Mathematician W. Hugh Woodin Explains Continuum Hypothesis 

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

Well, this is one of those occasions when you want to either remove all doubt or confirm it. So instead of guessing, watch the video, then you will have more information on which to form a better opinion.

@@countingfloats Remarkable. You've found a bijection between two countable sets. Certainly no one has ever done that before!!

I make an educated guess here that you are trying to be sarcastic and I will respond accordingly. Yes I have found a bijection between two countable sets (positive integers and positive floats) .Yes, after reading it, the consensus is that "hey that it is trivial, what is the big deal?" Indeed it is trivial after somebody discovered and wrote an algorithm which generates the two-way correspondence. What is not trivial however is that the best set-theory mathematicians and everybody else who has come in contact with the subject was and is still 100% certain that such bijection is not possible. Cantor started it and most everybody went along (Gauss and others of course did not !!! ) for over 140 years or so. I am getting endless comments along those lines i.e. : “the math experts came to a different conclusion, thus you must be an idiot.” Or that “Do you have a PhD in mathematics ? No, then how dare you to form an independent opinion ? I could go on and on. Examples of these conflicts over human history are well known. For example that the Earth was flat, that it was held afloat by a single giant Atlas; that the Sun rotated around the Earth (for that one the consensus was serious enough to burn dissenters at the stake ) The latest is that big government is good, if only we let it take care of us without esisting ( like Hitler, Stalin, Mao, Pol Pot, and Maduro etc ) everything will be peachy. So, other than a hit-and-run stab, do you have any constructive criticism to offer ?

User “PoppisFizzy” commented on my YT video on another site. He said “ Remarkable. You've found a bijection between two countable sets. Certainly no one has ever done that before!! “ My response : I make an educated guess here that you are trying to be sarcastic and I will respond accordingly. Yes I have found a bijection between two countable sets (positive integers and positive floats) . Yes, after reading it, the consensus is that "hey that it is trivial, what is the big deal?" Indeed it is trivial after somebody discovered and wrote an algorithm which generates the two-way correspondence. What is not trivial however is that the best set-theory mathematicians and everybody else who has come in contact with the subject was and is still 100% certain that such bijection is not possible. Cantor started it and most everybody went along (Gauss and others of course did not !!! ) for over 140 years or so. I am getting endless comments along those lines i.e. : “the math experts came to a different conclusion, thus you must be an idiot.” Or that “Do you have a PhD in mathematics ? No, then how dare you to form an independent opinion ? I could go on and on. Examples of these conflicts over human history are well known. For example that the Earth was flat, that it was held afloat by a single giant Atlas; that the Sun rotated around the Earth (for that one the consensus was serious enough to burn dissenters at the stake ) So, other than a hit-and-run stab, do you have any constructive criticism to offer ?

Numbers big big.

You don't need Cantor... You have me !!! 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

We don't need Cantor. I solved the mystery in a very elegant way. Incredibly enough, I discovered the proof of the CH 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.

Ultimate L and consequences of infinitely many Woodin cardinals.

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

@@countingfloats 💀

It doesn't asks for ANY cardinality, it asks whether the real no. cardinal is equal or greater than the cardinality associated to the first transfinite ordinal..

I'm sure you're not. And clearly you're all very profoundly wrong in the most fundamental possible sense.

​​@@adityamishra7711my understanding is that there is nothing between the cardinality of the integers and the cardinality of the real.

@@kusali11 do you have a proof ?

@@adityamishra7711 no proof, just my interpretation of the Continuum Hypothesis.

Tal Ibrahim Of course: CH asserts that every infinite subset of the real numbers is either countable or equipotent to |R itself. 😊 The Problem of the Continuum in a given Axiomatic Theory T in the language of which CH can be formulated is to determine if T proves CH, refutes CH or does not even decide CH.

