Infinity is bigger than you think - Numberphile
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- Опубліковано 5 лип 2012
- Sometimes infinity is even bigger than you think... Dr James Grime explains with a little help from Georg Cantor.
More links & stuff in full description below ↓↓↓
Minute Physics video on this topic • How to Count Infinity (somewhat more fast-paced... but we did film ours BEFORE his was uploaded, so similarities are coincidental... well actually, no they are not... we are all building upon Cantor's work!!)
NUMBERPHILE
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"We're going to talk about infinity..."
0:11 Draws a fish.
Daniel Yang wooosh
gradle no u
@@danielyang7366yes but it looks like a fish haven't you seen a fish before
No, it's a fibsh. Sea doggos love them.
Yes
Great, now i have something to talk about on a first date...
that will take place in approx Aleph0 days from now
@Hylian have you tried it? How did it go?
😂
Lol
How was it?
Cantor was one of the greatest geniuses of mathematics. Truly ahead of his time.
@ODIN Force I agree that infinity is not a number, however, these infinities are clearly different. In math you have to sometimes kind of "make up" numbers. I'm sure you agree with "i" as an imaginary number, and this is similar. If any of these imaginary ideas contradict, they are changed until they don't. Then, what difference is there really between imaginary concepts like this and standard math? They both have defined rules and can sometimes co exist
@@slurpnderp1838 This is where I have a problem. If you state infinity as the set of all numbers then you can't have different versions of it. If you state that you only have even numbers then it is not infinity. It is a subset of infinity with infinity as a limit.
@@pentachronic I think the concept of infinity isn't the set of all real numbers or real and imaginary numbers but just a set with an unending amount of elements an infinite amount of elements. If you define infinity as the way you did with it being the set of all numbers then you limit the abstraction that comes from the concept of unending amounts.
@@spooderdan9127 I understand it as being an infinite set, however you can't just take a subset and assume it is of infinite size). That defies all logic and mathematical rigour. The subset would be a smaller size than the original.
@@pentachronic "Infinity" here is a cardinality, not a set. The set of all natural numbers is isomorphic to the set of all even natural numbers, so they have the same cardinality and are thus both infinite.
4:25 Brady's "Do it, man" is one of the coolest things I've heard in a while
He sounds and is capable of being in Harry Potter series.
Bhai pubg khel na xD
MortaL Well, he's British.
Fir bhi iconic to legend hai🤣
@MortaL what are you doing here 😂 oh i know....you love MATHEMATICS right??
Magar tu to apna Harry Potter hai mortal 😂😂
“To infinity and beyond”
-Buzz Lightyear
See, he knew what he was talking about.
because there is something much bigger than infinity: STUPIDITY!
He’s referring to unconditionally infinite!
Philip Kuo deep 😳
The joke is, Buzz Lightyear didn’t know that it’s by definition impossible to reach infinity so he just kept flying and flying and was never heard from again
Maybe he was referring to a flat earth
Can you zoom in more plz I want to see the atoms
I have some heart shattering news for you. The size of an atom makes it so individual atoms can't be seen. Atoms are smaller then any wavelength of visible light.
@@cythism8106 you’ve ruined his dreams
@@_xndr7027 lol
@@cythism8106 r/whooosh
HAHAHAHAHAHAHAAH this comment made me laugh so much. Thank you hahhahahahaha
"Guys, it's a scam, -1/12 is actually the biggest number."
- Ramanujan
Thyron Dexter you don’t know your limits.😀
That's a scam because it's sum of all the numbers but not the biggest. -1/13 is actually bigger than that, see the difference is there between biggest number and sum of all numbers.
@@gentleman_gaming6529 it's a Ramanujan sum, so -1/12 is not the sum of all positive integers in any way that means anything to the average person.
0-800-????-???
Proove it
I wish I had the sparks in my eyes when I talk about my life as this guy does about numbers.
0:12 this is not a lemniscate this is a fish !
still better than Matt's infinity sign though XD
this guy is so passionate about it i love every second of this
True
I love it too!!! People won't listen and believe without passion.
7:01 he seems a little frustrated...
@@joelvansickle3623 That's true, it makes a whole difference for us viewers.
Have you seen Cliff?
this feels like an episode of the office
“Numberphile channel shuts down: forests of the world saved!”
Yeah,they spend too much paper!
Look up Smokey and the Bandit laugh and that’s me
I love the fact that are an infinite amount of number between 0 and 1....
