A classic example of pop culture mathematics. It's really easy to fool people who aren't mathematicians when it comes to these extremely advanced and unintuitive questions. Nice breakdown!
My understanding of his understanding: - The whole numbers are aleph_1 because they are separated by a space of 1 - The real numbers are aleph_0 because there is no gap between them (a space of 0) - So aleph_0.5 is the numbers separated by 0.5 (0, 0.5, 1, 1.5, etc) This is what he means by "density"
That seems like it might be it! That would explain why he swaps the aleph numbers around as well. Although he is stil very wrong since this notion of density still doesn't capture cardinality,, the rationals are dense in the reals and also densely ordered (No 'space' between them) so by his logic they'd have the same cardinality as the reals.
I think it's also hilarious that even the way he stated the continuum hypothesis. By definition, there is no set that has cardinality larger than א0 and smaller than א1. I mean, א1 is, by definition, the smallest cardinality larger than א0. There literally can't be nothing inbetween
9:40 Actually cardinals can be defined without the axiom of choice, but without the axiom of choice they are not totally ordered. You just have to be a bit more careful when deciding how to define Card(X); one possible way is to take α to be the minimal ordinal such that the Von Neumann set V_α has a subset that bijects with X (which exists because the Von Neumann hierarchy exhausts the universe of sets and therefore X ⊂ V_β for some β), and let Card(X) = {S ⊂ V_α: S bijects with X}.
Interesting! Yea, I was a bit sloppy with my language there. I did really mean "not every set bijects with an ordinal", and in particular not having a useful way to compare arbitrary cardinalities. Thanks for sharing!
@@themathyam This is usually called "the Scott trick", after Dana Stewart Scott, a modern set theorist and quite a nice guy if I might say so. :) For every proper class K, there is the lowest level of cumulative hierarchy K intersects, and for many of them (such as naive cardinalities--equipotence classes) the intersection itself can serve as a faithful representation of K. In fact the only thing choice gives you is the _unique_ element of that intersection, provided you require it is an ordinal. (I'm not necessarily explaining it to you, who probably know all of this and just misspoke, but to the wider audience.)
Meanwhile, at Randall Munroe's party: "The [diagonal argument] is a large spherical volume of water consisting of all the water content of a storm cloud dropped simultaneously."
ℵ₀ = |𝜔| = |𝜔∪{𝜔}|= |ℕ| = |ℤ| = |ℚ| ; 𝔠 = 2↑ℵ₀ = |ℝ| = |ℂ| ; ℵ₁ = |the set of all ordinal numbers which can be mapped 1-to-1 into 𝜔| ; so we know ℵ₀ ≺ ℵ₁ ≼ 2↑ℵ₀ ≺ ℵ_𝜔 but the question of which of the following holds: ℵ₁ = 2↑ℵ₀ or ℵ₁ ≺ 2↑ℵ₀ remains beyond the reach of the assumptions of ZFC set theory. Since the truth of the continuum hypothesis is independent of ZFC, we need some other framework to decide which assumption is "the best for our particular needs." // Edit -- 2↑ℵ₀ ≺ ℵ_𝜔 is NOT KNOWN, but there is a proof in ZFC that 2↑ℵ₀ ≠ ℵ_𝜔 which does not forbid ℵ_𝜔 ≺ 2↑ℵ₀ which is exhausting to even thrink about.
@@darkseid856 I think most OSes have a graphical display of Unicode characters to pick from. I use ℵ "ALEF SYMBOL" because the regular Hebrew character Aleph usually changes the writing direction to right-to-left which is inconvenient when typing English/math exposition.
2↑ℵ₀ ≺ ℵ_𝜔 is not known, so you shouldn't write it as a fact. The continuum may be essentially arbitrarily large in relation to the Aleph numbers. It may even be weakly inaccessible or weakly compact or beyond. The only bounds we have are using Shelah's PCF theory, but those require some assumptions that are also independent of ZFC as far as I'm aware.
He's very confused about the underlying definition of aleph... anything. I think MrSamwise25 said it best in this comment section, his understanding of aleph is: - The whole numbers are aleph_1 because they are separated by a space of 1 - The real numbers are aleph_0 because there is no gap between them (a space of 0) - So aleph_0.5 is the numbers separated by 0.5 (0, 0.5, 1, 1.5, etc) This is what he means by "density" Which is just funny. A complete misunderstanding of the subject matter.
@@themathyam Actually, the map you described sends the oddly even numbers (4n+2) to the even numbers (2n) and the evenly even numbers (4n) to the odd numbers (2n+1). 😅
I read that this is a standard behaviour from cranks. The moment that it was proved that it was impossible to square the circle thousands of circle squares appeared. That is because they are not a part of the community working on solving open problems, but because they want to be that guy who proved one thing and became famous.
kinda feel bad for the dude i hope people don't speak to him meanly kudos for patiently breaking it down and best of luck in your phd ^_^ i'm currently working on my B.A.
He shouldn't be miseducating people, he deserves all the mockery he gets. I wonder what the comments say on the video, they must be terrible, but has he taken the atrocity of a video down?
In a sense, the Russell's paradox can be seen as the statement that the axioms of ZFC, but with full Comprehension instead of restricted Comprehension, do not form a consistent theory. Though, when Russell coined his paradox, he was referring to Frege's logicistic formalization of set theory (that hence was exposed as inconsistent).
Between any two irrational numbers there is a rational number. Between any two rational numbers there is an irrational number. However the cardinality of Q is ℵ₀ and the one of R is larger. Hard to believe, but it actually is.
What I can imagine is meant at 4:45 is that there are countably many gaps (maximal intervals in the complement of the Cantor set) but uncountably many points (or connected components) in the Cantor set itself. This is true, though somewhat counter intuitive: at every step of the construction there is just one more component of the "remainder" of the interval then there are gaps; why does the difference become much more important in the limit? However, it is not fundamentally different from the fact that uncountably many irrational numbers are cut out from the continuum by removing countably many rational numbers. And the explanation given in the cited video ("there is two of those", vaguely waving their hands) doesn't even start to make sense just like everything else they say.
9:40 Reaction to reaction: cardinal numbers are defined and well-ordered without the axiom of choice. The only related thing that needs AC is that every set is equinumerous to some cardinal number.
What I meant to say is cardinality, i.e. assigning a cardinal to every set. My bad! As other commenters mentioned, one can also define cardinality for any set via scott's trick, but this isn't well behaved without AC, in particular, there are problems with comparability.
10:51 He got Aleph1 and Aleph0 backwards. He labeled the naturals Aleph1 (it should be Aleph0), and labeled the reals as Aleph0 (it should be Aleph1 (really it should be 2^Aleph0, but he's sort of conflating that with Aleph1)).
Not a mathematician here, but I couldn’t get past his basic errors that even I understand: eg, sloppy about Aleph-0 and Aleph-1 and mis-spelling Bertrand Russell’s name right at the end.
According to the logic of the guy in that video, the number of natural numbers is twice the number of even natural numbers, but in fact the count in both cases is aleph null.
Kinda disappointing. I think there are almost certainly false proofs of the continuum hypothesis where it's genuinely difficult to find the flaw, but this one very much wasn't that. There's value to debunking nonsense, but I'd be more interested in response to more impressive false proofs.