As eloquently as @VRB Blazy, expressed it, there is a simpler way to explain it. The natural numbers, 0,1,2,3,4... (meaning no decimals or irrationals or negatives ) is a set of numbers that goes onto inifinity. However, this set has a size, and is infinitely huge. we call this the cardinality, or the size, or even the number of elements it has (elements being 0,1,2... in natural numbers). The real numbers too, has a cardinality. So what CH asserts, is that there is no set that has a cardinality that is both greater than the cardinaltity of the natural numbers, and less than the cardinality of the real numbers. This also poses the question about the size of infinite sets, or more generally, the size of inifinities. Are their larger infinities than others? are some infinites the same size? these are some some of the questions that arise when studying set theroy, and more specifically countable and uncountable sets. A wonderful example would be the integers and natural numbers. Though the integers has negative numbers, which could be intuitively thought as twice the size of the natural numbers, they are actually the same size. They are the same size of inifinity.

Thank You

Of course :)

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

There is light at the end of the tunnel... 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

You are right, but there is light at the end of the tunnel. Incredibly enough, I discovered the proof of the CH 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.

That's not true. The continuum hypothesis is undecidable for a simple reason understood by Cohen, which shows the whole question was moot.

u human u You still cannot say (and actually cannot know if) it is a truth then... 😉

The waiting is over !!! 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

It is definitely exciting, see below Incredibly enough, I discovered the proof of the CH 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.

Never mind who the other people are... From this moment on I am the sole authority on the CH. 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

I one of them is Stephen Strogatz

Set theoretic definitions, like those of probability and measure theory, are definitely used in theoretical physics but things like infinities, let alone uncountable ones, have not yet been proven to exist in the physical world. There is no proof that the universe is infinite in any real sense since we're bounded by the observable horizon, and there is no proof of particles or distances that can be smaller than the planck length, let alone arbitrarily small.

Thanks this helps. I don't think the universe is infinite or continuous to begin with. But a disproof is also hard in this regards. Hence I will keep mum!

@@newwaveinfantry8362 great explanation 👍

Incredibly enough, I discovered the proof of the CH 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.

@@aligator7181 Indeed, that's literally not credible. A bijection between the integers and the rationals has been known since Cantor. Cohen's theorem isn't an "abandonment" of the CH, it's a provable result. Unlike your own, which is either a whole series of rookie blunders in maths, or an exercise in YT spamming.

I'm afraid your description is vague. The set of all rationals between 0 and 1 forms an infinitely dense line of infinitely small dots, but it's a pseudo-line and isn't closed. Almost every sequence of dots that converges to a position in that line will converge to a position that is not filled with a dot. The real interval [0,1] is the closure of that line, and that one has the cardinality of the continuum. The cardinality of [0,2] is the same as that of [0,1] despite the fact that one is twice as long. That's what measure theory is all about. If this still isn't clear, I suggest reading an introduction to Cauchy sequences of rationals, measure theory, as well as Suslin lines. All of those are relevant here. Wikipedia has all you need.

@@newwaveinfantry8362 yah its clear, there is infinity with in infinity. But my point is there infinity only till the time its not measured. also maths is a way to define physics and the real world... and for a tool built for this purpose will show such discrepancies as its suppose represent the real world. I mean the uncertainty principal and concepts like singularity infinity and nothingness...

You should look the concept of lebasgue measure... it formalises the concept of measurs, like the measure of a a line , a plane etc... And you should definitely check the video " How the axiom of choice gives sizeless sets | Infinite series " Any line made out of discrete points is not a line at all.... even if you include Infinite points its size remains ZERO... a real number line has a length cuz it has an UNCOUNTABLE number of points.... So your theory sounds about the same philosophyicaly... with the above formalised versions but you assume the concept of uncertainty.... can you formalise it..? And if that can give the answer to the "" apparent "" paradox mentioned in the video above ( or you can give an alternate set of axioms which do not assume unnecessary amount of things then you can prove that continuum hypothesis is true.... or false...

Anyway, I do think the subject is meaningless anyhow.

That *is* the case. See Goedel. It's also very much implied in what Woodin just said.

There are questions you can answer, and question that don't make sense. The continuum hypothesis isn't an absolute statement, it just doesn't have a truth value.

@@annaclarafenyo8185 Why do you think it doesn't have a truth value?