Well there is a infinite number of numbers between every number :D
really...are you sure? what about lets say 99.5
and 100?
sure there is
hmmm... dont know if i trust that
You are indeed correct. Here's my proof generalised to any two distinct real numbers a and b.
Proof. Suppose a and b are two real number. Without loss of generality, say a
The camera is way too close to his face.
Zoomed in XD
The camera is too close to your face on your google+ / youtube avatar.
I want to see him closer
JESUS IT ZOOMED IN EVEN CLOSER!
IT'S NOT CLOSE ENOUGH
"There is no infinity infinite enough to describe how infinitely many different infinities are there." Quote from my introduction to mathematics lecture.
Poor Cantor... It was one of my greatest moments in math when I understood many of these concepts, what a legacy.
You can't understand something which does not exist. Infinity is one of them.
You only pretend to understand which is infinity away from the truth.
@@countingfloats muggle
You sure didn't understood a lot; like how to spell his name.
'CANTOR'.
George Cantor.
@@autumnicleaf Ok, given the cardinality of N is ℵ0, state the cardinalities of Q and R.
The real truth is the human brain cannot grasp the concept of infinity, which is why everyone including Cantor went stark raving mad trying to do so over the last 2500 years.. I’m not saying you can’t have some contextual knowledge, but we will never understand it.. And it’s not because it doesn’t exist..
"Infinity is not a number, it is a fish." 0:15
Connor Yost that’s exactly what I was thinking!
Connor Yost I
Connor Yost fish live in water not on paper(if you take it like this)
😂😂😁
But you can catch fish not infinity.
Here is a mindblowing fact for you: No one number has an exact previous number or an exact number after. For example 3 can not have a previous number because the decimals never end. 2.999... cannot be one either because you cannot put a number bettween 3 and 2.999... so 2.999... is 3 written in a different way. Also 3 does not have a number after it because 3.000... continues to infinity and as a result you can not put 1 nowhere.
Dimitris Bekiaris To make this idea a bit more solid, assume there is a number X that comes right after 3. Then S = (3+X)/2 is also a number, but S is between 3 and X, a contradiction.
Dimitris Bekiaris There is something irrational in the use of numbers. After what I know, Gödel's incompleteness theorem is only valid when counting with numbers; pure logical mathematic systems can be complete, but are also very hard to do advanced maths with. In our reality we only have order, logic, relations, proportions and geometry -- numbers is a construction we use as a help, and they only confuse us when they lead us to incomprehensible things like infinity (which probably not exist in reality, either).
If i understand right you say that infinity does not exist but the universe is infinite..
Dimitris Bekiaris No, the universe is probably not infinite. Probably it don't even going to expand forever, because everything that exist (matter, particles) fall apart and dissolve into vakuum.
Dimitris Bekiaris if you are hesitating about that 2.999... should be followed by 3, i have a nice tip (or proof as you will) for you : lets do some simple math --> 2.999...=x ---> lets make another equation like previous one, but ten times bigger ---> 29.999...=10x ---> lets substract the smaller one from the bigger one ---> 27=9x ---> x=3 ---> from the original statement we get ---> 2.999=3 Also you can do this with every infinite repeating decimals, not only with the 0.333... ones but also with difficult ones, like 0.123123123... only here, you have to multiply by 1000, so the decimals line up and substract without problems. And with this method you can convert every infinite repeating decimal into fraction
Oh reeeallly liked how he came up with proving its incountable. This is a true beauty of math
Negin ? Me too. I just smiled for 5 minutes after that
@@nilsdula7693 yeah I also thought it's a really beautiful proof. But the fact that no one believed him and put him in mental institutions is really sad
This just something absurd
I like the idea of calling things "listable" instead of "countable". I have spent some time trying to come up with better names for things than the original names we've given them. In particular, I've tried re-naming "real", "imaginary", and "complex" -- and I've even come to realize these aren't even "names" (we always talk about "an integer" or "a fraction" but never really "a real" or "an imaginary" or "a complex"). It's a *lot* harder than it looks!
Bruh
Yep, they're not *names* because "real", "imaginary", and "complex" aren't nouns; they're adjectives, meaning they are *descriptions* of nouns. The noun described being "numbers". Try not to think about that too deeply, it's just how grammar works. 😅
Something being an adjective doesn't mean it's not a name. By that logic, the Dominican Republic isn't a name because it's an adjective + a noun and we don't say the adjective on its own. Or even United States isn't a name then because we don't just say "United" on its own and it has to be with the noun. That's nonsense. Names don't have to be nouns. Names can be noun phrases as well, including adjective+noun.