You forgot to mention that the Cantor-set is important in measure theory. Also Georg is not George. The first close by in your definition of topological density is superfluous as well, you are already covering the entire space. Would be better to just say "for every real number there is a rational number arbitrarily close by". Video was fun because I felt the exact same frustrations, however I doubt its validity as a teaching tool because anyone who knows what a model of an axiom system is, is gonna come even close to falling for whatever it was what that guy did.
I am finishing my maths bachelor, and I have legitimately no idea what he means. Like I know what the words mean, but this feels like an AI generated proof to me
Update: th-cam.com/video/odPWvCPcqNk/w-d-xo.html "proves" that the speed of sound is the square root of the speed of light. This. Is. Dimensionally. Impossible. But all you need to show this is a compass! Wow.
Saying that you proved the continuum hypothesis is like saying you solved the halting problem or found an algorithm to calculate the Busy Beaver function. It just means you have no idea what you're doing. Please understand the difference between unsolvable and unsolved
Even if everything he says were correct, he'd have proven there is another infinity below Aleph_0 (i.e. Aleph_0 is not the smallest infinity), not that there is one between Aleph_0 and Aleph_1.
Agree the "in 2 infinity" guy is talking nonsense. But remember a lot of the math you know disproved part of what went before. So don't be surprised when some one (not likely to be the "in 2 infinity" guy) comes along and disapproves some of the math that you know.
I’m sorry. Math isn’t a science. This isn’t “oh we have a theory and someone found better evidence so we moved on”. You don’t work with falsehoods. No one will “come along and disprove math”. Math is a set of things that have been proven in various systems with stated axioms. When’s the last time you saw a news item about a big piece of math being proven wrong?
I think that the funniest part is that he started by claiming there was two countably infinite sets, where one was "larger" than the other in the cantor set (5:55). But doesn't refer to it at all later on. He just invents a *new* mapping, just to keep things interesting. 🙃
Sorry if I sound stupid, but isn't the cardinality of the Quotient set of numbers or the Rational set of numbers smaller than the cardinality or the Real set, yet bigger than the cardinality of the natural numbers set?
This is a common misconception, but it is actually not! It is possible to match the rational numbers one-to-one to the real numbers, so they have the same cardinality (In other words we can "enunerate" the rationals, and there's actually explicit ways to do this)
Two sets are the same size, have the same cardinality, if we can produce a bijection between them. If we can label the elements of an infinite set as r_1, r_2, r_3,... and never miss an element out (a.k.a. given any element, we can identify which n it will be so that our number is r_n), then it's in bijection with the natural numbers. Just send n→r_n, and r_n→n as your function and inverse function. We can do that with the rationals. Label them *all* as r_1, r_2, and so on. There are good visual proofs out there, using diagonals.
@@sable7922to show two infinite sets are equal you construct a method of mapping between the elements of the sets. The mapping must use every element in each set and not have overlaps In this sense “countably infinite” is really just counting. You point at a fraction and say “one”, then “two”, etc. Someone says “will you reach this fraction?” And you always say “yes I can show it will be reached” It’s possible to assign a number to every fraction and not miss any or count them twice.
The little detail that's commonly missed is the presumption of a 'natural' order which allows the counting from zero[none]/one/two/etc (see the odds/evens example). Without that specific order (i.e. with some other arrangement) it might not be possible to do the counting (linear order of rationals?). We lay people don't always realise that the 'order' (method of arrangement) really matters, and that some of the axioms (that are talked about separately) are actually equivalent - lots of circularity. Lots of "arbitrarily large" recursion in some of the arguments ;-)
wdym the 'presumption' of a natural order, the naturals (as von neumann ordinals, and cardinalities thereof) are literally defined in order. no further assumptions are needed.
@@rarebeeph1783 The natural order presumption, is that you can 'obviously' count the rationals in value order, as opposed to the expanding diagonal diamonds order, and thus be able to determine the 'next' rational number after say 113/355, or 33102/103993, etc. In the diagonal argument a specific order is used just to side-step the classic infinitesimals problem of 'what's next'.
@@philipoakley5498 why would anyone presume that? it's obviously false. between any rational a/b and another rational c/d, for integers a,b,c,d (where b and d are positive), there is a third rational (a+c)/(b+d) between them.
FYI: I have a Math PhD and a Masters in Math, both from Rutgers University, and 6 published papers in 3 separate peer reviewed journals. I know how insanely hard it is to come up with deep novel proofs. #1 sign of bullshit: presenting proofs of major unsolved conjectures in math on TH-cam instead of the standard accepted proven way that all actual real professional mathematicians do: peer reviewed math journals. Sure - I might prove some little lemma that only I am interested in, and post it on TH-cam. I may or may not publish it in a serious journal. It's up to the journal to decide who novel and groundbreaking my theorem and proof are, even if my proof is 100% correct. However, #1 sign of crockery is that the crocks always think they can tackle the big famous conjecture that everybody has heard of, and do it ONLY on TH-cam. #2 sign of bullshit: filling up padding your "proof" with recaps of well-known history of math.
This must be incredibly frustrating to people who study this for fun/work, because, his proof, I think it completely misses the abstract complexity that the theorem is trying to define. How do you average a spacing unit of 1 to infinity with a variable infinite spacing unit to infinity? You need to define new mathematical mechanism. Different ways to describe numbers between other numbers, or something completely different all together to prove this theorem, right? I don't know this type of math that well. But I can tell his proof is not addressing the problem at all.
My question is: let's say we construct the ZFC + CH one, so there is an intermediate sized infinity - what would it look like? Basically, can you construct or describe of define in mathematical operations this intermediate infinity, assuming there is one?
the problem is that we can't. I think it's moreso that we know it exists, not what it would be. For example, according to ZFC, we can show that the well ordering theorem is true. (it fact the well-ordering theorem is precisely equal to the axiom of choice). This means any set can be well-ordered. However we can also prove that there is no way to find the well-ordering of the real numbers. So we know the real numbers have a well-ordering, but we cannot find it. (assuming ZFC).
ZFC + CH is that there is *no* intermediate infinity. Your question is instead about ZFC + ~CH, that is, "plus not CH". So then. In ZFC + ~CH, we have as a given that there is a cardinality between that of the naturals and that of the reals. That is, the cardinality of the reals is something other than Aleph_1. But Aleph_1 is by definition the cardinality of distinct well-orders of the naturals--in other words, the cardinality of "countable ordinals". So that is an example of an intermediate infinity. (Of course, in ZFC + CH, we would have that the cardinality of the reals is precisely Aleph_1. The same set can be put in bijection with the reals in one model, and be impossible to put in bijection with the reals in the other model.)
@@kazedcat It's possible that there are also many other cardinals between them considering 2^Aleph0 can be any aleph in the aleph hierarchy (with some caveats). But yeah, Aleph1 would be the smallest such infinity.
@@rarebeeph1783 sorry, sure I meant ZFC + NOT CH. But I still don't get it. If we say that in this case the cardinality of reals is less than 2^Aleph_0, then how to construct a set with cardinality 2^Aleph_0 ?
@11:23, you equate contradictions with paradoxes, as he did, but this is not quite correct. A paradox can be a provable statement that seems counterintuitive. An example is the Banach-Tarski Paradox. See the Wikipedia article here: en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox You’re correct, of course, that what he’s discussing isn’t a contradiction; however, it may be paradoxical in the same sense as the Banach-Tarski Paradox.
