Finally someone who can explain this without making me doze off into a mind numbing stupor! Thanks for helping me pass my math final nice British man!
It puts my mind at easy to think about it like this:
If anything was truly infinite we would know because it would take all energy, space and time in all dimensions and then more to hold it. There would be no room for us.
The set of integers is just a set of numbers that CAN go on as long as you keep counting, like a car that can go on as long as you keep putting fuel in it.
does that make a spaceship more infinite than 0 to 1 because it can work in 3 dimensions?
...find a point where you could not continue the diagonal, but the diagonal would continue indefinately between 0 and .1 alone.....okay never mind I give up
Okay, I think I've got it... I am absolutely not a math guy, but I have been watching your vid and similar vids and I only ask that you try to follow what I am going to try to explain:
If you were to take all of the decimal numbers between 0 and 1 they go on indefinately, but if you put them in order then the number of decimal places would not necessarily always increase. For instance 0.7>0.666... etc.
So at first thought, you might think that the diagonal wouldn't work because you would...
Between 0 and 1 there is an infinite amount of decimal places. There are an infinite amount of numbers, and bigger infinites beyond both of those infinities so shouldn't you be able to match the numbers?
All the decimilas from 0 to 1 = infinity. To put them in a list from 1 to infinity you need to order them to make them fit, you cannot just place them randomly like u have done in the video, 1=0.5 2=0.25 3=0.66... You can start with 1=0.1, 2=0.2, 3=0.3, 4=0.4...6456=0.6456... 584935034=0.584935034 and so on, and infinity = 1, So what ever number you are at you can just place a "0." in front of it, and there u go u have a list that fits to infinity from 0 to1 and they match,
@TheCeejReturns Yeah and i explained that, if you are going to infinity or forever then 2/3 can be written as 666,666,666,666,666,666,666,666.... all the way to infinity. So when we reach infinity, 2/3 will be in there. I think you have missunderstood something about infinity.
Infinity is a purely theoretical concept anyway. It works on paper but doesn't exist in practice. Just like negative numbers and the square roots of them.
Nope. I read somewhere that the number of atoms in the universe totals far less than a googol plex. So, it's highly unlikely that a googol plex will ever be used in practice, and infinity is impossible in practice.
Apparently you don't. Because I do know one thing about the size of the universe. I know that it's expanding. If it were infinite, it wouldn't be doing that, would it?
Okay. It has become clear that you are asking me because I don't know and you don't want an answer. If you wanted an answer, you would ask an expert in the field, not some comedian on YouTube.
@anticorncob6 infinity is not a natural number, so if ur going to list all the numbers down then, 666,666,666,666........ or 2/3 will be on it. its the same infinity
I've been fascinated by Infinity ever since I first saw Dangerous Knowledge on the BBC when it first aired. I remember googling frantically trying to understand Cantor's argument to no avail. Tonight's the first time I've finally got it. Thanks so much for making this video.
@RXP91 Exactly, i'm also who was inspired by the Dangerous Knowledge. Since then i've been trying to understand (a glimpse) of what Cantor's discovered.
I actually find it's hard to listen to an American talk about maths. Something about and English or Indian accent just makes math seem more digestible. By the same regard Aussies do have pretty absurd accents.
You actually CAN order NxN in the following manner: (1,1), (1,2), (1,3) ,.... , (2,1), (2,2), (2,3), ...
The fact that there isn't an explicit index for any duple which doesn't begin with one doesn't matter. It is an order, and it DOES prove them to be countable, it's just NOT an enumeration. It's a different type of order in the sense that it has a different ordinal. The subject of ordinals is fascinating and it infact answers a lot of difficult questions, such as how does one use induction on
@Tikeslar uncountable sets. This transformed into the concept of transfinite recursion, which is one of the most dominant building blocks on the advancement to modern mathematics, and a beautiful concept on it's own.
I believe that, along with Zorn's Lemma, it's the immediate result of Cantor's diagonal with the most profound implications.
@anticorncob6 It doesn't correspond to any number, not every ORDERING is a NUMBERING.