When he listed the integers in the video, did you notice what he was doing as he listed them? He was counting...
They are "countable" because you can always count them forever, just like you can list them forever
@@LAMarshallthis is numberphile, not letterphile
(sorry, couldn't miss that one)
Interesting, I've heard of this before.
There are more numbers between 1 and 10 than between 1 and 2, but they are both infinity.
31T3 1337 N008 The set of real/rational numbers between 1 and 10 has the same number of elements as the set of real/rational numbers between 1 and 2.
TimofAwsome Clearly not, as the set of reals between 1-10 encompasses every number in the set from 1-2, plus more.
The fact that the set of reals between 1 and 2 is a proper subset of the set of reals between 1 and 10 does not mean the have different cardinalities. Any interval of real numbers has the same number of elements as the entire set of real numbers.
TimofAwsome Oh turns out you're right :\
But how would you prove that each interval on the reals is bijective?
f: R --> (b,a+b) where f(x) = a/(1+e^x) + b is a bijection between the reals and (b,a+b) (you'd have to modify the codomain if a is negative as then a+b would be smaller than b). But this is a bijection between R and an (open) interval.
10 points to Gryffindor, Mr. Weasley.
The comment that I was looking for
666 likes
This is the comment you’re looking for- Obi-Wan Kenobi
@@thatssomethingthathappened9823 brave, but foolish,
poor Cantor ;-; thats really depressing, at least his story had a happy ending though even if it was after his death
0:13 “infinity is not a number” no it’s a fish :D
Though admittedly at first glance what he draws looks like a fish, in fact it is the infinity symbol.
Infishity
lol
@@photoshopguy4457 Infinishy
0:33 "What's the biggest number I can think of?" Answer = -1/12
I see what you did there.
:D
Clever... Very clever...
I don't know why I found that so funny hahaha
pi. 3.14159265358979... < that's all I know from the top of my head lol
You want to know what's also bigger than you think?
Your belief that this was going to be sexual.
Kek mate
+steve vansteenbrugge Trump?
What
+steve vansteenbrugge i thought of a 8=D
Your pepe picture didn't really help tbqh.
Human: *finds infinity
Also humans: *trying to count it with every possible way
The fault in our stars:
"Some infinities are bigger than other infinities"
I also thought of that
I’m not crying, you’re crying
Sorry but what do you mean?
@@alphaecho3875 you have to read the book to understand
@@alphaecho3875 Imagine the amount of decimals you could list between the numbers 1 and 2 that would be infinite right? of course. Now imagine the amount of decimals between 1 and 3 that would also be an infinite number. So the infinite decimals between 1 and 3 are greater than the ones between 1 and 2 but both are infinite numbers therefore making one infinity larger than the other.
I was talking to my intro to microeconomics professor after class and she was saying how I got a bit ahead of the class by realizing the significance of 1 in relation to elasticity of demand and then told me that next class we'll touch on trying to explain what infinity means and I was like "Yeah, and nobody understands infinity. Not even most math students really understand infinity. It's a direction, not a number." One of the accounting professors was nearby and he chimed in "Yeah, and some infinities are bigger than others so it gets even more confusing."
And that is how I came to watch this video again.
The first time he drew the infinity symbol I immediately thought, "That's a fish..."
ME TOO
Everyone did
His hand is almost bleeding from writing so many numbers. I love this guy!
I thought they were marker stains
@@Janken_Pro it is only your mind trying to protect you from reality.
I have always loved the concept of countable infinity. It is a math concept that truly does make sense and also is something many people don't know but can be described easily enough
Many crackpots have baulked at the idea of "more than" countable infinity, and many more will do so. It is not hard to see why: it is the intuitive concept of "more" that really breaks down. We mathematicians perceive that 1:1 pairing is a much more fundamental concept than is counting itself, and we are comfortable extending it to deal with the transfinite. But if one cleaves to everyday ideas of size, bigger-than-ness, and so on, this area of maths just seems very strange.
But the concept of countable infinity isnt real. Because u cant do it. Also, he said that 1 infinity can be bigger than the other. How does that make any sense? It does make sense on paper as was shown, but how does that not contradict infinity?
@@Zelchinho The concept of countable infinity is real. The mathematical definition for a set of numbers to be countable infinite is hinted at in the video, which is that the set of numbers is in bijection with N, the set of of non negative integers. (ie there exists a function from the set to N with a one-to-one correspondence). When we say that uncountable infinite sets are "bigger" than these countable infinite sets, there is no real "proof" or mathematical sense to that (to my knowledge) but is purely based on a intuitive/logical viewpoint.