@@xinpingdonohoe3978 Being an unintuitive conclusion is one kind of paradox. See the book by Sainsbury called Paradoxes. There are several kinds and this is one of them. Russell’s paradox was a contradiction in naive set theory (Cantor’s formulation) and Frege’s attempted axiomatization of the same. However, a modification of Frege’s formulation of class theory seems to have removed that contradiction. Thus, the only sense in which Russell’s paradox remains a paradox is that the efforts to remove it did not yield a proof that the modified axioms (Zermelo-Fraenkel Set theory) are consistent, even though no contradiction has been found. The Banach-Tarski paradox is so named because it appears to cause a contradiction in an area of applied mathematics closer to physics, in that it appears to lead to a violation of the law of the conservation of mass, because we can prove that there’s a decomposition of the unit sphere in three space into finitely many pieces that can be rearranged and reassembled using rigid motions to produce a sphere having twice the original volume. Physically this can be used to prove that if a unit sphere in 3 space comprised of a material of uniform density is subjected to such a transformation, then the result is a doubling of the material with no source of inflow of matter. To a physicist, this should be considered a contradiction, expect that a good physicist who knows sufficiently advanced measure theory would understand that the finite pieces into which the original sphere is decomposed cannot be measurable and so cannot be assigned a mass in the first place. That is, the theory of measures and its attendant result of Vitali that there exist non-measurable sets resolves the paradox to the satisfaction of the sufficiently informed physicist or mathematician, etc. In this case, another way to resolve the paradox is to modify the axioms for your set theory so to consider all sets of real numbers to be measurable. Solovay proved that if you commit to the consistency of the existence of inaccessible cardinal numbers, then you can prove the relative consistency of such an assumption. I don’t know of any result that removes that commitment. However, in set theory, it’s common to refer to “Solovay’s model” in this connection.
He's basically saying to consider the ratio between his half step set and the naturals. He's saying that if we start counting the elements, we find at step n of the Naturals we've counted 2n of his half-steps. the ratio is of course 2n/n which, if we take its limit, should give us the relative number of elements between the sets, which is 2, and so his set is twice as large as the naturals. But we could do the same with a set which goes twice as far as we have in our counting of the naturals, but is just the naturals itself given more time to count, so we can conclude from deploying the same argument as him that the Naturals themselves are twice as large as the naturals. So its ridiculous and as you said, he thinks he's talking about cardinality but hes not lol.. Hes talking about the rate at which we progress through any countable set. Not about the size of the set, but about the rate of "counting" which we choose...
This idea is actually worthwhile, and can be formalised using the notion of "natural density", but obviously as you pointed out it is very very different to cardinality!
@@themathyam Yes but I dont even think it amounts to anything in itself. I'm imagining two people with the natural numbers laid out in front of them, and we assign them to start counting progressing up the line but we make one of them move faster than the other somehow, say twice as fast, and when the slower one has marked the n'th number, the second would have marked the 2n'th.. So one would have a greater "natural density" than the other. And his 1/2 step set is basically just somebody moving twice as fast along a countable set, we can just replace those symbols with the naturals and thats what it would look like to travel faster or whatever in our experiment.. Anyway sorry for rambling and I know this wasnt rigorous at all lol.
I don't know what the hell was going on at the end, with the +0.5 thing and the aleph-0.5. He's literally as bad as Werner Hartl and his "proof" that cantors diagonal argument fails, its very painful to watch btw.
Anyone else notice he spelled “Russell” wrong. Look, I understand some people may not spell very well, but it seems a lot to ask me to believe a British-accented “mathematician” doesn’t have everything about Russell burned into his memory.
_"the world's most challenging problem of the last century"_ Immediately a bad start. I would surmise that most mathematicians would agree that the most prominent and challenging problem of the last century is either the Riemann hypothesis or the P vs NP problem. Most would probably choose the Riemann hypothesis if they had to choose between the two.
I would not fault him for making that subjective comment, frankly. I mean, yeah, I know what you mean. What is a bad start is his immediate naive use of elementary methods. That instantly killed his proof before he went any farther.
Lol, what a guy. Why a non mathematician would care about the continuum hypothesis is beyond me. It is very odd because he doesn't even understand the words he uses, and if he did, he would either go do the hard math required to understand CH, or realize that maybe that journey is just too much. Like, I don't even see how his video on the topic helps sell whatever it is that he is trying to sell. Very odd.
Not at all. He states quite clearly that you *can* construct axioms for set theory in which there can be something between aleph0 and aleph1. But you can also do it in such a way that there is not. If you're not familiar with it, it can come as a surprise, but mathematics is not universally True in the sense that it is commonly used. At more advanced levels mathematics can basically be the exploration of "what happens if we accept these rules (axioms)", and then "what happens if we accept THESE rules (axioms)?". From wikipedia, one of the axioms of ZFC is the "Axiom of extensibility", which is the rule that "Two sets are equal (are the same set) if they have the same elements." Now, there's different ways to choose those axioms which will be self-consistent and generally seem plausible, but there's nothing that tells us which rules (axioms) we *should* use. There's no "true" construction of axioms; there's just a choice between somewhat equal constructions. So what he's saying is that there are at least two different ways to create the axioms, such that in one construction there *definitely* isn't anything between aleph0 and aleph1, but in the other construction there *definitely* is.
😎 Great video. You need to be a lot more careful about the word “cardinal”. The cardinal numbers are things like one, two, three, four. If aleph null exists (it does! in theory!), then it is a cardinal too. I work with a type of set theory that uses the terms cardinal, ordinal, cardinality but we study finite sets only (very useful terminology in the real world!). Anyway, you’ll have a lot better luck persuading your audience about infinities if you acknowledge that infinite cardinals are the same kind of numbers as the natural numbers! Remember that not everyone is on board with this stuff. Folks I know in engineering and compsci either think you guys are blowing smoke, or that it’s completely irrelevant. Math unfortunately loses its influence when it’s out of touch with application. 👍 big thumbs up, thanks for the video.
lol as the phd said there is no such thing as aleph_{0.5} the video he is reacting to is just total nonsense, it's a long time since I did this stuff but but yeah I still remember enough to say NOOOO to this rubbish
Well done. All of the false statements he made to the side, it seems that all he has done is to construct a countable subset of the rationals, but it's hard to say for sure. My understanding is that it is possible to construct a model (via forcing) that places 2^ℵ0 anywhere you like in the normal sequence of alephs.
Aleph 0.5 is clearly counting all the cardinal numbers, but skip every other one. But Aha! You say, that's still alpeh null. But then only count half of those! Hey! This is easy! Russell was a dummy!
I can't blame you for critiquing his video "as is" and you've done a great job of remaining calm and explaining things patiently to interested viewers. However, if you'd looked up in2infinity, you'd find they're full of crackpot nonsense and it's probably a waste of time to critique any of their videos. I'm fairly sure anyone with reasonable maths understanding is not taking them seriously. Those that are in danger of falling for it are the ones that need reaching... and I doubt this video will do that. I thought their whole thing might be a prank, but I see no humour behind it or evidence of prankery. I honestly can't tell if the presenters believe their own cr*p thought.
@@themathyam And you've done a good job of making an informative video that I enjoyed and I'm sure others did. But after googling this guy even more since writing my 1st comment, I'm more convinced that he's a fraudster and people might be sucked in. I don't know what's the best way of calling him out and warning people. Ideas welcome.
Sure, there's no actual mathematics there, but he could help his credibility by at least spelling people's names correctly. "George" Cantor? "Russel" paradox?
