A Hierarchy of Infinities | Infinite Series | PBS Digital Studios
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- Опубліковано 7 гру 2016
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There are different sizes of infinity. It turns out that some are larger than others. Mathematician Kelsey Houston-Edwards breaks down what these different sizes are and where they belong in The Hierarchy of Infinities.
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Proof that the real numbers are a bigger infinity than the natural numbers:
germain.its.maine.edu/~farlow/...
www.people.vcu.edu/~rhammack/B...
Thanks James Barnes and Iian Smythe!
Hey all! Just wanted to give a clarifying not. There is no general consensus in math as to whether zero is a natural number or not. To precisely distinguish them, one can use the phrases "non-negative integers" or "positive integers." However (!) in set theory, which is the branch of mathematics being discussed in this episode, there is a consensus: Zero is a natural number. The set theoretic constructions of the natural numbers (e.g. in the Peano Axioms) includes zero.
logicians count 0 as natural number and analysts don't, it makes sense if you think about it
Gotcha. But one of the things that confused me is that she also referred to them as the "counting numbers", which is disjunct from that ISO 80000-2 zero-inclusive definition, sometimes used by logicians.
hi just whated to ask you something
what if the so called rule of bijection
dont apply to infinite sets..
+akhil prasad sebastian - Why wouldn't it? The definition of "bijection" is independent of what size the sets involved have. There is no reason to specifically exclude infinite sets from this definition.
Furthermore, the rule of bijection is the _only_ rule there is for determining if two sets are the same size. Even simple counting is merely a bijection between a subset of the natural numbers (1, 2, 3, 4, 5) and another set (the fingers on my right hand). Arbitrarily excluding infinite sets from that rule doesn't make sense.
Sure, you _could_ exclude them, but you would end up with much more boring and much less useful math. As it stands, the definition of bijection works just fine for infinite sets and so does the definition of "same size".
PBS Infinite Series
Could just say natural numbers include 0 and the positive integers doesn't (as 0 is a neutral not positive integer) to distinguish between the two sets...
I'm a math undergraduate student, and when I was 12 years old the subject of different infinite sets was one of the reasons I liked math so much. But I think this video is incomplete. I would have liked to see explanations for Cantor's diagonal proof and why the rational numbers are countable, and maybe an explanation of the power set and Cantor's theorem. It would be amazing if you could have that as a followup video next week. This topic is huge, and one of the most interesting ones for non-mathematicians.
There are vast of topics. I guess she could talk whole time only about infinity :D.
She says there's a link in the description to a rigorous proof, presumably that's Cantor's.
Totally! The whole series could be about infinity! Y'all have great suggestions for thing you'd like to see in follow-up videos
+ imaginary and complex planes?
He said next week. :)
"In english, this means that..."
Haha, I've used this phrase too when explaining concepts after a formal definition!
I just realized that the definition of cardinality actually relates really well to the Pigeonhole Principle.
Avg. length of each infinite series vid is about 8 min. 8 is like infinity standing up.
Standupmaths?
Lemuel Ogabang I initially misread that; I thought that you meant standing up for 8 minutes (as the presenter does here) feels like standing up forever.
Lemuel Ogabang This is so nerdy and great that I have to reply, but I don't know what to reply with except this.
What about 24.
Oh my god. How did we not notice this before. And if you add one to 8 you get 9. Taking the square root of 9 gives you 3. The same amount of sides of a triangle! PBS Infinite Series illuminati confirmed
How can you just *assert* the real numbers are a bigger set than the natural numbers? Cantor's diagonalization proof is so simple and elegant! Why leave it out?
+DontMockMySmock You should go read the comments in Vsaurce's video about it. Non-mathematicans get very angry when you tell them one infinitiy is larger than another. It's bemusing.
Cantor's diagonalization proof only proves that the infinite list of real numbers is not finitely complete, which is kind of pointless.
+Kai Kunstmann Say what now? What does 'finitely complete' mean here? I tried to look it up but the only context I could find was topology which I know almost nothing about. What does it mean for a set to be finitely complete? And why is it pointless?
