n is the same for the two sets. This is simply not the case for E and N, as they can't be counted. They can be matched, but matching is not the same as counting. It is better to say certain sets are 'match-able' , its elements can be matched with the set N by an appropriate function, or that the two sets are unmatch-able (no function can ever be found because if there was it would lead to a contradiction). Inserting the word 'countable' and 'uncountable' just leads to confusion, as neither N nor R are countable in the strict sense. I understand that Cantor may have thought it useful to draw an analogy between matching infinite sets and counting finite set , but as analogies go they can break down quickly if pushed too far. Counting is necessarily a finite process. It must terminate for it to be meaningful to say that someone counted something.

@allwanamar1 8 лет назад

This is true .Thank you for this precise thinking . I think Cantor didn't care about being ' philosophically' rigorous in his terminology. Thus his ideas still are meaningful if we accepted to use his own expressions and definitions.

@snatchngrab8262 7 лет назад

mark zender Defining objects is the purview of mathematicians, but ignoring axioms and theorems is not. Neither is making up unfounded premises as new axioms. Cantor was a joke, as are all that subscribe to the notions of him.

@DrSheldonLeeCooper97 6 лет назад

I actually think it makes sense to speak of size even with infinite sets. I don't see a reason why size should be tied to finite numbers. Also size with infinite sets is never used as an absolute but rather relative. Set A is smaller bigger or equal to Set B. And i actually agree that the amount of integers there are is equal to the amount of even numbers.

@Oners82 6 лет назад

Snatch n Grab What a truly vacuous remark.

@Oners82 6 лет назад

DrSheldonLeeCooper97 Agreed, I think his criticism is rather naive as size in no way implies finiteness. He also seems to muddy the water with regards to countable and uncountable infinities and makes the bizarre assertion that all infinities are uncountable which is complete and total nonsense. He tries to offer the alternative of match-able but this is no less clear than countable because both are impossible if you are naive enough to think that they refer to an actual physical process of counting or matching. All countable means is that the terms of a set have a 1-1 correspondence with the natural numbers and hence can be listed, not that they can physically be counted, and in this sense the natural numbers and even numbers are identical in size whether or not he agrees. His entire argument is nothing but a statement of "I don't understand it intuitively therefore it must be false", and arguments based upon intuition are almost always wrong. Any argument that says that there are twice as many natural numbers as even numbers is a complete misunderstanding of what infinity actually is. It is true for any FINITE set, but it is absolutely FALSE for an infinite set as both have a 1-1 correspondence with the natural numbers and hence have identical size.

@douglasstrother6584 4 года назад

"Transcendental numbers powered by Cantor's infinities" ~ Mathologer ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-3xyYs_eQTUc.html

@realbrush 8 лет назад

Maybe at the end of infinity you start at zero, iterating an infinite loop.

@ksmyth999 5 лет назад

This is a good point which I have also asked. The proof should also show that the natural numbers at the completed infinity can not be modular.

@haroldbn6816 4 года назад

Found the physicists.

@ToadRoach 8 лет назад

29:02 The real numbers being uncountable and integers countable, shame he never thought of using a mirror! Because if you were to take ANY irrational number and place a mirror on the decimal point what would you see in the mirror? Please tell me a way that you don't see its integer counterpart! If the above is true you could then perform the diagonal argument, add mirror and you get its integer counterpart too. So either the integers are also uncountable and there is a flaw in the basic principal or the real numbers are indeed countable and there is a flaw in the basic principal.

@ToadRoach 8 лет назад

***** I agree the diagonal argument works in creating new numbers, but why doesn't it work on both sides? If you think about the true nature of the diagonal argument, you are simply creating a new entry by making it different from every other entry using its corresponding decimal place, you could do the same to the other-side also. OK you say that the list would then be out of order, but the computational cost to put it in order would be at a maximum the length of the list, by the way this would also equal the computational cost of checking that one of the random number in the real number list was unique in the first place, something that you would be required to do for all. The speaker even goes to the effort of saying that the +1 rule for H.Hotel doesn't apply here, but this is more of a mistake than proof, because if it doesn't apply then that could also indicate that the ordered list was in fact finite and through the one to one pairing the real number list would also be finite. And as neither list was actually infinite, the whole proof is for nothing.

@fangming5173 7 лет назад

Toad Roach, Best argument I've read!

@Oners82 6 лет назад

Marvelous Whiskers It's a TERRIBLE argument and trivially easy to prove false.