@@Zelchinhocountable infinity means that you cna find a way to associate each natural integer (0,1,2...) to an element on the set (this association is known as surjection). If all elements of the set can be countable (using mathematical logic, there's a reason the quantificators exists), it is countable
Rela numbers are uncountable because there exists no such association (surjection)
James's voice and way of talking is so viewer-boosting
It's demeaning.
It's like he's talking to a 4 year old.
Shut up
@@sparhopper einstein said: if you cant explain it simply, you do not know it well enough
@@sparhopper If you interpreted it that way, then perhaps you do have the mind of a 4 year old
Some numbers are so big that you can't stop counting them. But others are so big that you can't START
I have started... and I have finished TWICE. The funy thing is: it's a true history. so.. Math becomes myself into a clone of Chuck norris.
not
@@willeemina Wait until see my math circular kicks...
Sets of numbers*
Yes, if you want to count from betwen 0 to 1, how many digits do you need? how many infinite combinations are on an infinite digit number? it must be a lot
Cantor's work on infinities is one of my favourite topics ever, where learning about them expanded my mind in a way I could never forget~ he's an absolute genius, and although this happened ages ago I'm still so angered at the discrimination and injustice he dealt with from his peers and society ..I hope he still found peace, in the end.
thanks Numberphile, for the passionate explanation :)
Infinity is HUGE!
"Some infinities can't be counted" -- Georg "Count-or"
Georg Cantor, not to be confused with legendary ubber-falsetto-voiced vaudevillian Eddie Cantor (1892-1964).
There's only one infinity
@@mysticwine there's actually infinite cardinalities of infinity.
@@12jswilson What's a cardinalitie?
@@mysticwine cardinality is the size of a set. For finite sets, it's easy. It's the number of objects. For infinite sets, it's more tricky but we say they have the same cardinality if and only if there can exist a bijection (1 to 1 correspondence) between the set. It's in this way that mathematicians say some infinities are bigger than others. Because there isn't a 1-1 correspondence from the real numbers to the natural numbers, we say there are "more" real numbers than natural numbers.
I can't take it any more, subscribed.
over the last two days, i've been going from video to video feeling the same way. I NEED MORE MATH
1:51 He clearly does a mistake and cuts the video (he forgot to put the negative mark in front of the 4)! YOU CAN'T FOOL ME WITH YOUR SOFT VOICE
Didn't see that lol
+Byakugan Sharing you got him down
+Byakugan Sharingan Yeah I noticed that too but he doesn't change it. You can see he just put the minus in front of it as it's very close to the comma compared to the other negative numbers he wrote down.
Recent studies demonstrate that 99.99% of The viewers of this video just paused it at minute 1:51 To See if your affermation was right
PS: I didn't
nope man, he lists positives first, then negatives
The concept of infinity stimulates the imagination which is what happened to me at the age of 4. I found in my father's garage a tin can with a picture on it. Within that picture was the same picture smaller, within which was the same picture even smaller. This captured my imagination for many days. I realised that I could imagine a series of even smaller pictures. I realised that this series that does not end, yet, I did not realise that it leads to infinity. Then I studied Cantor's discoveries of infinities at the university 16 years later. I was in awe.
Sir, please don't train young minds to wander too much or too far. The human mind has nothing to do with graciousness or mercy or love.
@@davidwest7299 Love? Are you sure your comment is in response to my comment?
Fractals seem like a very convincing illustration of infinity within a finite space.
@@lawrencedoliveiro9104 Indeed. In our minds only. Physically, infinity doesn't exist otherwise any region of space would be of finite energy.
Chuck Norris counted to Infinity.
Twice.
Ordinal infinity or Cardinal infinity. Or one of the many other variants of infinity. Assuming ordinal infinity as otherwise you couldn’t count it twice. And therefore he not only counted to infinity twice, but also three times and 100 times and infinity times.
Chuck Norris counted to Infinity.
Infinity times.
@@loganm2924 Awesome.
@@loganm2924 twice
@@loganm2924 *"...Chuck Norris counted to infinity...."*
And found Bruce Lee waiting for him and he said to Chuck *"...What was that?..."*
Logan McDonald are you the maker of true infinity? Are you Reinhardt-C?
why do I feel like a 7 year old child whilst watching this video?
That's funny. Me too!