M. Scott Veach: based on another comment that Kai Kuntsmann posted on this video, my guess is that he is trying to say that the diagonal argument merely shows that the real numbers are not in one-to-one correspondence with the set of natural numbers which have finite value. In this other post, he claims that there is indeed a one-to-one correspondence between the interval (0,1) and the set of natural numbers, but you need to use natural numbers with infinitely many (nonzero) digits to do so.
Of course, there is no such thing as a natural number with infinitely many (nonzero) digits, and all natural numbers have finite value, so Cantor's diagonal argument _does_ work.
+Muffins aha.. in other words, i can safely ignore.
"There're infinitely many sizes of infinities." Are there countably or uncountably many different infinities?
Uncountably many
Equivalent to continuum hypothesis tho
If you got enough time, then yes, countable. 😅
fun fact: both
Much much more than either
I truly love this series. Well put together , informative, and enjoyable. I appreciate all the work you all do to put these videos out. Many thanks, MM
Man I hope you go into the incompleteness theorem more!
^ This. I'd love to see a Infinite Series video on that, with cool animation and shit.
DekuStickGamer Ya, it is one of those areas I can't hear enough about because it is just so....odd.
Same here, that's one of the weirder parts of math and it's actually one I don't really understand. I would love to have someone explain it in a format like this.
I'd like it, too!
***** Gödel escher and bach would be a good place to start.
I didn't understand this when vsauce or vihart explained it, but I get it now! Thanks!
V’s always make things confusing
+Jan Coker
Yeah, he should just stop doing what he does. He shouldn't confuse people.
@@romanski5811 no he shouldn't, there are plenty of people who do get informed. There are other peopme you can go to but he doesnt need to stop
The real numbers contain irrationals while the whole numbers contain rationals or fractions. Since all fractions and whole numbers can be ordered as a list, they represent a smaller infinity than the irrationals which can't be ordered as a list. They are thus a greater infinity. Moreover, you can always create a bigger infinity by creating subsets of each whole numbered set infinity. You can also create infinite subsets of those subsets. Thus you can create an infinite number of infinities.
What a fantastic series! Thanks for helping me explain to my family what I love and want to spend my life doing!
Love this channel!
Keep up the good work!
Nice video! but I believe you could deliver more information in a single video. When I watch pbs spacetime, every video feels to have optimal information. Spacetime uploads a video of about 11-13 mins which is nice. I do believe you could do more.
Agreed! It is also possible to add information beyond what is spoken: I like that Space Time throws up text blocks in the background (generally from a Wikipedia entry) that are readable by pausing.
It's a new channel. They're still optimizing their style and, let's say, intellectual level of the audience. She sometimes repeat stuff seconds of having saying the same thing and such, I think that's what you're feeling.
Anyway, it's a great channel.
I had to watch the video many times over to truly appreciate the writing. Awesome stuff !
Who wrote the amazing intro/theme music at 10 seconds? I love it! The synthesizer has such a nice sound. Can you please attribute them in the description, I'd like to follow up and listen to more of their stuff (and hopefully a full length version of the theme).
Vsauese's video "count past infinity" is an other awesome video of the subject.
Bálint Áts but he actually to much talking rather give a argument.
Bálint Áts the vsauce video is better than this one.
Vsauce videos are non educated and just for 'show' like common drama.
I'd love to see some differential geometry, like Gauss' Remarkable Theorem or the Gauss-Bonnet Theorem, even if they take a few videos to build up to. I thibk it would look great animated.
Great suggestions! Thanks
The explanations are really clear. I actually understand the Video's context!
This is an excellent explanation of cardinality. Amazing job!
Are the complex numbers larger than the real number or the same size?
They are the same size
Are there larger infinities?
The class of all sizes of infinities is so big, it doesn't even fit in a set!
What's another example other than the two given in this show?
Such a good question! There are. So many of them! They just keep going up and up and up... which would be a great future topic.
It's hardly a matter of opinion. Pick a rule set. Is it true there?
Since it's independent of ZFC, you are free to make the choice and then figure out what happens if it's true and what happens if it's false. Then you can pick which system is more useful for what ever it is you are trying to do at the moment.
not really
Many important mathematical results are related to the specific ZFC model
Ever since finding this concept in like, first year math, I found it fascinating!!