@tamasvarhegyi8813 5 лет назад

It is the later, the floating point numbers are countable. The reals are not, since they include algorithms (such as sqrt(2), PI, and 1/3 ). Algorithms in turn cannot be counted because they cannot be defined as a set. There are an endless number of ways the number 1 can be generated using algorithms, e.g. 1/1, 2/2, 3-2, 0.2*5 and so on. But floating point numbers which are generated by algorithms can certainly be counted using the positive integer counting agents. Of course a random set of algorithms will be mindless and clumsy and land on the same floats repeatedly. But that does not make floats uncountable. Just acknowledge the first incoming hit, then ignore the rest. The essential principle is this : If you undertake counting you must have a plan and strict rules in place. For example if you are generous and allow floats to have both more than one leading and any trailing zeroes then your counting venture will stall at the very first float which is 0.1. Here is why : You would be obligated to count 0.1, 0.10, 00.1, 00.100 and so on ad-infinitum. When they finally ring the bell on you and you have to disclose how many floats have you counted you will have to confess that you are still working on the very first one. They won't be happy !

@countingfloats 5 лет назад

You are right, and I proved it not with mirrors but with an algorithm, acting on a very simple matrix. Incredibly enough, I discovered the proof of the Continuum Hypothesis by simply establishing 1-to-1 correspondence between positive decimal floats and positive decimal integers. For the last 125 years such a proof eluded the top set-theory experts of the planet. It was eventually abandoned and classified either as impossible or undecidable. Well, no more. Sit back and enjoy one of the most trivial and elegant proofs in set theory. It is published on RU-vid with the title : “Proving the Continuum Hypothesis”. I would appreciate if you watched and commented on this video. Your input will be given full credit. Tamas Varhegyi

@user-ec9rc8jj8q 6 лет назад

첫째줄 둘째줄 세째줄 넷째줄 다섯째줄 여섯째줄 일곱째줄 여덟째줄 아홉째줄 열째줄1 2 3 4 5 6 7 8 9 10 ? : 우리반 남녀 10 명 ?1 3 5 7 9 ? : 남자 5 명 ? 2 4 6 8 10 : 여자 5명 앞으로 전학생이 오면 빈 자리에 남자나 여자가 올 수 있는데, 그러면 우리반 남녀 총 원수는 늘어나는 거야.

@user-ec9rc8jj8q 6 лет назад

residual prlblem.

@user-ec9rc8jj8q 6 лет назад

residual problem

@user-ec9rc8jj8q 6 лет назад

subjects problem.

@vectorshift401 8 лет назад

The hotel is WRONG WRONG WRONG. When you tell someone to chance rooms they have to leave their room to go to the next. they will be without a room for that time. When the next person leaves to go to their new room they will be without a room. There will always be someone without a room! Forever!

@clare2385 8 лет назад

+Vector Shift they change the rooms at the same time, just imagine that the guests sync up in their movements

@vaderetro264 8 лет назад

Hilbert's hotel paradox buffles me too. It would take an infinite amount of time to accomodate all new guests, and at least one of them would always be temporarily without a room...

@vectorshift401 8 лет назад

Saskia H. They couldn't do it "at the same time." The speed of light is a limit on communication so there is no way to inform everyone to act simultaneously. There would always be at least one person without a room and it would stay that way forever and forever and forever ....

@clare2385 8 лет назад

Vector Shift oh now I understand what you mean you're absolutely right. but I'm personally using the hotel as a really simpel picture to prove infinity + 1, you couldn't put it into practice anyway

@Kalumbatsch 8 лет назад

+Vector Shift This is mathematics, not physics. If you mean that you will run into problems if you try to implement the idea in reality, well, NO SHIT SHERLOCK.

@PartVIII 5 лет назад

Infinite universes is a silly idea and needs to be snuffed. Infinities are within sets. Infinite number of sets within an infinite number of sets within an infinite number of sets.... Seems more like statistics than rigorous mathematics.

@countingfloats 5 лет назад

You are right,...sort of. Where you are mistaken is the admission that one can create a nested infinite within infinite number of sets. Infinites cannot exists in a set. At any rate we can abandon that losing argument because :Incredibly enough, I discovered the proof of the Continuum Hypothesis by simply establishing 1-to-1 correspondence between positive decimal floats and positive decimal integers. For the last 125 years such a proof eluded the top set-theory experts of the planet. It was eventually abandoned and classified either as impossible or undecidable. Well, no more. Sit back and enjoy one of the most trivial and elegant proofs in set theory. It is published on RU-vid with the title : “Proving the Continuum Hypothesis”. I would appreciate if you watched and commented on this video. Your input will be given full credit. Tamas Varhegyi

@awhodothey 6 лет назад

27:00 I can prove this is wrong. The same logic used to disprove the countability of real numbers also disproves the countability of rationals. Assume you have a list of every rational number with the numerator 1 and they are numbered x1... xn. Make a new number x0 such that the first digit of the denominator is different from the first digit of x1, the second digit of x0 is different from x2, etc. x0 cannot be on that list! Therefore the rationals cannot be counted, and that proof contains at least one logical error (it actually contains several). I don't see how they're could ever be any way to prove that any infinite number set is countable without subtly assuming it is true in the premise, as Cantor did.