Khulhu Cthulhu same. I'm actually seven
I am 5
Khulhu Cthulhu yeah thn i m not evn born
Because he's a young, smart guy and you don't want to feel lectured by him? Old fart
It's easy - infinity is an eight number written horizontally.
-8i
Interferencyjny infin8y
you could've put it in a better way, as in infinity is a sleeping 8
Infinity is what happens when 8 drinks a bottle of tequila.
How do you know it's horizontal? Why not vertical? What it actually is, is 90 degrees rotated.
“How long have we got?” So quick and subtle but so hilarious.
i learned them as countable as well, but my favorite word for them that i came across was "enumerable"
Not enough tight close-ups.
Help me-this hurts
Respectfully, I disagree. I contend that there are too many close-ups and that the cinematic quality would improve if they added more medium shots to the mix.
@@michaelerickson985 Either this channel has more than average number of trolls, or more than average number of people completely incapable of understanding the notion of a joke. I bet on the second
I prefer not to use terms like "bigger" when referring to infinities, since "big" implies a size - something infinities don't have by definition, being endless.
Rather, I think of some infinities as being "denser" than others.
I agree with this wholeheartedly. Think of infinity as a measure of density rather than a measure of size.
i also completely agree with u
+A. Joe Technically very true. As Vsauce explains, the list of whole numbers isn't "twice as big" as the list of even numbers - it's just denser.
+NoriMori The whole numbers are twice as dense as the even numbers, but they are both the same size, a countable infinity. Using density isn't really helpful, especially considering that the reals are (uncountably) infinitely more dense than the rationals which are (countably) infinitely more dense than the integers which are infinitely (still countable) more dense than the squares of integers and so on. Density means nothing in infinities, size is a much better determining factor.
Nukestarmaster But as you pointed out, if you only consider size, you wouldn't understand a key difference between - for example - the whole numbers and the even numbers.
Perhaps both have their uses.
You simply explain the idea of real analysis in such great way! So fun 😍
I was recently watching a video explaining why the speed of light is what it is. According to some physicists this existence has a limiter. They said light could go even faster than it does but the universe prevents it so in regards to infinity there may a realistic limiter to it. I wish someone would touch on this subject! If not I’ll have to become a brainiac and do it myself 😢
I don't know too many people who believe anything infinite can exist within the universe. Nevertheless, we can do mathematics with infinite sets and various concepts of infinity.
If you have a philosophy of mathematics where math actually exists in some metaphysical sense, then you might be concerned by this. Maybe dealing with infinity in math is wrong! There is nothing in the universe which is infinite, so how do we know we're correct about it in math? This isn't a death knell to the philosophy. One can believe that the concept of infinite things still is a valid concept and we can abstractly work with it even if nothing *physically infinite* exists.
But there are other philosophies of mathematics. Things like antirealism, where mathematics is seen not as a thing which actually exists in a metaphysical sense, but more of as a useful fiction which we might or might not use as a tool or which we might view as beautiful like art or intellectually stimulating like philosophy. There is certainly utility to some mathematics of the "infinite" under such a philosophy - some statements about an infinite set simultaneously encode infinitely many statements about finite sets and thus reduce the amount of work we must do to express those infinite families of statements. And some may just be for beauty and a desire to speculate.
@@MuffinsAPlenty Would space being "dense": any non-zero length can be divided into strictly smaller lengths, imply the 'physical existence' of infinite things? It kind of does to me, because then any *exact* representation of the universe would require infinite precision (countably many numbers/bits).
At the same time, one may never need the full exact representation. We could always work with an approximation that suits our purpose (as we do with pi, e, etc.). It would be interesting to find out if it is possible to set up an experiment which determines if space is dense in the above sense; or even rule such an experiment out (which would make the question unscientific...?).
I remember reading online that only one person (so far) counted to infinity, and that was Chuck Norris. In fact, he did it twice.
how's that possible
+Aniruddh Naganur It's an old joke where people would say that Chuck Norris could do anything. When I saw this video, I couldn't resist saying this joke.
Narata only true UA-cam users remember the classic chuck jokes:)
I heard he did it a third time, but this time he started from infinity and counted backwards to 1
Niels Unnerup Next time he'll start from infinity to negative infinity...
0:33 "Oooo it's 20." I laughed so hard at that probably since it's so true. 😅
😂
Two old men have a contest, to see who can come up with the bigger number. The first man deliberates long and hard, before he starts, and with a knowing smile proclaims: "78".
The second man smiles and nods, defeated.
@@tsawy6 WHAT?!?
@@tsawy6 what dat mean?
@@barritoothy i think because of age and that they feel old and it took a long time to get there.