Just subbed due to this amazing video and the clarity in which it was presented! Great work
I thought there was a natural ordering of the infinities which is that each higher infinity is the power set of the previous one.
Melinda Green I believe the continuum hypothesis implies that this is the hierarchy of infinities. But as they mentioned in the video, it would also be compatible with ZFC if there were infinities between any set and it's powerset.
Shawn Ligocki
What would such a fractional infinity even look like?
I don't think that anyone has proposed any description for a set that would would have one of these in-between infinities. But for finite sets, there can be sets between a set and it's power set. For example, between 2 and 2^2=4 there is 3.
Shawn Ligocki
I can't get my head around the idea. It sounds like someone saying that maybe round squares exist but nobody has found one yet.
No, it does not follow even if you assume continuum hypothesis. Your statement is the so called generalized continuum hypothesis.
No mention of aleph?
Sad.
@@zuccx99 Go right ahead and create a video - I am sure we all would be most interested. Post where you put your video in the comment section here. Looking forward to it.
No, it too Borges.
ooops (sic) it's
Great video. I had never really thought of the boundary between regimes in such a way. Though, it is obvious in our speech as Sean Carroll describes. But yeah, I suppose we have to (i)magine or invent new ways to talk about new regimes and discuss the unique maths that reside within them.
I love this channel. Keep up the great work! I can see you aren't - but please don't be afraid to get as technical as you like - it's awesome. Bijection - sweeeeeet.
Is there any proof that the natural number infinity is the smallest? Or is that just assumed?
What do you mean natural numbers and natural numbers are the same size? It's literally half of the natural numbers. How can that equal the same amount? Does she mean you have 100 percent of each set within each set? If that is the case, wouldn't odd numbers also be in the same hierarchy of infinities as even numbers?
The even numbers are just an example. Numbers divisible by 3 or 7, or numbers where the equation "n % 5 = 2" is true would all work. It's just about the bijection and not about the individual numbers.
It's essentially because infinity means there is no largest number. Infinity isn't really a number on its own, its more of a process that never ends. For every natural number that exists, you can pair it with an even number. If the sets ended eventually, they would be different sizes. However, since they are infinite, they must be the same size, because every element of one set can be matched to an element of the other set.
Ian Krasnow What sets definitely end?
Finite sets
Ian Krasnow some examples?
Glad she covered the continuum hupothesis! it's fascinating how it could true or false, but it doesn't matter!
awesome, great channel with a lot of potential in sense of added value (looking at the CC, PBS, CGP grey landscape).
This video is infinite confusion
that gap tho :o
Thank you Kelsey. Amazing show.
Mind Blown. Totally loved this.
What about complex numbers?
The complex numbers are the same size as R^2 (that's two-dimensional real space, like a plane). Here's the bijection: the complex number a+bi is paired up with the point (a,b) in 2D real space. Now, how big is R^2? It's the same as one-dimensional real space, the real number line! Can you find the bijection?
I'm gonna think about it a little bit. By the way, the videos are awesome and the content is extremely interesting, congratulations.
It's a while since I did Maths at this level but this question looks _hard_. Is there an elegant answer? I ended up Googling and was constantly referred to the Cantor-Schröder-Bernstein Theorem. Am I missing something?
Luis Vasconcellos The Hilbert curve does exactly that. A bijection betwen 1D and 2D space.
Actually not quite. The Hilbert curve is surjective, so it covers all of R^2 and does what we need. But it's not bijective because it's not 1-to-1. If you really want a bijection you can't use a continuous mapping.
I think, infinity is *evil*.
In many senses.
1) Induction proofs, as part of infinity.
2) Hilbert's Infinity Hotel.
3) Infinity in space and time. (related to Achilles paradox, and Arrow paradox)
4) Infinity in integrals, or series.
5) Banach-Tarski paradox.
Everywhere it's evil. Even in Pi.
Btw, fractal - is evil with evil inside.
Without evil, you can't even calculate the area of a circle. we need the evilness
Yes. We don’t need this concept in mathematics with computer today. All are finite and human being is keep advancing ahead!