@sumanthnani777 5 лет назад

Unfortunately, the rational numbers are limited kind of infinite decimals, i.e. only recurring kind. So the new number you create have to be proved to be necessarily of this kind for you to claim a proof is given by the way of contradiction. With reals it's not a problem as the resultant newly constructed number may be rational or irrational is always real.

@countingfloats 5 лет назад

I agree. Incredibly enough, I discovered the proof of the Continuum Hypothesis by simply establishing 1-to-1 correspondence between positive decimal floats and positive decimal integers. For the last 125 years such a proof eluded the top set-theory experts of the planet. It was eventually abandoned and classified either as impossible or undecidable. Well, no more. Sit back and enjoy one of the most trivial and elegant proofs in set theory. It is published on RU-vid with the title : “Proving the Continuum Hypothesis”. I would appreciate if you watched and commented on this video. Your input will be given full credit. Tamas Varhegyi

@MuffinsAPlenty 4 года назад

I know your comment is quite old, but I wanted to address it nonetheless. Your argument here is, more or less, a more complicated version of a common critique of the diagonal argument, which attempts to show that the natural numbers are uncountable. The argument goes as follows. List the natural numbers: 1 2 3 ... We now construct a new number by looking at digits and moving to the left (since it doesn't make sense to move to the right). If the number is a 0, replace it by a 1, for instance. And if the number is not 0, replace it by 0. So in the ones place, you compare the ones place of the first number, 1. It's a 1, so you replace it with 0. For the nth number (n > 1), the 10^(n-1) place is 0, so you make it a 1. The number you construct is "...1110" So you conclude you have constructed a natural number not on your list of natural numbers, meaning the list of natural numbers is incomplete! This is clearly nonsensical, so the argument must be invalid. The issue with this attempt at invalidating the diagonal argument is that "...1110" is not a natural number. Each natural number has finitely many (nonzero) digits. So while it is true that the thing you constructed is not on the list, the thing you have constructed is not a natural number. Thus, you cannot conclude that the list is incomplete since you have _not_ demonstrated that there exists a natural number which is not on the list. Your argument about rational numbers suffers from the same affliction. The "rational number" you end up with will be 1 in the numerator with the denominator being a "number" with infinitely many (nonzero) digits. This is not a valid rational number since the numerator and denominator must both be natural numbers, but the constructed denominator is not one. The reason that Cantor's diagonal argument works for _real_ numbers is because real numbers _can_ have infinitely many nonzero digits to the right of the decimal point. And indeed, _every_ infinite sequence of digits to the right of the decimal point actually does represent a real number. So the newly constructed infinite decimal number is a valid real number which is not on the list.

@awhodothey 4 года назад

@@MuffinsAPlenty That's only true if you arbitrarily define real numbers as being uncountable and define rational numbers as only having countable denominators (AND even then, only if you assert that each specific, individual denominator produced by my method is "uncountable", simply because the "proof" assumes the set is). But if you are going to say rational numbers have to be countable, why not assume that real numbers have to be countable too? Why assume infinities exist within either set? The main logical flaw of this proof is its ignorance of time. Countability, and infinites, are time focused definitions- not necessary properties of numbers, and certainly have nothing to do with the time dependent, first order logic of traditional arithmetic. This doesn't mean that everything Cantor said was wrong. It just means that his math is a separate logic, which doesn't really have anything to do with other maths.

@awhodothey 4 года назад

@@sumanthnani777 My "proof" assumes that the numerator is always 1. Any number whatsoever in the denominator makes the fraction a rational number.

@euphratesjehan 4 года назад

Does anyone have info on using Cantor's theory to prove that God is infinite? I found some notes I jotted from having heard or read something but I don't know when or where. For example G = God F = Father S = Son H = Holy Ghost G = {1,2,3,4,5,6,7,...} F = {3,7,12,...} So on Has anyone seen this kind of formula to express the theological infinity of God and The Trinity? Your help is appreciated

@user-ec9rc8jj8q 6 лет назад

In my view points, Cantor is wrong.Cantor's Even and Odds' problem is not a infinity problem.he confused between 'an ordinal number' and cardinal number'.God bless Cantor... Convergence and Divergence' problem is the problem about solutions -expection: dependent variables. Cantor's numbers -Even or Odds, an Ordinal number or Cardinal number- is independent variables in fuction.so, according to syllogismCantor's independent variables did not definitioned, his theory is denied.