When talking about ORDERINGS (opposed to NUMBERINGS) you distinct between elements which have a direct predecessor and limit elements. In this case, (2,1) is a limit element.
You can then use transifnite induction (which is similar to mathematical induction, only it takes limit elements into account as well). However, since there are countably many limit elements, and countably many elements between each two...
How can you know that two sets of things are the same size? What does it mean for them to be the same size? Does this mean that they have the same number of things in them? If so, then how can you possibly say that one infinity has a larger number of numbers, if infinity is not a number?
@sk8rdman This is why I go to the trouble of all the definitions in section 1. Our usual use of counting numbers is bogged down with the assumption that things are finite, which, while perfectly sensible in the real world, does not allow it to generalize to infinite sets. The "size" of an infinite set does not fit well with our common understanding of arithmetic.
@sk8rdman That's why the term "size" isn't used. We use "cardinality" instead. And for a good reason, "size" can have different intuitive meanings which depend on context. I find size to be more compliant with the concept of measure then of cardinality (which is more parallel, to, say, amounts, than to sizes).
Perhaps this proof of infinities of different sizes is only successful because of the limits of our numerical system? Perhaps it is possible to pair up all numbers between two points with some other system for counting.
Isn't it true to say that the because the infinite number of points between any two real numbers is infinitely larger than the infinite number of real numbers, that one could say something like, (infinity)*(super infinity)=super mega ultra infinity?
@sk8rdman Your first point is quite interesting. I suppose we are being slightly circular, because we (more or less) define the real numbers as being representable in this form. All maths comes with the disclaimer that we define objects with particular properties and there is no guarantee that they correspond to reality or any other sort of "universal" truth.
I don't understand your second point though - it seems to me that the first set you refer to is a subset of the second.
@rasmusfribble What you are saying here does not seem right. We define real numbers in some way, first like saying, look, sqrt(2) is the smallest number x which is at least as big as any number that satisfies x*x being at least 2 Then we look and check that 1.42 is not the answer, since 1.415 is smaller but it still works, and so on. We do not define x as some number with a prescribed represetation, but a number with a specific property, that one has to figure out, possible with some trouble.
@tommyrjensen Sure, that's how we define the real numbers, but I can still associate a decimal expansion with any such number and then follow the diagonal argument.
@sk8rdman It makes no difference in your first point. If we counted in 13ths the same patterns we see in our current Mathematics is exactly the same as if it were in 13ths. The values 0 and 1 were chosen primarily because they don't deal with n. something but always 0. .
With the second point infinity*infinity is still infinity. We can match up a finite length to an infinite area if you so wished.
@sk8rdman Nope, that proof can be applied in various contexts without depending on numerical systems. For example, there's a very profound theorem in computer science called "the halting problem". The problem is devising an algorithm that, given any computer code, could determine whether it gets stuck or not. The theorem states that such an algorithm could never exists, the proof is essentially applying the same trick shown here.
You were doing just great until the line about "snooker, darts and golf" ..... It IS possible to engage your mind one day AND enjoy that sort of stuff another day .... I'd even say it's a must in order to remain a balanced individual ...... but well done anyway :-)
but at 2:20 , wont the second set always have exactly the same numbers as the first one, and then zero? But then again, since they both go into infinity, they technically have the same amount. hmph.
If I have 10 babies and 10 chainsaws and then you come along and say... well, actually there is another chainsaw. Then I have the right to reorder the list and say, well actually I have another baby.
If you can continually add numbers in a list that falls between 0 and 1 , and you can also continually add numbers to a list of 1+ then they are two equal lists that can always be added to and reordered.
But what you did say was half true. Yes infinities can be different sizes, which is what I thought you were going to talk about, but they can not be different amounts.
I can have two different sized infinities that contain the same amounts.
This is why I can have an infinity of airplanes which are much larger in size than an infinity of babies.
This is why cantor was confused when he pared up a big circle outside of a small circle and extended lines between them
@ecclesiologyproper Cantor was using the wrong thinking tools; Galileo was closer to a practical answer except he probably had difficulty explaining Node density on the radial vectors.