I love these stupid "proofs" that just immediately make no sense at all! They're terrible if you don't know they make no sense, but quite funny if you do know They also love jumping from basic stuff to super complicated stuff with no preparation or definitions or anything that would help anyone follow their "logic"
I think your introduction might be wrong? We say that 2^ℵ0 is the cardinality of R because R should comes after ℵ^0. That's precisely what 2^ℵ0 means and is nothing between ℵ0 and 2^ℵ0, If there was, let's say, some intermediate set A, then |Z|=ℵ^0, and |A|=2^ℵ0, and then |R|=2^ℵ1
"We say that 2^ℵ0 is the cardinality of R because R should comes after ℵ^0". No, we say that 2^ℵ0 is the cardinality of R because there is a bijection between R and the power set of N (the set of all its subsets), which is known to be strictly greater than ℵ^0 because of Cantor diagonal argument What comes after ℵ0 is (by the very definition) ℵ1, and the video correctly states that the continuum hypothesis is the affirmation "ℵ1 = 2^ℵ0" Then your example would just be |Z|=ℵ^0, and |A|=ℵ1 and |R|=2^ℵ0 with ℵ1 being of cardinality either ℵ^0 or 2^ℵ0 or not calculable (because if it were strictly in between it would be a proof of CH)
The existence of a paradox doesn't mean the existence of a logical contradiction. It can also just mean the existence of something that seems like a contradiction at first. This is more of a philosophy thing than a math thing usually. The position of accepting the counterintuitive seeming contradiction is commonly referred to as biting the bullet.
#1 sign of bullshit: presenting proofs of major unsolved conjectures in math on TH-cam instead of the standard accepted proven way that all actual real professional mathematicians do: peer reviewed math journals. Sure - I might prove some little lemma that only I am interested in, and post it on TH-cam. I may or may not publish it in a serious journal. It's up to the journal to decide who novel and groundbreaking my theorem and proof are, even if my proof is 100% correct. However, #1 sign of crockery is that the crocks always think they can tackle the big famous conjecture that everybody has heard of, and do it ONLY on TH-cam. #2 sign of bullshit: filling up padding your "proof" with recaps of well-known history of math.
EXCELLENT AMERICAN YOUNG MATHEMATICIAN WHOSE NAME UNFORTUNATELY WASN'T POSTED!! CONGRATS ANYWAY. th-cam.com/video/JUVYBBlxw7E/w-d-xo.htmlsi=AJaGS3OyNsRHnW6k BTW the confusion between density and cardinality was also the root of the misunderstanding of a famous political and Christian Brazilian philosopher called Olavo de Carvalho in his best seller "O Jardim das Aflicoes". Olavo de Carvalho has died recently in the State of Virginia. He was a great Aristotelian philosopher and a political figure involved with lots of political controversy (he was an anti communist fella and a true iconoclast); Also in 8:15 this smart boy also catched the other layman confusion: the absence of domain/definition of sets and functions (there are no numbers or points or whatever between integer numbers in the Natural numbers. They also confound the sequence of symbolic representation of quantities as the quantities themselves) - we mustn't say that any competent philosopher would be also capable to understand math and the natural sciences. Sadly, the greek intellectual philosopher like Euclides or the own Aristotle is a very hard specimen to be found in the complex modern world. Perhaps Carl Sagan was such a good example or perhaps not.😊 PS. In the high school teachers usually say that the integers is an infinite set which is a subset of the infinite set of the reals (it creates a cognitive bias: as the reals would be just only a denser set of integers). 😊
the "proof" starts at 11:33
what he has done is not find a number between aleph_0 and aleph_1. he just found a number between 0 and 1.
just multiply both sides by aleph and u are set
@@blahblahblah23424 no, the ordinal number goes on the bottom-right of the aleph. that means you have to find the aleph-th tetration of the number
@@blahblahblah23424 Q.E.D.
@@brightblackhole2442now you’re doing REAL math
A classic example of pop culture mathematics. It's really easy to fool people who aren't mathematicians when it comes to these extremely advanced and unintuitive questions. Nice breakdown!
what sort of pop culture talks about the continuum hypothesis
😂@@ryzikx
@@ryzikx the video you literally just watched.
These aren’t “advanced”. This is first semester undergraduate math. Any 18yo math student would know this is nonsense, it doesn’t take a PhD.
@jasonmp85 Yeah, but first year undergraduate mathematics is certainly advanced mathematics to the average person.
wait until he discovers Aleph-0.25 and Aleph-q for all rational q and and and ...
😂😂😂
What about p-adic Alephs? Aleph-p for all prime p… ;)
It gets really funky once you get to Aleph-Aleph-0.
Aleph pi. Aleph i.
I thought I was bad at math, but this makes me feel better.
Bro’s thought process in a nutshell:
1) Is there an infinite cardinal between aleph_0 and aleph_1?
2) the “proof” goes: well 0 < 1 and 0 < 1/2
no, he's clearly saying aleph_1/2 is _bigger_ than aleph_1 and _smaller_ than aleph_0
@@galoomba5559 yea I forgot he confused them xddd
@@galoomba5559 Yeah, he kept saying that, making his inane arguments even more mindless.
My understanding of his understanding:
- The whole numbers are aleph_1 because they are separated by a space of 1
- The real numbers are aleph_0 because there is no gap between them (a space of 0)
- So aleph_0.5 is the numbers separated by 0.5 (0, 0.5, 1, 1.5, etc)
This is what he means by "density"
That seems like it might be it! That would explain why he swaps the aleph numbers around as well. Although he is stil very wrong since this notion of density still doesn't capture cardinality,, the rationals are dense in the reals and also densely ordered (No 'space' between them) so by his logic they'd have the same cardinality as the reals.
If it's to do with infinity and it's intuitive, it's probably wrong.
I think it's also hilarious that even the way he stated the continuum hypothesis. By definition, there is no set that has cardinality larger than א0 and smaller than א1.
I mean, א1 is, by definition, the smallest cardinality larger than א0. There literally can't be nothing inbetween
Wow. That actually makes sense...
He also has aleph-zero and aleph-one swapped.
9:40 Actually cardinals can be defined without the axiom of choice, but without the axiom of choice they are not totally ordered. You just have to be a bit more careful when deciding how to define Card(X); one possible way is to take α to be the minimal ordinal such that the Von Neumann set V_α has a subset that bijects with X (which exists because the Von Neumann hierarchy exhausts the universe of sets and therefore X ⊂ V_β for some β), and let Card(X) = {S ⊂ V_α: S bijects with X}.
Interesting! Yea, I was a bit sloppy with my language there. I did really mean "not every set bijects with an ordinal", and in particular not having a useful way to compare arbitrary cardinalities. Thanks for sharing!
@@themathyam This is usually called "the Scott trick", after Dana Stewart Scott, a modern set theorist and quite a nice guy if I might say so. :) For every proper class K, there is the lowest level of cumulative hierarchy K intersects, and for many of them (such as naive cardinalities--equipotence classes) the intersection itself can serve as a faithful representation of K. In fact the only thing choice gives you is the _unique_ element of that intersection, provided you require it is an ordinal.
(I'm not necessarily explaining it to you, who probably know all of this and just misspoke, but to the wider audience.)
He must have heard a mathematician at a party: "Cantor's [dubstep drop] proves that there are infinities of different sizes".
Meanwhile, at Randall Munroe's party:
"The [diagonal argument] is a large spherical volume of water consisting of all the water content of a storm cloud dropped simultaneously."
@4:30 *Clearly*, this is the most painful proof by intimidation.
Him: He's not doing math.