Cauchy Riemann In fact, the area we got was always not right but just approximation only.
Hoson Lam but the whole analysis (calculus) will collapse without axiom of infinity, I deem ita as necessary for now, for we don't have alternative choice now
Cauchy Riemann Agree all previous continuous calculus cannot be defined rigorously. However discrete version of the calculus with help of today computer will provide similar powerful tools to solve similar problems alternatively. If Newton has today’s computer, he may not necessary invent/define differential calculus at all.
I love this video. the fun stuff we did in my proof classes long ago.
Part of the 'problem' [esp for communicating with the lay person] is that 'infinity' itself isn't that well understood in the first place. The set 'goes on and on' aspect, and the separate 'counting' aspect are distinct concepts that get confounded when the set is the 'integers' that appear to match the countings.
The bijection between the positive integers and the evens is between _different_ sets (and their particular orderings). Both sets 'go on and on' in a definite countable order so are of the same 'countable size'.
For the rationals, the ordering isn't (for the purpose here) by linear value, rather by one of the diagonalization orders. It is that ordering which makes the set 'countable'.
Having decided that one _can_ count the rationals, there is a flip to an order that doesn't appear to have the countable property (but is the same set) that is then used to show that the reals are definitely larger even though we get into the 'alternating' vs 'between' problem of reals and rationals (i.e. reals having smaller infinitesimals that the rationals ;-)
If you want to further confuse the issue you get into the 1.000000... being preceded by 0.999999... for some arbitrarily small infinitessimal ! Monty-Hall had it easy.
She didn't even get into surreal numbers, because she didn't want to get brain matter all over your keyboard.
kj01a The surreal numbers don't form a set; they form a proper class. So, they don't have a cardinality. But, one could say that they are too big to make a set, which is tough to think about.
This is my favorite topic in mathematics of all.
It would be great if you could recommend a book at the end of each video that goes into greater depth over whatever topic you covered. In this case, "Uses of Infinity" by Leo Zippin is something I've heard a lot about.
0:57 It took literally less than 1 minute to blow my mind this time!
Just finished by linear algebra final, now I finally get to relax while learning math : ))))
How did it go? I loved linear algebra!
I got a B after trying like hell towards the end. I coulda woulda shoulda got an A, if I had worked a bit better from the beginning, but overall, since my professor was great, I actually really enjoyed it. : )
As a Harvard professor once said, we teach Linear Algebra like a throw-away class for the Physicists but it turns out ALL of Math is Linear Algebra--you can never know too much Linear Algebra.
That makes a lot of sense, haha.
I was deeply confused until I realized you were talking about cardinality and not the actual size of infinity or how fast something approaches infinity. Please make a note somewhere about it but I'm looking forward to seeing followup videos!
Good thing Physics Girl mentioned this channel or I'd've not found it this early. Thanks, PG! To my point: It's a choice to accept that a bijection may define the equivalence of magnitudes of infinite sets. I for one reject this definition and hence also Cantor's diagonal argument that assumes actual infinities. I believe there's one potential infinity of one infinite magnitude that can not be actually reached even with a supertask or any thought experiment, and that an absence of a bijection is no proof that two infinities would have different magnitudes. The pigeon hole argument holds for all finite numbers, but applying it to the infinity is an error. I think that a countable infinity is an oxymoron simply because you cannot count up to a potential infinity. Infinite hierarchy of infinities is a nice playground, but alas, as already Aristotle said, "infinitum actu non datur" (there is no actual infinity). Despite mine and Aristotle's opinion I do enjoy all kinds of maths and I want to send a big thanks to PBS for creating this show!
One quick note, the set of natural numbers, denoted N, actually doesn't include 0. The next set, the integers, is when 0 is added.
See the pinned comment.
The set of natural numbers is called countable or countably infinite, while the set of real numbers is called uncountable or uncountably infinite. There are other surprising results about countability and uncountability. For instance, the cartesian product of two countable sets (defined as if a is in set A, and b is in set B, then (a,b) is in the cartesian product AxB) is countable. The union of two countable sets is countable. The rational numbers are countable. Therefore the irrational numbers are uncountable. However, it turns out there are countable and uncountable partitions of the real numbers which include some infinite sets of irrational numbers; for example, the algebraic numbers include the irrational algebraic numbers, and are countable.