@user-ec9rc8jj8q 6 лет назад

Contrast to your check list.do you consier that Implicit function and Explict fuction's relation?algebra and matrix need theirs' own position and direction.

@peanut12345 5 лет назад

Hilbert's Grand Nonsense, Once infinite always Infinite. Make up Sets, any set can be the sum of other sets. Create Real numbers but then say "that's not all". How many sheep are in the barn? All of them, I guess.

@niveditham1692 4 года назад

Give the definition for finite then

@psilopsybr 9 лет назад

I'm not convinced. There is a fundamental flaw in the reasoning, which defines infinity in terms of math. Infinity is not a number. To demonstrate this, simply start counting. You will never reach infinity, no matter how high you count. He's using apples to describe the flavor of oranges. The root problem is that we always think of infinity in terms of size because it is so... big. Really, really big. Hmm. Think about that for a moment. It is a mathematical impossibility for anything with a numerically defined size to fit within a smaller version of itself. But it can easily be shown that infinity can fit within the finite. This is because infinity has no size, no matter how big it is. You contain the Infinite within you. If you want to understand the Infinite, look within yourself. You won't find it with pencil and paper.

@Chris-5318 9 лет назад

Premliana Mathematical infinities are not real or natural numbers. They are a different kind of number. Cantor called them transfinite numbers to distinguish them from the ordinary numbers.

@psilopsybr 9 лет назад

Chris Seib Infinity is not a number of any kind. It is a substitute to indicate that the answer is unknown and can not be expressed as a number. Those three dots at the end of an infinite series tell the story. They explain that although the next number in the series will always be a number, there is no number that can end the series. When a calculation returns ∞ as a result, it is not saying "the answer is infinite." It is saying "the answer is beyond my capacity to calculate." The mathematician is saying "the answer is beyond my capacity to comprehend." For this reason, a calculation that either calls on infinity or returns it as a result should be disallowed in mathematics as nonsensical, just as division by zero is not allowed because it returns a meaningless result. Stop trying to squeeze infinity into a finite box. It can't be done with logistical reasoning. Put down the pencil and close your eyes. Touch the Infinite within and you will understand the futility of trying to express it in a formula.

@royradax 8 лет назад

+Premliana Can you then suggest, what might be the unit- digit of five raised to 'x' limit 'x' tending to infinity ?

@psilopsybr 8 лет назад

Radax Roy I have no idea what you just said. That's a mathematical question and I'm not a mathematician. That's my whole point. Infinity can not be defined with numbers. It can't be understood with an abacus. The only role of the infinity symbol in math is to state that the result is bigger than any number no matter how big a number you use. Shut your computer off. Close your eyes. You can't calculate infinity but you can experience it. You can appreciate the awesome magnificence of it. I guarantee that you'll get much more satisfaction than solving an equation.

@Chris-5318 8 лет назад

Premliana Of course infinity is a number of a kind - that's why it's called a transfinite *number* . The natural and real numbers are not the only things that mathematicians call numbers. Try Googling instead of spouting nonsense. The smallest infinity is aleph-0 and it is the *number* of natural numbers. The next infinity is aleph-1 and is the *number* of real numbers. Despite the use of the word *number* , the alephs are not like natural numbers.

@JohnDee0 5 лет назад

Cantor theory is a set of contradiction piled one on top of the other ...there is no such thing as infinite set ...that in itself is a contradiction!!! That theory of different infinite belongs to the loony bin...

@countingfloats 4 года назад

You are absolutely right, it is total nonsense. All conjectures and a lot of dancing around but nothing is ever proven.

@MikeRosoftJH 4 года назад

@@countingfloats Some things can be proven; for example, that natural numbers can't be mapped one-to-one with real numbers (and, more generally, that no set can be mapped one-to-one with the set of all its subsets). Other things can be proven to be impossible to prove true or false, assuming set theory itself is consistent; for example, set theory without additional axioms can't decide whether or not there exists a set whose cardinality is strictly between natural and real numbers. It's your proof that is total nonsense. You have correctly proven that natural numbers can be mapped one-to-one with the set of all real numbers with a finite decimal expansion. That's not a new result; all such numbers are rational. But this has *absolutely nothing* to do with the continuum hypothesis. (And conversely, continuum hypothesis has nothing to do with "floats".)