If you are old you will probably think of your age first.
I absolutely love this channel !! The passions of those mathematicians radiating through each video is something so inspirational; thank you for this amazing content!
It’s funny that this was so widely rejected at the time. To me, this is one of the most intuitive things I’ve seen in this channel, and so clearly true.
Why not just use a white board instead of wasting all that paper.
They're sold on ebay to raise money for charity :)
Neil McMahon Hippy.
edadou lol
Ed Gein Noticed by Ed Gein, I feel so special..
Andy Merrett Maybe the best system from a bad lot.
how did i end up here i started from sneaker collections
Sneakers are costing so much nowadays that they are raising the price to infinity
I got here by a video how much you click on your keyboard and the fun was, When you click space 600, 000 peoples do that at the same time you do
XD
You've probably watched similar videos so it got recommended
+Alpha XenoGenesis (TBNR) I got here from the Rogue One trailer, funnily enough.
You may think it's a long way down to the shops, but Infinity is big, really big ...
The title should be.. *"To infinity and beyond"*
That makes perfect sense
Sponsored by the fault in our stars
+Vic Pownall yeah, there is a running theme in the book about 'some infinities being larger than other infinities' relating to how an infinity of love between two people with cancer isnt as long as it would be if they didnt have cancer, but it is still an infinity. Or something like that, tbh i didnt pay much attention while watching it
XDDDDDDDDD you won
+Karim Shoaib When I heard that I was thinking TFIOS
Kane Bell That was kind of rude...
+Karim Shoaib best comment ever
This type of concept is called the cardinality of the set. It's something you learn in real analysis, modern/abstract algebra and other courses that deal with numerical concepts.
And Vsauce
Its Cantor Set , cardinality = aleph-0
You learn it in any discrete mathematics class.
@@tay_piss_saucer_mk.400 no no no. The cantor set is actually uncountable because it contains all binary numbers which is uncountable.
RIP Cantor.
I want a list of all the people that were ahead of their time but treated badly while they were still alive.
Unlistable!
guys based on research 80% of comments are random, and 20% is actually about the vid
So what they're saying is that there is an infinite type of infinities?
Yes YoshiFace.
Cantor, that did all that demonstration and created what is known as the set theory, demonstrated that there are infinities of infinity and that concept is really not a concept, it's the simplest part of what he did.
Let make things clear first... it's difficult to talk about infinity when we misuse the vocabulary.
"infinity" is the concept of infinity
"cardinal" is the "number" of element(s) of a set
"infinity" can also designate an "infinite number", they are called by Cantor (I think it's him) : transfinite numbers.
Your question then being :
"Is the cardinal of the set of transfinite number infinite ?"
... again, the answer is yes.
Cantor proved that if you have a set A, infinite or not, the magnitude of the cardinal of the set P constituted of the "parts" of A, is a magnitude higher than the cardinal of A.
Intuitively, we can write that card(A) < card(P)
But you must understand what it means when A and P are infinite sets...
In the case of A being an infinite set, its cardinal is a transfinite number. And Cantor proved that you cannot match all the elements of P uniquely with an element of A.
Thus, cardinal of P is higher (bigger) than cardinal of A.
Now, that's the beauty of math : you can continue infinitely with the P again, and construct the set constituted of parts of P. That set will then be of a cardinal bigger that the cardinal of P itself.
You can then start with the set A0 being the set of all natural number of size aleph-zero (the first transfinite number) and build a list of transfinite numbers :
{A0, A1, A2, A3, ...} with :
A1 = the set of parts of A0
A2 = the set of parts of A1
...
and that construction has no end itself
and each A* is a transfinite number different (higher) that the ones before it
Thus, there are infinite number of "infinites" (= transfinite numbers).
(sorry if I'm wrong about the name "transfinite number" and also about the construction of the sets using the "set of parts"... maybe it's an other construction Cantor used. But the idea is that one, roughly.)
Have fun ^_^
Though true, this particular proof only shows 2 types of infinities.
Actually I don't agree with this concept. For example: The infinity of integers is the same size as the infinity of decimal types. It might be slightly filosofical, but I compare it to the speed of light.
Something that has the twice the speed of light is as fast as something that has once the speed of light, there are a couple of reasons for thing which are hard to comprehend in the accepted system we use in our society, but with infinite numbers it's the same in my opinion.
A infinite amount of integers can go on forever, so can a infinite amount of decimals, therefor they are the same.