@ecclesiologyproper I don't reorder the list. The proof is basically "If we were able to construct a full list, then I'd be able to construct a number not on the list. This is logically impossible, therefore no such list can exist." No reordering of the list is necessary. (The reason the list appears to change in the video is to point out that we can find an extra number regardless of what candidate list we consider.)
You made a list of fractions and paired up whole numbers with them and then you put a new fraction inside the list and forgot to put a new whole number in.
There are just as many fractions as there are whole numbers
@ecclesiologyproper First of all, I agree there are just as many fractions ("rational numbers") as whole numbers. The numbers in that list are "real numbers", and cannot all be expressed as fractions (some will be "irrational numbers").
Secondly, I did *not* add that number (0.44542856...) to the list. It would be impossible to assign a natural number to it, as any natural number you could think of would already have a real number assigned to it. Please reread my previous comment.
@ecclesiologyproper You can't do the diagonal argument on whole numbers! If you do, you get an infinitely long decimal which is not a whole number. If you pair up the rationals correctly, you will get an IRRATIONAL number when applying the diagonal argument.
If you pair 0.44542856 with 44542856, you won't be able to pair 0.666... with anything but infinity. Cantor's diagonal argument is right and there's nothing you can do about it.
This is actually a very good introduction to the idea of infinity (assuming the n00bs are not thrown by the whimsical yet perhaps distracting comments by the presenter - it made it more fun for me at least). Good job.
I remeber thinking about this topic for years. My Control systems teacher used to argue with me that infinite numbers had no tangible use, while I argued that they were like circles or squares, just an idealization (and a very useful one).
Loved this video, to be honest it made me feel dumb which is good because that means it made me think. I would really like to hear more about this or other similar topics.
@BigGordonLips Besides, pi is a well-defined infinite number, as is 2/3 = 0.6666... . Just because you can't write out its decimal expansion doesn't mean you can't use that number.
@creuynni Well, I did say that infinity isn't a number, so now I am indeed forced to come round to Calon's house and eat all of his biscuits one by one!
good response, guess they are abstract things think your video thou proves there not normal things by any stretch, variously unknowable, unlimited and contradictory but never the less the essence of natural logic
could it work like this" 1=0.1 2=0.2 ... 9=0.9 10=0.01 11 = 0.11 ... 19=0.91 20=0.02 21=0.12 ... 26=0.62 and so on... You reverse the integer's digits, then put a 0. before them. I think thats the same as puting it to the power of -1 or something.
I can assure you that I have thought about it. You cannot pair every non-terminating decimal with "infinity", firstly because "infinity" is NOT a natural number, and secondly because even if it were a natural number, you cannot pair it up with more than one decimal - that's not a bijection!
"You cannot pair every non-terminating decimal with "infinity", firstly because "infinity" is NOT a natural number"
Just because infinity is not a natural number doesn't mean that you can't pair natural numbers leading up to it. There are an infinite amount of natural numbers, so you could pair them trying to get to infinity until you lost your mind. But what if 0 was paired with infinity?
@adamkyler You may as well have asked "what if 0 was paired with a croissant?" as neither infinity nor a croissant is in the set of natural numbers.
Infinity. Is. Not. A. Natural. Number.
(A bijection, as I defined in the video, is a pairing up of two sets such that each element is paired with precisely one element in the other set. The rigorous definition is that it's a function which is both surjective and injective.)
I don't know. It seems to me that for every number you can find between 1 and 2, there will always be an integer available to match up with it. Which is to say: you can keep counting these numbers between 1 and 2 without limit, infinitely. So it seems meaningless to say there are more of them than there are integers. There would be more of them only if you would run out of integers and still have some numbers left between 1 and 2. But you will never run out of integers.
It intuitively seems meaningless, yes, but the diagonal argument shows that it's true!
You will never run out of natural numbers if you pair things up one by one. But if you do all of the pairings at the same time then you will use all of the natural numbers, and there will be real numbers left over.
@rasmusfribble Is there a proof of the diagonal argument somewhere? I don't consider it obvious enough to be an axiom, and every demonstration I've seen of it lacked the rigor I'd prefer.