"All Brontosaures are thin at one end, much much thicker in the middle, and then thin again at the far end." - Anne Elk
I was about 15 when I encountered Aleph Null, Beth...etc... Mind blowing stuff (in a book in Leeds University library)
The background music is beautiful.
Thanks gurl. I learned a lot of math (8 semesters just from calculus) and never could grasp the continuum hypothesis. But now I think I do.
My only question is, is there an alef omega infinity? An infinity which as infinitely many infinities smaller than it?
I've heard proper forcing axiom would allow you do prove continuum hypothesis in a way that makes sense. Maybe episode on that? tried reading about it but its complicated
Excellent, I want more!
More!
Also, if the continuum hypothesis is independent of ZFC, how would an infinity in between the integers and the reals look like?
the same way you can have a space with 2.5 dimensions
I would like to commend your great efforts.
It would be a nice idea if you introduced Codel Model's of Sets and P.Cohn model's of sets. The method of Forcing, introduced by Cohn, would be a nice topic. Best Regards,
There is a different concept on how to compare the magnitude of infinite sets called "numerosity" which preserves Euclid's principle "The whole is greater than the part". Maybe you could also do a video on that? (See recent papers by the italian mathematician Vieri Benci).
The intro music to this channel is kind of badass
Great video; I hope you shed some light on the axiom of choice, which is the main reason why Continuum H. is unprovable
AC is not the reason. It will still be unprovable without AC.
Where do you get that type of background music?
The rational numbers and the natural numbers are both size Aleph Null right? I believe I saw a diagonal-style pairing back in a math class.
10 seconds in and my brain already exploded
I think a more certain definition for describing sizes of infinities can be by their dimensions. Simply, if 'natural numbers' is a one dimensional infinity, then 'real numbers' is a two dimensional infinity because between every possible interval of the natural numbers set, there is an infinite set of numbers. Similarly, we can define a 3 dimensional infinity by describing an infinite set of numbers between every possible real number. Like the coordinate system but only more linear: Instead of describing an infinite set of numbers for every number, describe an infinite set of numbers 'between' every two numbers. In fact these can be translated to each other: for every interval you can define a regular multidimensional array by associating the infinity set to one of it's interval's endpoints
There are infinite number of rational numbers between each pair of natural numbers. But still the all the rational numbers can be paired with natural numbers - they have the same 'level of infinity'.
Eneri Giilaan are they? if so doesn't that mean any infinite series can be paired the same way? Because any infinite series should theoretically be indexable, meaning they can be paired with natural numbers
Agreed that this is a quite unintuitive - but still 1st year university math.
For a relatively decent explanation search for UA-cam video: Infinity is bigger than you think - Numberphile.
very sorry for the late response. I got exams in college, I had watching that video in mind all the time but I didn't feel like it :D these exams are really exhausting, just wanted to say I will watch it sometime :) (yeah, still haven't ^^)
Really helpful! Thx!
Oddly enough, when my professors covered set theory back in college (early 1980's for me), they didnt use the term bijection ... I like it.
So for the bags of pennies analogy, since every penny has standard size & material (pure copper) and thus identical volume & mass, we could use any difference in either weight or displacement of water as forms of bijection in a proof that the contents of one bag was larger than another (caveat: exempting 1944 pennies made of steel instead of copper).
In the '70s they used "bijection".
On the curved interval between 0 and 1 would infinity have a ray parallel to to the line or an infinitely small angle or are they the same?
It was described as an open interval. So the endpoints are excluded.
I want a full version of the intro music. That beat is fire.
I have thought about this for a while, it seems like you paired the interval between 0 and 1 with every integer and number between
I was disappointed that you never mentioned Cantor's diagonalization argument, because that was what got me to understand the difference between countable and uncountable infinities. Also, how it can be used to prove very intuitively that the rational numbers are countable. Also the fact that mathematicians call them countable and uncountable infinities, not "The sets of natural numbers and the set of all real numbers".
Can we segregate infinities of the "same" size by how fast the elements in the sets approach infinity? Meaning use "size" and "rate" to define different infinities?