Niels Jan van de Pol> you cannot say "an infinite amount of integers can go on forever, so can an infinite amount of decimals" and the conclude that "THEY are the SAME".
No, there is logical implication there... you have to proove what you say.
And indeed Cantor proove that there are at least two "size" of infinities... and in fact there are an infinite size of infinities also.
But all infinities, however "small" they can be "goes on forever" by "definition" ^_^
Marc Ly I think people are just getting confused by what the word "bigger" actually means in this video. Just think of the "bigger infinities" as 'encompassing' more dimension. for example: one line that looks like this - is smaller than two lines that intersect like this + but all lines go on for ever in each direction in both cases. The smaller case takes in to account left and right while the other has left and right but also up and down.
That first infinity looks like a fish.
I'm glad Georg got recognised in the end 😢
A video about math with the ups and downs of a great drama!
I struggled with understanding math concepts and prime/integers/rational numbers etc in high school, until today in my 30s it dawned on me that there were infinite numbers between 0 and 1. And all of a sudden my brain understood and watching these videos made so much sense. Why am I such a late learner 🤦
Everyone learns at their own pace, don't worry about it
I'm a teenager and I don't know how to socialise properly, while my peers pull it off gracefully
0:15 that's no infinity (car) ... That's a fish
Why am I watching this at 2am
Why kind of videos someone watches at 2 am.... This ofc
2:26 am for me atm, while doing calculus homework LOL
do what I did, chug some NyQuil...
2 week ago...
1:59 here 😂😂
Infinite infinity.......absolutely superb video. A great mix of practical demonstration and historical detail presented in an interesting and engaging way! Well done infinity......................................
Now he’s on Numberphile, THE GREATEST ACCOLADE OF ALL! Well done!
There is infinity between every single decimal, e.g. 0,13209832 and 0,13209833
xooperz FBF that's true for rational numbers as well. that's not the difference between reals and rationals.
+xooperz that's not to say that there isn't a bijection between those two numbers and the real numbers. there's a bijection between any two real numbers and the set of all real numbers
That's a comma...
Chica the Chicken What?? What are you talking about?
+xooperz Oh...never mind, just saw it. Sorry to cause trouble...😵
I first watched this video when I was in high school and now I'm in university studying cardinality in my math course. It feels amazing.
And now?
Hey I want to delve deep into this. Can you suggest some textbook to get the feel of it :)
I watched this in high school for fun like 6 years ago now I'm in uni having to learn this and its very intuitive thank you Dr James Grime
For every natural number there are two related integers. The set of integers seems therefore twice as large as the set of natural numbers. But if there is an infinite number of natural numbers, and you can't have more than an infinite number of something, then there can't be more integers than there are natural numbers. There can only be more integers than natural numbers when we're talking about the finite. For example, if we have a trillion natural numbers, there are 2 trillion related integers ( for 1 there is 1 and -1, for 2 there is 2 and -2, etc). What does this tell us? That we pay too much respect to the idea of infinity? That it is a mere idea, a mere concept, that it's not real?
This is the first numberphile video I watched and now I love numberphile 😀
"It's a big topic" woh, them puns...
johnytest464
A-THE- 1ST
Define space and define the universe. That will make your question answerable.
They came at the same time...?
0:12 Is it a fish?
because there is something fishy with the very concept of infinity (not the potential one, but the REAL one)
Well you know what’s bigger,
Infinity+1
Edit: it’s a joke, jokes are supposed to be understood. But you people don’t understand this
∞ + 1 = ∞
:>
You can't triple stamp a double stamp!
I don't think so. You are implying that infinity ends.
You could say, I have a number so big that it is infinite, because you can always add 1 to it.
You can actually do that. Check the Vsauce video on "How to count past infinity".
Great vid! Thanks expanding the decimal distances of irrationals and rationals to infinite!
at 1:23 it shows 1, 2, 3 on the paper...woahh
Josh Werner that's actually more interesting than this video
your observation tho
Math illuminati confirmed
“You can divide infinity an infinite number of times, and the resulting pieces will still be infinitely large. But if you divide a non-infinite number an infinite number of times the resulting pieces are non-infinitely small. Since they are non-infinitely small, but there are an infinite number of them, if you add them back together, their sum is infinite. This implies any number is, in fact, infinite.” - Uresh, character from The Wise Man's Fear by Patrick Rothfuss ^^
timewasteland Except that that's nonsense.
Bon't blame me, Blame the author of the book the quote is from. Math was never my thing, anyway.
I haven't read that kind of nonsense since I was 6 years old.