@theboombody Which step in the proof do you disagree with? There is a minor point about decimal expansions which I skip over in the video (namely that some numbers have two decimal representations, e.g. 0.1 = 0.0999... - this can be fixed by forbidding change of a digit to 0 or 9.) - other than that it's pretty watertight.
@rasmusfribble I'm not disagreeing with the diagonal argument, I just think that since I never see it presented in a sentential logic format, I don't consider it to be a pure proof.
The point is for each number in one of the sets there's a corresponding number in the other set.
From a mathematic perspective this video is poorly made, since by applying definitions its a lot easier to see that the following operators (+)a, (*)a, (^)a for any complex number a doesn't apply, we can further prove that 2^x (x |-> y)for any infinite set x is grander than x.
@FundamentalPhysicist I was more concerned with the diagonal argument itself than the size of the sets. I just posted a response to this video about set size. It's poorly made too. I show that there's a 2 to 1 ratio and a 1 to 1 ratio between the sets (0,2] and (0,1], if that means anything.
@theboombody The logic is sound, it's just reductio ad absurdam. (i.e. suppose a bijection f from N to R exists, now construct a real number that cannot be in the domain of f - this is a contradiction with f being bijective, so our initial assumption is false.)
Do you insist that any proof be entirely in sentential logic format before accepting it? The whole point of a language of mathematical proof is that we don't have to lug hundreds of sentential logic statements around the whole time.
@rasmusfribble I figured a proof of the diagonalization argument could be done with the few symbols that already exist, and I think there are less than ten. There's a symbol for "and" "or" "not" "for every" "for some" and I forget what else, but there aren't very many. And you can just use plain old letters or Greek letters to represent particular statements.
i have a theory i thought of how many number are between 1-2 (yes i made this)
so if decimals go on for infinity the nubers would be 1/i (i= infinity)but if you divided any number (only using digits) infinity times like 1/infinity and 40/1 it would be reduced to nothing so in a sense there are 0 number between any 2 numbers
Interesting. Its one of those things, you either get it or you don't. Ive done some A level maths but not too much, however I do understand most of the principles behind what you have said. The way I would pair up the final numbers (on the vid) would be simply to remove the zero and the decimal point, and reverse the numbers. So 1 would be infinity, 0.00000-----(infinate zeros) followed by one would be 1... etc... however the only difficulty is, labeling 0.
Good idea - unfortunately you again run into trouble with numbers such as two-thirds, i.e. 0.666... - how are you going to reverse an infinite number of 6es?
Also, 0.000...001 doesn't make any sense... how can we have an infinite number of zeros if they eventually end?
You're not stupid. Don't ever feel stupid for asking questions, even ones that may seem basic. It is only by asking questions that we can make progress in mathematics. :)
you should have covered the the size of the irrational numbers versus the size of the rationals. a bijection can be formed between the rational numbers and the natural numbers, therefore rationals have the same cardinality as naturals. then notice that irrationals unioned with rationals = real numbers, which has greater cardinality than natural numbers. thus irrationals have same cardinality as the reals, and thus there are more irrationals than rationals(even though they're both dense sets)
When I first read Cantor's Diagonal argument I couldn't believe it. It seem amazing that there are more irrational numbers between 0 and 1 than there are natural numbers!
two things what is the implications of not being able to pair up decimals, you said it would take forever to do so which it would to with integers too. also if you assume that the infinite number of zeros before a decimal are accounted for surely you can treat decimals as integers for the purpose of pairing them up, as integers also have an infinite number of zeros but the base starting point is different..
Yes, it would take an infinite amount of time to write out a mapping from integers to naturals, for example, but the point is that a mapping does exist even if it can't be done physically. No mapping exists for reals to naturals.
Are you suggesting e.g. 0.002 pairs up with ...000002? Problem there is that 0.02, 0.2 also would pair up with ...00002 which means it's not a pairing (it's not "injective").
I almost wanted to say that an infinite number of babies or chainsaws would take up the entire universe (and all of nothingness as well), but then I reminded myself that it was a gag for comedic effect and used solely to explain how things can be paired up.