What could you subtract from infinity to result in a natural number once you have reached infinity? Is it possible to add the inverse of any real or imaginary number to result in a natural number? Would infinity plus negative infinity still be infinite in set size?
Excellent content! keep going! :)
Bijection alone doesn't convince me that the complex nums (or quaternions, etc.) are
the same size as the reals. Only when I think of the Hilbert curve does
it seem plausible. Because you're mapping a higher dimension object onto a lower dimension one, you see, and Hilbert showed there is a bijection between points in R^2 and points in R^1. Just having a beer here. Thnks for the beautiful video, please keep it up.
the definition of "having the same size" is "there exists a bijection". Any other thing than that would be a completely different concept of "size"
Ah, no mention of Omega Omega.. Corvaire is disappointed. :O(-
I'm surprised there was no mention of the rational and irrational numbers? Rational numbers as a dense subset of the reals is quite amazing
1-1/(x+1) is also a bijection of the interval from 0 to 1 and 0 to infinity.
4:48 Why cannot I extend the (0,1) semicircle into a 3/4th circle or a complete circle excluding two points to argue that all the real numbers have a one to one correspondence with just the semicircle and I still have infinitely many points left over in the interval (0,1) to show that (0,1) is bigger than the set of all real numbers?
Can you please speak more to the other models of infinity towers and what infinities they place in-between Uncountable and Countable infinities?
That there is no set whose cardinality lies strictly between natural and real numbers is the continuum hypothesis; and this proposition can't be proven true or false in set theory, assuming set theory itself is consistent. In other words, the set theory axioms are insufficient to settle the question one way or other - the proposition is true in some models of the set theory and false in others.
The set of all countable ordinals has a cardinality of aleph-1, therefore if the continuum hypothesis is false, it's an example of a set bigger than the naturals, but smaller than the reals. If the continuum hypothesis is true, then it's the same size as the reals.
I understand that we don't know whether the continuum hypothesis can be proven (or disproven) under other set of rules (other than ZFC). Am I right?
Is the cardinality of the set of primes equal to the cardinality of the natural numbers? It would make sense because the primes can be paired with their indices (which are natural numbers).
I really liked the video, but I wished you went into some levels of the infinity tower more. There are more levels than just 2 (Natural Numbers and Real Numbers) which you could go into.
Amazing Channel!
I would love for you to go more in depth on this topic. Miniseries?
I'll start by saying I'm new here, and I love the show.
Now, I'm not a mathematicion, but I think I've found a way to prove the existance of infinaty bigger than the natural numbers, but smaller than the real numbers, that this infinite number of infinities between it and the real numbers infinity, and that it shows that there are infinate infinities bigger then the real numbers infinities. My guess I'm not as smart as most mathematitions, so I'm probably wrong about something - but maybe you can help me understand what went wrong.
I added a conjecure, that wasn't specified in this video, but I find kinda obvious. If I can pair all members of group A to the group B - both are the same size. However, if I can only map all of group A only to part of group B, and I can't do this the other way around - group B is bigger than group A. For example, if there are 3 sits and 2 kids, I can map one site for each kid, and still be left with one sit available. 3 > 2.
First, instead of the integers group, I've decided to talk about the integers on a number line. You can easly see that I can map every number to a point on the number line, so they are the same size.
But the number line, is just the X axis of a graph. I can imagine this graph having a Y axis too. Now, I may be lacking in imagination, but I can see no way of paring all of the points on the number line, to all possible points on the 2D graph. I can pair the other way around, by just choosing a constant value on the y axis, like this 1 => (0,1), 2 => (0,2) and so on. This method will still leave me with infinate points on the graph.
Now, the same way I did this, I can build a 3D graph. It will be bigger then the group of all integer points on the 2D graph, and naturally, bigger than all the integers group ( that is (1,1) => (0,1,1), (4,5) => (0,4,5) and so on). If I have some imagination, I can build a 4D graph, and get an even bigger group, then a 5D and a 6D graph - up to infinity. My guess, which I cannot prove right now, is that the real numbers infinity, is the size of the infinite dimantions integers, but this is just a guess. I don't reallt know if this can be proven.