Me: Yeah, he's doing meth.
So really it is a bit like when an inventor claims to have made a 'perpetual motion machine,.
"In this house we obey the laws of thermodynamics!"
And then unveiling a hamster on a wheel
Responding to the intro before watching the "proof": If he found an inconsistency in ZFC, then he can prove the Continuum Hypothesis.
In fact: ZFC can prove the continuum hypothesis, if and only if ZFC is inconsistent. (The same is true for the negation of continuum hypothesis.)
ℵ₀ = |𝜔| = |𝜔∪{𝜔}|= |ℕ| = |ℤ| = |ℚ| ; 𝔠 = 2↑ℵ₀ = |ℝ| = |ℂ| ; ℵ₁ = |the set of all ordinal numbers which can be mapped 1-to-1 into 𝜔| ; so we know ℵ₀ ≺ ℵ₁ ≼ 2↑ℵ₀ ≺ ℵ_𝜔 but the question of which of the following holds: ℵ₁ = 2↑ℵ₀ or ℵ₁ ≺ 2↑ℵ₀ remains beyond the reach of the assumptions of ZFC set theory. Since the truth of the continuum hypothesis is independent of ZFC, we need some other framework to decide which assumption is "the best for our particular needs."
// Edit -- 2↑ℵ₀ ≺ ℵ_𝜔 is NOT KNOWN, but there is a proof in ZFC that 2↑ℵ₀ ≠ ℵ_𝜔 which does not forbid ℵ_𝜔 ≺ 2↑ℵ₀ which is exhausting to even thrink about.
How to write these spaghetti letters ?
@@darkseid856 I think most OSes have a graphical display of Unicode characters to pick from. I use ℵ "ALEF SYMBOL" because the regular Hebrew character Aleph usually changes the writing direction to right-to-left which is inconvenient when typing English/math exposition.
@@darkseid856 Google the description of the character then copy and paste from the first website you find.
@@darkseid856I believe I can disagree with you sir, but these are spaghetti symbols rather than letters.
2↑ℵ₀ ≺ ℵ_𝜔 is not known, so you shouldn't write it as a fact. The continuum may be essentially arbitrarily large in relation to the Aleph numbers. It may even be weakly inaccessible or weakly compact or beyond. The only bounds we have are using Shelah's PCF theory, but those require some assumptions that are also independent of ZFC as far as I'm aware.
Yeah, I was looking for an explanation of PEMDAS, …..I’ll show myself out.
I love this example
It irritates me to no end that this is the thing most people remember about “math”. Our system is bad.
I hope it is a very sopisticated trolling of some sort, but I'm most probably wrong, but I still want to believe.
It's at least 1 million times more intelligent than Terrence Howard's "1*1=2", I gotta give him that. :D
If you knew nothing about math, you might take this as a credible video.
there is a much simoler, equally valid proof:
aleph-1 / 2 = aleph-0.5
QED.
why not the square root of aleph-1? Just as awful
Even beter:
aleph0 + aleph1 + aleph2 + ... = aleph(-1/12). QED.
I wondered why the "proof" looked so familiar and noticed that i commented on the original video 6 months ago lol. Nice video!
Aleph e, Aleph sqrt(-1), Aleph vector, Aleph matrix, Aleph group. Aleph HUMAN
Aleph pro max
15:23 "It's not even wrong" - Wolfgang Pauli
I always thought that was Woit, I didn't know that he was quoting Pauli.
at 10:00 he writes aleph1 < infinity < aleph 0, which is definitely not true regardless of CH haha
He's very confused about the underlying definition of aleph... anything.
I think MrSamwise25 said it best in this comment section, his understanding of aleph is:
- The whole numbers are aleph_1 because they are separated by a space of 1
- The real numbers are aleph_0 because there is no gap between them (a space of 0)
- So aleph_0.5 is the numbers separated by 0.5 (0, 0.5, 1, 1.5, etc)
This is what he means by "density"
Which is just funny. A complete misunderstanding of the subject matter.
The increasingly aggressive tone throughout the video is so funny to me, but I’m honestly amazed you managed to keep such a good temper
13:37 Your bijection sounds needlessly complicated. The canonical bijection from naturals to even numbers is mapping each n to 2n, right?
That would be a map from N to 2N. I described a map from 2N to N. In all honesty i'm not sure why I overcomplicated it 😂
@@themathyam Actually, the map you described sends the oddly even numbers (4n+2) to the even numbers (2n) and the evenly even numbers (4n) to the odd numbers (2n+1). 😅
@@cannot-handle-handles Oops! Arithmetic is not my strong suit
At 10:00 and all over his "proof", he has Aleph 1 < Aleph 0 🤣🤣
when this goes uncorrected by the uploader it makes me perplexed and confused i thought the problem is in me 😂
his channel and website is a whole rabbit hole it's wild
I read that this is a standard behaviour from cranks. The moment that it was proved that it was impossible to square the circle thousands of circle squares appeared. That is because they are not a part of the community working on solving open problems, but because they want to be that guy who proved one thing and became famous.
kinda feel bad for the dude
i hope people don't speak to him meanly
kudos for patiently breaking it down
and best of luck in your phd ^_^ i'm currently working on my B.A.
Don't feel too badly. He scams people out of money with his "4D mathematics".
He shouldn't be miseducating people, he deserves all the mockery he gets.
I wonder what the comments say on the video, they must be terrible, but has he taken the atrocity of a video down?
14:15 'outside' AND inside of the specified infinite set 😂
In a sense, the Russell's paradox can be seen as the statement that the axioms of ZFC, but with full Comprehension instead of restricted Comprehension, do not form a consistent theory.
Though, when Russell coined his paradox, he was referring to Frege's logicistic formalization of set theory (that hence was exposed as inconsistent).
When the video in the quarter of the screen has better quality than the full-sized background video.
Between any two irrational numbers there is a rational number.
Between any two rational numbers there is an irrational number.
However the cardinality of Q is ℵ₀ and the one of R is larger.
Hard to believe, but it actually is.
What I can imagine is meant at 4:45 is that there are countably many gaps (maximal intervals in the complement of the Cantor set) but uncountably many points (or connected components) in the Cantor set itself. This is true, though somewhat counter intuitive: at every step of the construction there is just one more component of the "remainder" of the interval then there are gaps; why does the difference become much more important in the limit? However, it is not fundamentally different from the fact that uncountably many irrational numbers are cut out from the continuum by removing countably many rational numbers. And the explanation given in the cited video ("there is two of those", vaguely waving their hands) doesn't even start to make sense just like everything else they say.
18:18 »not even wrong« - my sentiment exactly.
Subbed, cheers!
This guy who thinks he has a "proof" is the Terrance Howard of mathematics.
Ah ah ah, not so fast good sir! Terrance Howard is the Terrance Howard of mathematics
that was...quite a journey
I was completely baffled by why anyone would even do this, until I saw the "4D Yoga" comment. "Aaaah", says I.
Your introduction was excellent.
9:40 Reaction to reaction: cardinal numbers are defined and well-ordered without the axiom of choice. The only related thing that needs AC is that every set is equinumerous to some cardinal number.
What I meant to say is cardinality, i.e. assigning a cardinal to every set. My bad! As other commenters mentioned, one can also define cardinality for any set via scott's trick, but this isn't well behaved without AC, in particular, there are problems with comparability.