I was about to say that, actually, if the universe is infinite, then you could fit the babies and chainsaws into one "half" of the universe. But then I realised your comment was posted for comedic effect...
mhmm just stating that there are an infinite amount of numbers says that there is always an infinite amount of any type of number and all about in between. no matter how small or big
My friend, i am from Portugal,and i speak genuine Portuguese and i have a funny Portuguese accent....i think that for you to listen to an american to speak would be something like me, listening to a brazillian?? (and it is funny) ohww! by the way...i am not at mathematics...but i know alot about fractal imaging...and i know that in a Mendelbrot set for instance...you always find a paired sequence until the never coming infinity...??? i know i am stupid!! but what i say it is true :)
No no no!! We NEVER evaluate double(infinity) or prime(infinity). There is no such thing.
For example, when bijecting 0,1,2,3,... with 1,2,3,4,..., I used the function f(x) = x+1. Nowhere did I evaluate f(infinity). Instead I note that this is defined for all x, and it is bijective (pairs up numbers uniquely). Give me a number and I'll show you what its partner is.
I repeat, in this field we never evaluate a function at infinity!
So, is the prime() a bijection even if it is defined it with an algorithum? For prime() the range is the all of primes. So, can you still prove the bijection. Because, that seems to put the cart before the horse. How do you know the set of all primes, which is the range of the function?
Again, please please please don't worry about algorithms - it makes no difference if a complicated algorithm rather than a closed form is required or not. Besides, "multiply by 2" is an algorithm as well. Existance is all we care about.
The prime() function works because there is always a "next" prime number - e.g. 7 follows 5, there are no primes in between. This allows the function to be defined. There is no such thing with the reals, for example - what is the next number after pi?
If the "next" prime number is "infinitely far" away, then that means there are no more prime numbers. Take a look at the Wikipedia article "Euclid's Theorem" for a proof that there are an infinite number of prime numbers. :)
Maybe math depends on locality. What do I mean by that? That means if the calculation is made in a different space-time-situations it will produce different results. For example in the blackhole 2x2=5. Maybe these meaningless calculations will affect the way we think (I can't tell you how). Anyways, just the process of setting up the project will require a lot of interesting work. It might be more interesting than hogging the computer screen for visuals.
The last one that was ingenious...awesome..although it is kinda hard to believe that the number of integers and natural numbers are the same..
Amandeep26983 2 weeks ago
Finally someone who can explain this without making me doze off into a mind numbing stupor! Thanks for helping me pass my math final nice British man!
lecoffeebandit 1 month ago
I loved this video.
It puts my mind at easy to think about it like this:
If anything was truly infinite we would know because it would take all energy, space and time in all dimensions and then more to hold it. There would be no room for us.
The set of integers is just a set of numbers that CAN go on as long as you keep counting, like a car that can go on as long as you keep putting fuel in it.
does that make a spaceship more infinite than 0 to 1 because it can work in 3 dimensions?
faceshed 2 months ago
This was beautiful...Keep up the great work!!
MISTERASMODEUS 2 months ago
...find a point where you could not continue the diagonal, but the diagonal would continue indefinately between 0 and .1 alone.....okay never mind I give up
bridgemaniii 3 months ago
Okay, I think I've got it... I am absolutely not a math guy, but I have been watching your vid and similar vids and I only ask that you try to follow what I am going to try to explain:
If you were to take all of the decimal numbers between 0 and 1 they go on indefinately, but if you put them in order then the number of decimal places would not necessarily always increase. For instance 0.7>0.666... etc.
So at first thought, you might think that the diagonal wouldn't work because you would...
bridgemaniii 3 months ago
Giving racist babies chainsaws... not a smart idea.
andysido 3 months ago
Between 0 and 1 there is an infinite amount of decimal places. There are an infinite amount of numbers, and bigger infinites beyond both of those infinities so shouldn't you be able to match the numbers?