@@newwaveinfantry8362 It's not an opinion. There are different axiomatic systems which purport to describe the reals, and to answer the continuum hypothesis you have to decide which one is correct in the Platonic sense. To see which system is best, you should have some idea of what real numbers are. One property of real numbers is that they can be an infinite list of random digits. By that, I mean you flip a coin for each successive binary digit of a real number (or roll a 10-sided die for each decimal digit, whatever). If you can do this, it is easy to prove the real numbers don't have a cardinality at all. First, I should prove that if the continuum has cardinality c, then a random number in the interval [0,1] has zero probability of landing in a set of smaller cardinality. The reason? Suppose there was a probability p of landing in the set. You can translate the set by a random amount (move the set over, treating the interval [0,1] as a circle), and then the probability of landing here is again p. If you do this countably many times, you have 100% probability of hitting the point. So the continuum is completely covered by a union of countably many copies of a smaller cardinality, which is a contradiction. Given this, well order the continuum [0,1] in whatever cardinality it has, and pick two random numbers x and y. The cardinality of all w

@@annaclarafenyo8185 No, I'm not at all convinced that there can be mathematical questions with no truth value. I also think that you are mixing up undecidability with independence results. Just because ZFC can't settle CH doesn't mean that no theory can. ZFC + CH obviously settles CH, but that's cheating. You're assuming the thing you want to prove. ZFC + V=L settles CH, but you are assuming something even less likely to be true than CH. It's possible that someone comes up with an axiom or set of axioms that are "obviously true" and also imply CH, thus settling it. It's not a truly impssible problem the way that the halting problem is. The halting problem can't possibly be settled by any theory as an algorithm for solving all algorithms would be paradoxical and thus can't exist regardless of your axioms. Also, just because there are theories that prove CH true and ones that prove it false, doesn't mean its truth value is maluable. There are theories that prove 1+1=3. You can even make a consistent one that does that if you try hard enough. There is a difference between consistency and soundness. Your third paragraph about probabilities of hitting numbers is completely wrong.

@@newwaveinfantry8362 I explained to you WHY it doesn't have a truth value--- because the real numbers can't be well ordered. That means ZFC IS PLATONICALLY FALSE, you can't well order R. That settles the CH question permanently, with no ambiguity. The question needs to be rephrased. When it is rephrased as "do the real numbers match up to countable ordinals one-to-one" it has an answer--- it's false. When the question is "If a subcollection inside R cannot be matched to R then it can be matched to Z" the answer is Platonically true. That's the complete Platonic answer to the question. There is no future progress possible, the question is resolved.

his answer WAS the continuum

Master Terraformer That's the Choice 😜

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

It’s so you can understand it

@@countingfloats i watched your video. Not only are you wrong, you're also incredibly arrogant.

Denying importance of science (theoretical math) because of some arbitrary views on infinity is religious.

You mean Heisenberg or is it Hey Seen Burger ??? (yummy ) 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

Yessir I’ve thought that as well as a Muslim. I think the concept of infinity is just amazing.

But which infinity is god?

You are asking a question related to time, time does not affect God. God created time.

@@Life_42 The very statement "created -- past tense -- time" presupposes the concept of time. Therefore it's just self-contradictory religious nonsense. As per.

What? That made no sense. R and R\{1.5} both have the cardinality of the continuum. A bijection between them exists, but isn't explicitly definable, as it would require a wellordering on R which can't be set-theoretically defined in a finite amount of time or with a finite amount of symbols. The union of your f1 and f2 is no longer a bijection as it now contains two seperate reals mapping to the same one - 1.5.

Now apply the same Formula for natural numbers ..... do you get a different result ? And i don't know if the union will be inconsistent cuz on performing union of f1 and f2 u r basically creating 2 sets on - S union 1.5 and other is R ... and then establishing a bijection , since S union 1.5 is R thus there is no inconsistency

@@newwaveinfantry8362 The bijection is not even hard to define: you just define some Hilbert's-hotel style map on the half-integers larger or equal to 1.5, and use the identity for the rest. Simple homework exercise if you ask me...