Why isn't anyone talking about aleph sub i (being i the imaginary unit of course)
10:51 He got Aleph1 and Aleph0 backwards. He labeled the naturals Aleph1 (it should be Aleph0), and labeled the reals as Aleph0 (it should be Aleph1 (really it should be 2^Aleph0, but he's sort of conflating that with Aleph1)).
sort of not a good thing to conflate 2^aleph0 with aleph1 when you're trying to prove the hypothesis that says 2^aleph0 = aleph1
You know your stuff and explained it very clearly. Thank you for this video
I knew it was fake when he started writing with his left hand.
I almost feel sorry for the guy. He's making a fool of himself.
Ok but I’m BOPPING to this bg stock music
Not a mathematician here, but I couldn’t get past his basic errors that even I understand: eg, sloppy about Aleph-0 and Aleph-1 and mis-spelling Bertrand Russell’s name right at the end.
Misspelled and mispronounced Georg Cantor's name too.
It's a bit like Terrance Howard who wants us to believe he "refuted math" but cannot even pronounce "orthogonal" or "circumference".
According to the logic of the guy in that video, the number of natural numbers is twice the number of even natural numbers, but in fact the count in both cases is aleph null.
By _nineteen hundreds_ he probably meant 1900 to 1909.
Kinda disappointing. I think there are almost certainly false proofs of the continuum hypothesis where it's genuinely difficult to find the flaw, but this one very much wasn't that. There's value to debunking nonsense, but I'd be more interested in response to more impressive false proofs.
I'm always open to suggestions if you have any ideas
You forgot to mention that the Cantor-set is important in measure theory. Also Georg is not George. The first close by in your definition of topological density is superfluous as well, you are already covering the entire space. Would be better to just say "for every real number there is a rational number arbitrarily close by".
Video was fun because I felt the exact same frustrations, however I doubt its validity as a teaching tool because anyone who knows what a model of an axiom system is, is gonna come even close to falling for whatever it was what that guy did.
I am finishing my maths bachelor, and I have legitimately no idea what he means. Like I know what the words mean, but this feels like an AI generated proof to me
ChatGPT told me the proof of the prime twin conjecture is trivial and then refused to write it down for me. :D
Update: th-cam.com/video/odPWvCPcqNk/w-d-xo.html "proves" that the speed of sound is the square root of the speed of light. This. Is. Dimensionally. Impossible. But all you need to show this is a compass! Wow.
Saying that you proved the continuum hypothesis is like saying you solved the halting problem or found an algorithm to calculate the Busy Beaver function. It just means you have no idea what you're doing. Please understand the difference between unsolvable and unsolved
Even if everything he says were correct, he'd have proven there is another infinity below Aleph_0 (i.e. Aleph_0 is not the smallest infinity), not that there is one between Aleph_0 and Aleph_1.
Agree the "in 2 infinity" guy is talking nonsense.
But remember a lot of the math you know disproved part of what went before. So don't be surprised when some one (not likely to be the "in 2 infinity" guy) comes along and disapproves some of the math that you know.
Would love an example to carry home with me
I’m sorry. Math isn’t a science. This isn’t “oh we have a theory and someone found better evidence so we moved on”. You don’t work with falsehoods. No one will “come along and disprove math”. Math is a set of things that have been proven in various systems with stated axioms. When’s the last time you saw a news item about a big piece of math being proven wrong?
I think that the funniest part is that he started by claiming there was two countably infinite sets, where one was "larger" than the other in the cantor set (5:55). But doesn't refer to it at all later on. He just invents a *new* mapping, just to keep things interesting. 🙃
Sorry if I sound stupid, but isn't the cardinality of the Quotient set of numbers or the Rational set of numbers smaller than the cardinality or the Real set, yet bigger than the cardinality of the natural numbers set?
This is a common misconception, but it is actually not! It is possible to match the rational numbers one-to-one to the real numbers, so they have the same cardinality (In other words we can "enunerate" the rationals, and there's actually explicit ways to do this)
@@themathyam thank you for the reply. I should probably learn a bit more about this topic, because I don’t fully understand your reasoning.
Two sets are the same size, have the same cardinality, if we can produce a bijection between them. If we can label the elements of an infinite set as r_1, r_2, r_3,... and never miss an element out (a.k.a. given any element, we can identify which n it will be so that our number is r_n), then it's in bijection with the natural numbers. Just send n→r_n, and r_n→n as your function and inverse function.
We can do that with the rationals. Label them *all* as r_1, r_2, and so on. There are good visual proofs out there, using diagonals.
@sable7922 think he meant one-to-one with the natural numbers btw
@@sable7922to show two infinite sets are equal you construct a method of mapping between the elements of the sets. The mapping must use every element in each set and not have overlaps
In this sense “countably infinite” is really just counting. You point at a fraction and say “one”, then “two”, etc.
Someone says “will you reach this fraction?” And you always say “yes I can show it will be reached”
It’s possible to assign a number to every fraction and not miss any or count them twice.
The little detail that's commonly missed is the presumption of a 'natural' order which allows the counting from zero[none]/one/two/etc (see the odds/evens example). Without that specific order (i.e. with some other arrangement) it might not be possible to do the counting (linear order of rationals?).
We lay people don't always realise that the 'order' (method of arrangement) really matters, and that some of the axioms (that are talked about separately) are actually equivalent - lots of circularity.
Lots of "arbitrarily large" recursion in some of the arguments ;-)
wdym the 'presumption' of a natural order, the naturals (as von neumann ordinals, and cardinalities thereof) are literally defined in order. no further assumptions are needed.
@@rarebeeph1783 The natural order presumption, is that you can 'obviously' count the rationals in value order, as opposed to the expanding diagonal diamonds order, and thus be able to determine the 'next' rational number after say 113/355, or 33102/103993, etc.
In the diagonal argument a specific order is used just to side-step the classic infinitesimals problem of 'what's next'.
@@philipoakley5498 why would anyone presume that? it's obviously false. between any rational a/b and another rational c/d, for integers a,b,c,d (where b and d are positive), there is a third rational (a+c)/(b+d) between them.
FYI: I have a Math PhD and a Masters in Math, both from Rutgers University, and 6 published papers in 3 separate peer reviewed journals. I know how insanely hard it is to come up with deep novel proofs.
#1 sign of bullshit: presenting proofs of major unsolved conjectures in math on TH-cam instead of the standard accepted proven way that all actual real professional mathematicians do: peer reviewed math journals.
Sure - I might prove some little lemma that only I am interested in, and post it on TH-cam. I may or may not publish it in a serious journal. It's up to the journal to decide who novel and groundbreaking my theorem and proof are, even if my proof is 100% correct. However, #1 sign of crockery is that the crocks always think they can tackle the big famous conjecture that everybody has heard of, and do it ONLY on TH-cam.
#2 sign of bullshit: filling up padding your "proof" with recaps of well-known history of math.
This must be incredibly frustrating to people who study this for fun/work, because, his proof, I think it completely misses the abstract complexity that the theorem is trying to define. How do you average a spacing unit of 1 to infinity with a variable infinite spacing unit to infinity? You need to define new mathematical mechanism. Different ways to describe numbers between other numbers, or something completely different all together to prove this theorem, right? I don't know this type of math that well. But I can tell his proof is not addressing the problem at all.
His math skills look exactly like those of Flat Earthers.