This graph method, can be used to handle the real numbers too, and create a group bigger than the real numbers - using a 2d graph of all real numbered points on a graph.
I hope my horrible english, and lack of formalation, didn't get in the way of me explaning this idea, but I'd love to hear a mathmatition rebattle of it.
Thanks in advance
Yair Eshel Cahansky there are many known ways to pair 1d with 2d or 3d. One way would be to do a spiral from origin. (0,0), (1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1), (1,-1), (2,0), (2,1), ....
I think that it’s interesting that the infinite amount of complex numbers is somewhat ambiguous. I have not explored the idea yet, but space filling curves could provide an answer. Perhaps because an interval of length 1 can be stretched to be infinitely long, the logical jump is being able to use space filling curves to prove that a line of length one can fill in a square of area 1. Because of the loose definition of a rule, we can see that we are adding up a countable amount of uncountable infinities. This makes it clear that when the curve is thinned out, it is a whole number line. Thus the number of complex numbers is the same as the number of real numbers. Dimensions higher than 2 can be rather interesting though. If there is a space filling curve that can fill a hypercube of dimension-n & can be easily tiled, then the size of those infinities is the exact same. Maybe exploring 3D and higher dimension Hilbert Curves could give some type of answer.
There's nothing ambiguous about the cardinality of complex numbers; there are precisely as many complex numbers as real numbers (the two sets can be mapped one-to-one). Sure, spacefilling curves give you an idea that this should be true; but note that a continuous spacefilling curve can't be one-to-one - it intersects itself infinitely many times. We can instead use decimal expansions of real numbers - x=0.(x1)(x2)(x3)..., y=0.(y1)(y2)(y3)..., and construct from any pair of real numbers another real number: 0.(x1)(y1)(x2)(y2)(x3)(y3)... (or vice versa). But again, we need to be careful when doing a formal proof - some real numbers have two different decimal expansions, such as 0.1999...=0.2000... . (A formal proof may use the Cantor-Bernstein theorem that if set X can be mapped one-to-one with a subset of Y, and Y with a subset of X, then there exists a one-to-one function between the two sets.)
Hey just saw this video today. Not sure if you are still answering questions on this. But what about the complex number set which has all real numbers and numbers like square root of -1 and so. Can we identify a bijection between Complex numbers and Real numbers ? Or is the complex number set a bigger infinity than the real number set ?
The complex number set is indeed in bijection with the real number set. The easiest way to do this is to prove that the complex number set is in bijection with ordered pairs of real numbers (where a+bi is paired with (a,b)), and that the set of pairs of real numbers are in bijection with the set of real numbers.
There are many different, but tricky, ways to prove that R is in bijection with R^2. I don't know of any "direct" proofs of this fact (in that they don't ever seem to explicitly establish an actual bijection, but rather use set theory facts to show that a bijection must exist).
For example, using cardinal number arithmetic, you can prove that the cardinality of R and R^2 are the same:
|R^2| = |R|^2 = (2^(ℵ0))^2 = 2^(ℵ0*2) = 2^(ℵ0) = |R|
Or some people use the existence of space filling curves to show that there exists a surjection from R to R^2. And we also know there exists an injection from R to R^2. A set theory fact is that if there exists an injection and a surjection from one set to another, then there exists a bijection between them as well.
And there are other proofs too.
What happens with infinity between reals and complex numbers, quaterinions, octonions and so on......also, can you make a video for ZFC with quantifiers (thanks a lot. Jan Pahl from Caracas, Venezuela)
thanks a lot
Hey guys, any chance on the music, used in this episode? its really good!
Your proof for the intervals being of the same size as the real numbers seems a bit ambiguous.
When you drew the semi circle, the end points of the semi circle would have a slope that is paralell to the real numberline and therefore the rays from the center of the semi circle would never intersect with those two points (the numbers at the end points).
So my question is whether or not a closed interval is of an equal infinity of an open interval.
I'd really appreciate a response!
What sort of axiom(s) would need to be added to ZFC, such that they don't create inevitable contractions with known theorems, but would allow Cantor's conjecture to be proven or disproven?