Minnie4life1 4 months ago
you can have all kinds of fun with a circle when it comes to infinity :)
lmos26 5 months ago
Comment removed
lmos26 5 months ago
All the decimilas from 0 to 1 = infinity. To put them in a list from 1 to infinity you need to order them to make them fit, you cannot just place them randomly like u have done in the video, 1=0.5 2=0.25 3=0.66... You can start with 1=0.1, 2=0.2, 3=0.3, 4=0.4...6456=0.6456... 584935034=0.584935034 and so on, and infinity = 1, So what ever number you are at you can just place a "0." in front of it, and there u go u have a list that fits to infinity from 0 to1 and they match,
Typho0n86 5 months ago
@Typho0n86 0.6666... doesn't have a natural number to go with it.
rasmusfribble 5 months ago
@rasmusfribble yes it does, 6, 66, 666, 666, 6666, 66666, 666666 .... all the way to infinity
Typho0n86 5 months ago
@Typho0n86
I believe he was referring to the decimal for 2/3. The sixes repeat forever.
TheCeejReturns 3 months ago
@TheCeejReturns Yeah and i explained that, if you are going to infinity or forever then 2/3 can be written as 666,666,666,666,666,666,666,666.... all the way to infinity. So when we reach infinity, 2/3 will be in there. I think you have missunderstood something about infinity.
Typho0n86 3 months ago
@Typho0n86
Maybe I have, but I was under the impression that you never would reach infinity due to the nature that it keeps going forever.
TheCeejReturns 3 months ago
@TheCeejReturns You do understand! Guess what!? 2/3rd's goes on forever too :D
Typho0n86 3 months ago
@Typho0n86
So, the decimal for 2/3 has the same amount of digits as there are natural numbers. But, that wasn't exactly what he was explaining in the video.
TheCeejReturns 3 months ago
@TheCeejReturns I know he is trying to put them in lists. So i made a listing system for the decimals, which he said that even Cantor couldnt do!
Typho0n86 3 months ago
@Typho0n86
If Cantor couldn't do it, then it can't really be as simple as you make it out to be.
TheCeejReturns 3 months ago
@TheCeejReturns Simplicity is the key ;)
Typho0n86 3 months ago
@Typho0n86
Infinity is a purely theoretical concept anyway. It works on paper but doesn't exist in practice. Just like negative numbers and the square roots of them.
TheCeejReturns 3 months ago
@TheCeejReturns Well then you have shut ur self in a very small universe. :P
Typho0n86 3 months ago
@Typho0n86
What does that even mean?
TheCeejReturns 3 months ago
@TheCeejReturns The universe can be infinite. Not just a theoretical concept, infinity Can exist in practice.
Typho0n86 3 months ago
@Typho0n86
Nope. I read somewhere that the number of atoms in the universe totals far less than a googol plex. So, it's highly unlikely that a googol plex will ever be used in practice, and infinity is impossible in practice.
TheCeejReturns 3 months ago
@TheCeejReturns The number of atoms in our universe would be finite. But how far does space go?
Typho0n86 3 months ago
@Typho0n86
A finite amount of space. For a more accurate answer, ask an astrophysicist.
TheCeejReturns 3 months ago
@TheCeejReturns Your ignorant, They would say they didnt know! Cause there is no way of measuring it
Typho0n86 3 months ago
@Typho0n86
Yes. I'm ignorant as to the amount of space in the universe. This is why I told you to ask an astrophysicist.
TheCeejReturns 3 months ago
@TheCeejReturns you ask them then
Typho0n86 3 months ago
@Typho0n86
You're the one who wants to know.
TheCeejReturns 3 months ago
@TheCeejReturns I already know that they dont know! Your the one thats putting words in their mouth "A finite amount of space".
Typho0n86 3 months ago
@Typho0n86
Apparently you don't. Because I do know one thing about the size of the universe. I know that it's expanding. If it were infinite, it wouldn't be doing that, would it?
TheCeejReturns 3 months ago
@TheCeejReturns Expanding into what?
Typho0n86 3 months ago
This has been flagged as spam show
@Typho0n86
Okay. It has become clear that you are asking me because I don't know and you don't want an answer. If you wanted an answer, you would ask an expert in the field, not some comedian on YouTube.