My question is: let's say we construct the ZFC + CH one, so there is an intermediate sized infinity - what would it look like? Basically, can you construct or describe of define in mathematical operations this intermediate infinity, assuming there is one?
the problem is that we can't. I think it's moreso that we know it exists, not what it would be. For example, according to ZFC, we can show that the well ordering theorem is true. (it fact the well-ordering theorem is precisely equal to the axiom of choice). This means any set can be well-ordered. However we can also prove that there is no way to find the well-ordering of the real numbers.
So we know the real numbers have a well-ordering, but we cannot find it. (assuming ZFC).
ZFC + CH is that there is *no* intermediate infinity. Your question is instead about ZFC + ~CH, that is, "plus not CH".
So then. In ZFC + ~CH, we have as a given that there is a cardinality between that of the naturals and that of the reals. That is, the cardinality of the reals is something other than Aleph_1. But Aleph_1 is by definition the cardinality of distinct well-orders of the naturals--in other words, the cardinality of "countable ordinals". So that is an example of an intermediate infinity.
(Of course, in ZFC + CH, we would have that the cardinality of the reals is precisely Aleph_1. The same set can be put in bijection with the reals in one model, and be impossible to put in bijection with the reals in the other model.)
ZFC +CH means Aleph1=2^Aleph0
ZFC-CH means Aleph1
@@kazedcat It's possible that there are also many other cardinals between them considering 2^Aleph0 can be any aleph in the aleph hierarchy (with some caveats).
But yeah, Aleph1 would be the smallest such infinity.
@@rarebeeph1783 sorry, sure I meant ZFC + NOT CH. But I still don't get it. If we say that in this case the cardinality of reals is less than 2^Aleph_0, then how to construct a set with cardinality 2^Aleph_0 ?
@11:23, you equate contradictions with paradoxes, as he did, but this is not quite correct. A paradox can be a provable statement that seems counterintuitive. An example is the Banach-Tarski Paradox. See the Wikipedia article here:
en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
You’re correct, of course, that what he’s discussing isn’t a contradiction; however, it may be paradoxical in the same sense as the Banach-Tarski Paradox.
Is that really a paradox though, or just an unintuitive conclusion?
@@xinpingdonohoe3978
Being an unintuitive conclusion is one kind of paradox. See the book by Sainsbury called Paradoxes. There are several kinds and this is one of them.
Russell’s paradox was a contradiction in naive set theory (Cantor’s formulation) and Frege’s attempted axiomatization of the same. However, a modification of Frege’s formulation of class theory seems to have removed that contradiction. Thus, the only sense in which Russell’s paradox remains a paradox is that the efforts to remove it did not yield a proof that the modified axioms (Zermelo-Fraenkel Set theory) are consistent, even though no contradiction has been found.
The Banach-Tarski paradox is so named because it appears to cause a contradiction in an area of applied mathematics closer to physics, in that it appears to lead to a violation of the law of the conservation of mass, because we can prove that there’s a decomposition of the unit sphere in three space into finitely many pieces that can be rearranged and reassembled using rigid motions to produce a sphere having twice the original volume. Physically this can be used to prove that if a unit sphere in 3 space comprised of a material of uniform density is subjected to such a transformation, then the result is a doubling of the material with no source of inflow of matter.
To a physicist, this should be considered a contradiction, expect that a good physicist who knows sufficiently advanced measure theory would understand that the finite pieces into which the original sphere is decomposed cannot be measurable and so cannot be assigned a mass in the first place.
That is, the theory of measures and its attendant result of Vitali that there exist non-measurable sets resolves the paradox to the satisfaction of the sufficiently informed physicist or mathematician, etc.
In this case, another way to resolve the paradox is to modify the axioms for your set theory so to consider all sets of real numbers to be measurable. Solovay proved that if you commit to the consistency of the existence of inaccessible cardinal numbers, then you can prove the relative consistency of such an assumption. I don’t know of any result that removes that commitment. However, in set theory, it’s common to refer to “Solovay’s model” in this connection.
Good news, I've also solved the Continuum Hypothesis. They've never heard of Alef 0.25
The countable set is countable because we use it to count.
Seems like the inequalities being the wrong way at 9:58 and the alephs being the wrong way at 10:55 wasn't even worth remarking.
He's basically saying to consider the ratio between his half step set and the naturals. He's saying that if we start counting the elements, we find at step n of the Naturals we've counted 2n of his half-steps. the ratio is of course 2n/n which, if we take its limit, should give us the relative number of elements between the sets, which is 2, and so his set is twice as large as the naturals. But we could do the same with a set which goes twice as far as we have in our counting of the naturals, but is just the naturals itself given more time to count, so we can conclude from deploying the same argument as him that the Naturals themselves are twice as large as the naturals. So its ridiculous and as you said, he thinks he's talking about cardinality but hes not lol.. Hes talking about the rate at which we progress through any countable set. Not about the size of the set, but about the rate of "counting" which we choose...
This idea is actually worthwhile, and can be formalised using the notion of "natural density", but obviously as you pointed out it is very very different to cardinality!
@@themathyam Yes but I dont even think it amounts to anything in itself. I'm imagining two people with the natural numbers laid out in front of them, and we assign them to start counting progressing up the line but we make one of them move faster than the other somehow, say twice as fast, and when the slower one has marked the n'th number, the second would have marked the 2n'th.. So one would have a greater "natural density" than the other. And his 1/2 step set is basically just somebody moving twice as fast along a countable set, we can just replace those symbols with the naturals and thats what it would look like to travel faster or whatever in our experiment.. Anyway sorry for rambling and I know this wasnt rigorous at all lol.
I don't know what the hell was going on at the end, with the +0.5 thing and the aleph-0.5.
He's literally as bad as Werner Hartl and his "proof" that cantors diagonal argument fails, its very painful to watch btw.
"I can make the steps between 0 and 1 infinitely small"
Anyone else notice he spelled “Russell” wrong. Look, I understand some people may not spell very well, but it seems a lot to ask me to believe a British-accented “mathematician” doesn’t have everything about Russell burned into his memory.
There's a reason they deducted points for spelling mistakes in math tests back in school. Sloppy work always shows itself in such ways first.
Great explanation. Thank you.
How you can try to prove cardinality of infinite sets without understanding bijection is beyond me
_"the world's most challenging problem of the last century"_
Immediately a bad start. I would surmise that most mathematicians would agree that the most prominent and challenging problem of the last century is either the Riemann hypothesis or the P vs NP problem. Most would probably choose the Riemann hypothesis if they had to choose between the two.
I would not fault him for making that subjective comment, frankly. I mean, yeah, I know what you mean.
What is a bad start is his immediate naive use of elementary methods. That instantly killed his proof before he went any farther.
was that video proving Aleph 1.5 or 0.5 (not sure, which), perhaps satire?
Lol, what a guy. Why a non mathematician would care about the continuum hypothesis is beyond me. It is very odd because he doesn't even understand the words he uses, and if he did, he would either go do the hard math required to understand CH, or realize that maybe that journey is just too much. Like, I don't even see how his video on the topic helps sell whatever it is that he is trying to sell. Very odd.
so... @1:17 people are still on the fence as to weather or not there really is anything between 1 and 2? so... no fractions, or infinity fractions?
Not at all. He states quite clearly that you *can* construct axioms for set theory in which there can be something between aleph0 and aleph1. But you can also do it in such a way that there is not.
If you're not familiar with it, it can come as a surprise, but mathematics is not universally True in the sense that it is commonly used. At more advanced levels mathematics can basically be the exploration of "what happens if we accept these rules (axioms)", and then "what happens if we accept THESE rules (axioms)?". From wikipedia, one of the axioms of ZFC is the "Axiom of extensibility", which is the rule that "Two sets are equal (are the same set) if they have the same elements."