I have a question.
You define the Natural numbers as 0, 1, 2... and you say that evens and natural numbers (counting numbers) are the same size, adding zero to the set that includes 2, 4, 6....
However, I had learned that the Natural numbers are actually 1, 2, 3.... (no zero, as that was what I understood as "whole numbers"), meaning that the infinite sum in a physical construct as defined via the Zeta Riemann function series sums to -1/12, with a cardinality of aleph null. Am I missing something, as I have little doubt you have a broader and deeper math background than I do?
As the pinned comment at the top explains, set theory (the realm of this video) counts 0 as a natural number, where other areas of mathematics may not.
The term "natural numbers" originally meant the "counting" numbers {1,2,3,4,...} .These are the numbers that children use to play the game of "hide and seek" and have been around for many millennia. The number 0 essentially didn't exist for the Greek mathematicians such as Euclid, whose book "Elements" was the mathematical standard from 300 BCE until abut 1850 CE. The number 0 was incorporated into Euclidean math in India after the Roman conquest of Greece. I personally wish that modern mathematicians had invented a new name for the set {0,1,2,3,4,...} because, for me, 0 is not a natural number.
0:07 ... Well what is the size of the set of all infinities? What type of infinity do we assign to the "number" of infinities?
The beginning. There are "Bigger infinities". It sounds like semantics for just saying, "Infinity is Infinitely bigger than itself."
if there is cardinality in the infinities lowest (natural numbers) and highest (real numbers)
then there is a cardinality in binaries the highest significant bit and the lowest significant bit?
However in order to know highest and lowest bit it is not infinite any more ??
Do you know how to tie the lose ends together in retrospect of the continuum hypothesis?
(and use non-standard analyses as a tool)
Why can't I n++ my like if I watch the video more than once. This is a beautiful video (not just because the host is beautiful). I want to hit like more than once.
I've a question. we know that the distance between two physical points on earth namely x0 and x1 is real infinite. And we also know that the difference between two point of time namely t0 and t1 is real infinite too. And we know that we can travel from x0 to x1 in (t1-t0) time with a constant speed. Is that means division of two real infinites may be a natural number or we are not actually moving and living in inception world (!)?
In the function y=a/x, the value soars to infinity when x approaches 0, but it happens with different speed, depending of the value of a., hence many infinities. A handy method to handle this, is to define a unit of infinity. Call it inf, and define it as such: imf=1/x when x approaches 0. Then any other infinity can be expressed as a*inf. Defined that way inf can to some extent be regarded as a number and used in calculations, even though it will not work universally due to some deficiences in the axiomatic foundations of mathematics.
No. Just no
Yes, just yes. But you must handle it with care because mathematic is cracky in its basic foundings, so you cannot use it without precautions.
So what does 2*inf stand? 1/x^2 as x go to zero? Or 1/(2x)?
What does it has to do with infinity? Do you want it to be negative or positive infinity considering lim 1/x as x goes to zero is not defined because it both goes to infinity and negative infinity?
Does inf*inf exist? And inf^n? And inf^2?
And most important..what the helld does this has to do with infinity? Can you find a connection with the amount of things in sets?
My point is this. Infinity is not a well defined concept in standard mathematics, and hence it cannot be used to calculate with in an easy sense. What i show you is that this problem can partly be remedied by defining 1/0 as a mathematical unit called for example "inf". Then every a/x for any a will have a meaning. There will alo be a negative version defined as -1/0. which will be -inf. Inf and its negative version can then be treated as a number and calulated with, provided you take certain precautions. In standard mathematic these precautions are taken care of by saying tha 0 does not have a reiprocal. But this is only an arbitrary precautions that make mathematics very clumpsy in certain situations.But you cannot avoid certain precautions when calculating with 0 and this inf. This is because the number 0 also is in some ways logically ambiguos and the very foundation of mathematics lacks something.
The precautions I am talking about can be seen in expresions like 0/0. It can be evaluated as 0/0=1/0*0=inf*0=0. But then you must avoyd evaluating it as 0/0=1.
Hey Knut. Let's chat about your idea of defining 1/0. I've studied those attempts for a long time and feel close to a useful framework.