Forget ZFC... it has nothing to do with CH or cardinality of integers vs. floats. 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

For example: Axiom of constructibility implies the continuum hypothesis (the generalized continuum hypothesis, too). On the other hand, there are some forcing axioms which imply the negation of continuum hypothesis - in fact, they imply that there is exactly one cardinality strictly between natural numbers and the continuum. (Therefore, if we were to add both the axiom of constructibility and the forcing axiom to the axioms of ZFC, we'd get an inconsistent theory. :-) )

We need infinities. Without the infinity axiom in ZF we woulnt have the real numbers, nor calculus, engineering, statistics and many many other subjects. We need to explore what is beyond the realm of finite sets, and to advance in mathematics we need to explore set theory itself and how does the V universe work

Us. 😍

Often there's a snowball effect in mathematics where one discovery leads to many other discoveries. That's why every part mathematics should be given some thought (even if the concepts are beyond the comprehension of our minds) because who knows what they may lead to. Another thing is that maths that was once only theoretical can now be used to accurately model many different parts of our universe which is just amazing. Even imaginary numbers have some real world applications.

Incredibly enough, I discovered the proof of the CH 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.

Casual infinities expressed in our Cartesian system are useful for derivatives and integrals which are fundamentally made up of infinite limits, not to mention infinite summations such as Taylor Series and Convergence. True, Aleph Null may have no real world application... yet. Infinite cardinality is only in theory, however mathematics aren't bounded by science. Aleph Null and Omega are merely notations used to describe unending amounts that extend past the naturals. Whether or not we find a use for them in the real world doesn't really matter.

Scott Gulliford Nope, it is undoubtedly subtler than that 😜...

I solved it.Incredibly enough, I discovered the proof of the CH 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.

Just think of how many years science would have been set back if Richard Feynman hadn't been paid thousands of dollars to sit on his ass doing drugs and contemplating the universe.

Drugs?, weed? mathematicians actually mostly use amphetamines.

A joint takes your thoughts outside the universe. Infinity becomes graspable once you understand the opposite (finite), 'Zeno's infinite steps in a finite period'.

Alright, then you do it.

I discovered something which works on either side of infinity. On the right side of it I don't even need a joint to explain it. I only crave it when I have to endure some totally boneheaded comments on my discovery...see below : Incredibly enough, I established the proof of the CH 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.

why is it fiction?

We don't really know whether we even invented the continuum question, we could just be discovering the effects of how our math is designed around the natural sciences and how it buckles when you try and find the relations between different sizes and classes of infinite cardinalities in set theory.

The set of things that are true are equal only to the set of axioms? Under that thinking there essentially is no mathematics or truth at all. Or do you have some way of dealing with the possibility that the other assumptions of set theory might imply an answer to the question of whether there is any transfinite set bigger than the integers and smaller than the reals that we haven't figured out yet and that is difference from the axiom specified?

That isn't what Winson said. If S is a statement that can't be proven from ZFC, and can't be disproven from ZFC, you can create two new sets of axioms, ZFC + S and ZFC + not S, both of which are consistent (iff ZFC is consistent). There are a lot of *theorems* that are true in ZFC, that need not be considered axioms in addtion to ZFC, because doing so would be redundant. The Continuum Hypothesis belongs to the former type of statement, not the latter. So we can make it an axiom, or we can make its negation an axiom. Or we can leave it out, saying it's unsettled. Of course, another mathematician could rightly ask you to justify your choice.

If CH is not included in the axioms of set theory, then that does *not* mean it's false. It just means it's left as an unsettled question, and could be either true or false. It's only decidably false if its negation is included as an axiom.

That's not quite right either: the negation can follow from other set-theoretic axioms (in addition to ZFC), without being itself an axiom. (But at least you were much closer to the mark than Winson.)

I have no idea what you are talking about but that did not stop me to solve the enigma if you can call that...maybe it is only a case of boneheaded fantasizing. Incredibly enough, I discovered the proof of the CH 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.

Why do you say that?

Neither is he a software design engineer ...lucky for us I am. Incredibly enough, I discovered the proof of the CH 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.

A 1:1 correspondence between reals and integers would both *disprove* CH (which says the reals have cardinality omega-1) and contradict Cantor's proof that no such correspondence exist. You are wrong two different ways.

He's literally a logician.

@@rsm3t The spammer hasn't even grasped the distinction between the reals and the "floats".