'If you divide a non infinite number an infinite number of times the result isn't infinitely small'- simply not true. 1/infinity =0
That man clearly did not grasp the idea of a limit.
Finally, the great Cantor, got mentioned in numberphile, his soul must be in peace now...
Simply amazing! Thanks for great content!!!
The way he said "There are different kinds of infinity" killed me 0:51
Now, we are having a war on Infinity, *Infinity War* .
when you get to the Berkeley cardinal: we're in endgame now
Hey what does this mean I don't get it
@@NathanielCoran Avengers Infinity War
@@andrewgrebenisan6141 Yeah what's that?
endame is -1/12 lol is it
I love how all these videos start
Was watching World Science Festival and came here for a better explanation and got it.. well done.
That paper looks so damn uncomfortable to write on...
same
I know right?
I'm 5 years old and I'm offended. I can almost count to 30 not 20
Are you sure? it's past your bed time
+Ben J hold this L
Keegster Z L
+Keegster Z hold this L too bruh
+Ben J *Tosses L*
He looks like a Weasley brother. 👍🏼
I don’t regret watching this with French subtitles
I am not a number...I am a free man
Nothing to do with the vid but okay.
+JYE - all hail Lelouch
Now now, number six...
That is what every number thinks
Its not about the number standing alone numbers are only value when they come together. Same with humans. One human is weak. Alot of humans are strong. We need eachother like numbers do
What about complex/imaginary numbers aren't they a whole new infinity?
The way i interpret this is that the infinities are not bigger or smaller than one another but that some of the counts or lists use more fractions or possibilities (calculations) faster than the others so what we are really talking about is the rate of change of the calculations with the calculator making multiple calculations for the last one at one time while it would only decide on one at a time for the first example.. you could argue that the more complicated one is 'bigger' only in terms of the number of concurrent calculations rather than the size which is still the 'idea' that the size is limitless - the same as the more complicated infinity..
how did I get here from Gangnam Style 7 years ago?
that recommendation changed how I watched UA-cam
Just like the idea of zero. How nothing can something be?
I've always wondered how time works with this concept. If you start at 2:45, how can we possibly make it to 2:46. We have to go through the sixty seconds first, and between each of those seconds there's so many milliseconds, and between those, nanoseconds, and so on. So, in order to actually advance in time, we would have to at some point skip forward. Either that or there is a smallest possible unit of time.
Actually there is a minimal unit of time, which is called the planck-time. It is the time which you need to cross the minimal unit of distance (planck-lenght) with the fastest velocity possible (lightspeed). But don't trust me too much, better google it ;)
Sorry didnt see your comment
Willie Numbers are not the same as time.
Time is not numbers, we created numbers and link them with this palpable time units we have in nature to measure the advancing of "time"
***** time is a dimension and how you feel it passing is not how fast time is going
TheMrFloorball Planck's length and Planck's time are the defining numbers for the highest resolution physically/mathematically possible from what humans know so far. No distance or time can be smaller than those numbers.
Which doesn't make sense really :D
The first thought I get after watching this channel's videos is that, if they upload a video on 1 April, they can literally be explaining something which does not even exist in the video, and then when u finally understand that they break it to u at the end of the video : IT WAS A PRANK ! 😂😂🤣🤣I mean I would literally fall for that
My love for numbers is growing through this channel I think my favorite term is transcendental numbers.
In my maths lessons when I do my Infinity talk, I demonstrate that a set of numbers that is infinitely small has the same number of elements of a set of numbers that is infinitely large.
I love how this channel talks about the pioneer mathematicians that surface the specific theories that their videos talk about
thanks almost had my head cracked in my math class trying to understand cantors diagonal argument
7:01 I think a better way to explain it is from the POV of computer science:
If you coded a Turing Machine to enumerate all Reals, you can always code another TM to enumerate the Reals in such a way that *at least 1 number* in the sequence is different.
And because the set of algorithms to enumerate Reals is enumerable itself, this means we cannot use an infinite set of algos+TMs to enumerate ALL POSSIBLE Reals
We can barely understand this, how could we look at it from the POV of computer science. We are all noobs here.
@@fern_fern_fern true. I couldn't think of a simpler explanation.
What I was trying to say, is that simply appending more numbers to our generated sequence is "cheating", since we're *modifying the algorithm* to include more numbers, instead of letting it run as usual.
Even if we defined a complex algorithm to enumerate all Reals, it would be *infinitely long,* which _breaks the very definition of "algorithm"_ (a *finite sequence* of instructions)