TheCeejReturns 3 months ago
@Typho0n86 666,666,666,... is not a natural number, and you have to pair up the NATURAL NUMBERS with the real numbers.
anticorncob6 2 weeks ago
@anticorncob6 infinity is not a natural number, so if ur going to list all the numbers down then, 666,666,666,666........ or 2/3 will be on it. its the same infinity
Typho0n86 2 weeks ago
@Typho0n86 2/3 is not infinity.
anticorncob6 2 weeks ago
@anticorncob6 good work
Typho0n86 2 weeks ago
@Typho0n86 so 1 = 0.1
What number would 0.000001 be?
MrRSheep 5 months ago
@MrRSheep -100000
Typho0n86 5 months ago
I've been fascinated by Infinity ever since I first saw Dangerous Knowledge on the BBC when it first aired. I remember googling frantically trying to understand Cantor's argument to no avail. Tonight's the first time I've finally got it. Thanks so much for making this video.
RXP91 6 months ago 3
@RXP91 Awesome! So glad I could be of assistance.
rasmusfribble 5 months ago
@RXP91 Exactly, i'm also who was inspired by the Dangerous Knowledge. Since then i've been trying to understand (a glimpse) of what Cantor's discovered.
@rasmusfribble Thanks for your video!
enjoy2204 2 months ago
I actually find it's hard to listen to an American talk about maths. Something about and English or Indian accent just makes math seem more digestible. By the same regard Aussies do have pretty absurd accents.
VividVelvet 6 months ago
You actually CAN order NxN in the following manner: (1,1), (1,2), (1,3) ,.... , (2,1), (2,2), (2,3), ...
The fact that there isn't an explicit index for any duple which doesn't begin with one doesn't matter. It is an order, and it DOES prove them to be countable, it's just NOT an enumeration. It's a different type of order in the sense that it has a different ordinal. The subject of ordinals is fascinating and it infact answers a lot of difficult questions, such as how does one use induction on
Tikeslar 7 months ago
@Tikeslar uncountable sets. This transformed into the concept of transfinite recursion, which is one of the most dominant building blocks on the advancement to modern mathematics, and a beautiful concept on it's own.
I believe that, along with Zorn's Lemma, it's the immediate result of Cantor's diagonal with the most profound implications.
Tikeslar 7 months ago
@Tikeslar Which number does the pair (2, 1) pair with? I don't understand what you were saying.
anticorncob6 6 months ago
@anticorncob6 It doesn't correspond to any number, not every ORDERING is a NUMBERING.
When talking about ORDERINGS (opposed to NUMBERINGS) you distinct between elements which have a direct predecessor and limit elements. In this case, (2,1) is a limit element.
You can then use transifnite induction (which is similar to mathematical induction, only it takes limit elements into account as well). However, since there are countably many limit elements, and countably many elements between each two...
Tikeslar 6 months ago
@Tikeslar But can't you use the same logic to show you CAN pair the natural numbers with the real numbers?
anticorncob6 6 months ago
@anticorncob6 most definitely not. If you think you can, tell me how and I'll tell you if I spot your flaw, I'm pretty sure I will.
Tikeslar 6 months ago
This was brilliant. You are both intelligent and creative. Awesome combination! Thanks!
MISTERASMODEUS 7 months ago
Are you seriously still commenting a video you made... 4 years ago!??!
Meh, besides that, and besides the horrible jokes, it was OK. Stop talking and do some others!
fl260 10 months ago
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anticorncob6 11 months ago
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anticorncob6 11 months ago
@rasmusfribble i belief cantor went through his insanity due to pibolar disorder not the critics of his colleges
marwanxyz123 1 year ago
Good work, math guy. Here's looking at Euclid....
Jhopnik 1 year ago 3
you are great kid,keep it up,i wish my child is as brilliant like you
amickonk 1 year ago
No, I didn't know, you PRICK.
FrenchExpat1 1 year ago
I don't understand this video at all. But its calming to watch... :)
keychainere 1 year ago
you should get around to making another video.