Now, there's different ways to choose those axioms which will be self-consistent and generally seem plausible, but there's nothing that tells us which rules (axioms) we *should* use. There's no "true" construction of axioms; there's just a choice between somewhat equal constructions. So what he's saying is that there are at least two different ways to create the axioms, such that in one construction there *definitely* isn't anything between aleph0 and aleph1, but in the other construction there *definitely* is.
@@Multihuntr0 yeah, i already knew all of that. it was a rhetorical question to show interest, and to provide a footpath for responses, like yours :)
😎 Great video. You need to be a lot more careful about the word “cardinal”. The cardinal numbers are things like one, two, three, four. If aleph null exists (it does! in theory!), then it is a cardinal too. I work with a type of set theory that uses the terms cardinal, ordinal, cardinality but we study finite sets only (very useful terminology in the real world!). Anyway, you’ll have a lot better luck persuading your audience about infinities if you acknowledge that infinite cardinals are the same kind of numbers as the natural numbers! Remember that not everyone is on board with this stuff. Folks I know in engineering and compsci either think you guys are blowing smoke, or that it’s completely irrelevant. Math unfortunately loses its influence when it’s out of touch with application. 👍 big thumbs up, thanks for the video.
lol as the phd said there is no such thing as aleph_{0.5} the video he is reacting to is just total nonsense, it's a long time since I did this stuff but but yeah I still remember enough to say NOOOO to this rubbish
He wouldve been more believable if he had just said the proof was too big to fit in a youtube video
Well done. All of the false statements he made to the side, it seems that all he has done is to construct a countable subset of the rationals, but it's hard to say for sure. My understanding is that it is possible to construct a model (via forcing) that places 2^ℵ0 anywhere you like in the normal sequence of alephs.
Well, the printed models are pretty.
Aleph 0.5 is clearly counting all the cardinal numbers, but skip every other one. But Aha! You say, that's still alpeh null. But then only count half of those!
Hey! This is easy! Russell was a dummy!
His video is boring in its wrongness. I think the real question is why anyone asked you to watch such nonsense.
why?
why not
@@nicob9279 Y₀
I can't blame you for critiquing his video "as is" and you've done a great job of remaining calm and explaining things patiently to interested viewers. However, if you'd looked up in2infinity, you'd find they're full of crackpot nonsense and it's probably a waste of time to critique any of their videos. I'm fairly sure anyone with reasonable maths understanding is not taking them seriously. Those that are in danger of falling for it are the ones that need reaching... and I doubt this video will do that. I thought their whole thing might be a prank, but I see no humour behind it or evidence of prankery. I honestly can't tell if the presenters believe their own cr*p thought.
I mostly just wanted to laugh at how insane this is, but if you have suggestions for more plausible sounding cranks I'm all for it!
@@themathyam And you've done a good job of making an informative video that I enjoyed and I'm sure others did. But after googling this guy even more since writing my 1st comment, I'm more convinced that he's a fraudster and people might be sucked in. I don't know what's the best way of calling him out and warning people. Ideas welcome.
I enjoyed your video and subscribed. Now I need to look at this dude's channel!
Do this with terrence howards "theory" that 1×1=2 haha
Okay. I'll once give you one dollar; then you're going to have two dollars. Is that correct?
Brilliant 😊
Sure, there's no actual mathematics there, but he could help his credibility by at least spelling people's names correctly. "George" Cantor? "Russel" paradox?
Cool video :-)
My one thing would be that you should definitely get a better mic! Sound is a bit inconsitent throughout :)
I did! Next video will hopefully have better sound.
I love these stupid "proofs" that just immediately make no sense at all! They're terrible if you don't know they make no sense, but quite funny if you do know
They also love jumping from basic stuff to super complicated stuff with no preparation or definitions or anything that would help anyone follow their "logic"
I think your introduction might be wrong? We say that 2^ℵ0 is the cardinality of R because R should comes after ℵ^0. That's precisely what 2^ℵ0 means and is nothing between ℵ0 and 2^ℵ0,
If there was, let's say, some intermediate set A, then |Z|=ℵ^0, and |A|=2^ℵ0, and then |R|=2^ℵ1
"We say that 2^ℵ0 is the cardinality of R because R should comes after ℵ^0".
No, we say that 2^ℵ0 is the cardinality of R because there is a bijection between R and the power set of N (the set of all its subsets), which is known to be strictly greater than ℵ^0 because of Cantor diagonal argument
What comes after ℵ0 is (by the very definition) ℵ1, and the video correctly states that the continuum hypothesis is the affirmation "ℵ1 = 2^ℵ0"
Then your example would just be |Z|=ℵ^0, and |A|=ℵ1 and |R|=2^ℵ0
with ℵ1 being of cardinality either ℵ^0 or 2^ℵ0 or not calculable (because if it were strictly in between it would be a proof of CH)
Who is this guy?
Ha ha, just like the God Hypothesis 😅
The existence of a paradox doesn't mean the existence of a logical contradiction. It can also just mean the existence of something that seems like a contradiction at first. This is more of a philosophy thing than a math thing usually. The position of accepting the counterintuitive seeming contradiction is commonly referred to as biting the bullet.
Sorry but you take such a long time explaining things that you sap my patience before you're finished.
since the real numbers is Aleph 0, the slope is clearly 0
#1 sign of bullshit: presenting proofs of major unsolved conjectures in math on TH-cam instead of the standard accepted proven way that all actual real professional mathematicians do: peer reviewed math journals.
Sure - I might prove some little lemma that only I am interested in, and post it on TH-cam. I may or may not publish it in a serious journal. It's up to the journal to decide who novel and groundbreaking my theorem and proof are, even if my proof is 100% correct. However, #1 sign of crockery is that the crocks always think they can tackle the big famous conjecture that everybody has heard of, and do it ONLY on TH-cam.
#2 sign of bullshit: filling up padding your "proof" with recaps of well-known history of math.
EXCELLENT AMERICAN YOUNG MATHEMATICIAN WHOSE NAME UNFORTUNATELY WASN'T POSTED!! CONGRATS ANYWAY.
th-cam.com/video/JUVYBBlxw7E/w-d-xo.htmlsi=AJaGS3OyNsRHnW6k
BTW the confusion between density and cardinality was also the root of the misunderstanding of a famous political and Christian Brazilian philosopher called Olavo de Carvalho in his best seller "O Jardim das Aflicoes". Olavo de Carvalho has died recently in the State of Virginia. He was a great Aristotelian philosopher and a political figure involved with lots of political controversy (he was an anti communist fella and a true iconoclast); Also in 8:15 this smart boy also catched the other layman confusion: the absence of domain/definition of sets and functions (there are no numbers or points or whatever between integer numbers in the Natural numbers. They also confound the sequence of symbolic representation of quantities as the quantities themselves) - we mustn't say that any competent philosopher would be also capable to understand math and the natural sciences. Sadly, the greek intellectual philosopher like Euclides or the own Aristotle is a very hard specimen to be found in the complex modern world. Perhaps Carl Sagan was such a good example or perhaps not.😊
PS. In the high school teachers usually say that the integers is an infinite set which is a subset of the infinite set of the reals (it creates a cognitive bias: as the reals would be just only a denser set of integers). 😊
It's like this guy is spitting on Paul Cohen's legacy. What an asshole.