TheWordsNotSpoken 1 year ago
How can you know that two sets of things are the same size? What does it mean for them to be the same size? Does this mean that they have the same number of things in them? If so, then how can you possibly say that one infinity has a larger number of numbers, if infinity is not a number?
sk8rdman 1 year ago
@sk8rdman This is why I go to the trouble of all the definitions in section 1. Our usual use of counting numbers is bogged down with the assumption that things are finite, which, while perfectly sensible in the real world, does not allow it to generalize to infinite sets. The "size" of an infinite set does not fit well with our common understanding of arithmetic.
rasmusfribble 1 year ago
@sk8rdman That's why the term "size" isn't used. We use "cardinality" instead. And for a good reason, "size" can have different intuitive meanings which depend on context. I find size to be more compliant with the concept of measure then of cardinality (which is more parallel, to, say, amounts, than to sizes).
Tikeslar 6 months ago
Perhaps this proof of infinities of different sizes is only successful because of the limits of our numerical system? Perhaps it is possible to pair up all numbers between two points with some other system for counting.
Isn't it true to say that the because the infinite number of points between any two real numbers is infinitely larger than the infinite number of real numbers, that one could say something like, (infinity)*(super infinity)=super mega ultra infinity?
sk8rdman 1 year ago
@sk8rdman Your first point is quite interesting. I suppose we are being slightly circular, because we (more or less) define the real numbers as being representable in this form. All maths comes with the disclaimer that we define objects with particular properties and there is no guarantee that they correspond to reality or any other sort of "universal" truth.
I don't understand your second point though - it seems to me that the first set you refer to is a subset of the second.
rasmusfribble 1 year ago
@rasmusfribble What you are saying here does not seem right. We define real numbers in some way, first like saying, look, sqrt(2) is the smallest number x which is at least as big as any number that satisfies x*x being at least 2 Then we look and check that 1.42 is not the answer, since 1.415 is smaller but it still works, and so on. We do not define x as some number with a prescribed represetation, but a number with a specific property, that one has to figure out, possible with some trouble.
tommyrjensen 6 months ago
@tommyrjensen Sure, that's how we define the real numbers, but I can still associate a decimal expansion with any such number and then follow the diagonal argument.
rasmusfribble 5 months ago
@sk8rdman It makes no difference in your first point. If we counted in 13ths the same patterns we see in our current Mathematics is exactly the same as if it were in 13ths. The values 0 and 1 were chosen primarily because they don't deal with n. something but always 0. .
With the second point infinity*infinity is still infinity. We can match up a finite length to an infinite area if you so wished.
noionxion 1 year ago
@sk8rdman Nope, that proof can be applied in various contexts without depending on numerical systems. For example, there's a very profound theorem in computer science called "the halting problem". The problem is devising an algorithm that, given any computer code, could determine whether it gets stuck or not. The theorem states that such an algorithm could never exists, the proof is essentially applying the same trick shown here.
Tikeslar 6 months ago
I find this kid really anoying, but his explanation of infinity is pretty darn good for a 10 minute video.
xenoepist 1 year ago
You were doing just great until the line about "snooker, darts and golf" ..... It IS possible to engage your mind one day AND enjoy that sort of stuff another day .... I'd even say it's a must in order to remain a balanced individual ...... but well done anyway :-)
SearchBucket 1 year ago
nice video
Lifeplay 1 year ago
Great video!
maniac7777 1 year ago
Thanks to you i finally understand the cantor thing at the end , that i road in a book :)
I do my endwork about infinity at school :D
krikke117 1 year ago
but at 2:20 , wont the second set always have exactly the same numbers as the first one, and then zero? But then again, since they both go into infinity, they technically have the same amount. hmph.
maxx1231 1 year ago
@maxx1231 Yup. Adding another number into the set doesn't make it any bigger.
rasmusfribble 1 year ago
Ofcourse you can put all the numbers between 0 and 1 on a list.
If you reorder your list of 0 - 1 then it is only fair that you be allowed to reorder your list of 1 +
ecclesiologyproper 1 year ago
@ecclesiologyproper
If I have 10 babies and 10 chainsaws and then you come along and say... well, actually there is another chainsaw. Then I have the right to reorder the list and say, well actually I have another baby.
If you can continually add numbers in a list that falls between 0 and 1 , and you can also continually add numbers to a list of 1+ then they are two equal lists that can always be added to and reordered.
ecclesiologyproper 1 year ago
@ecclesiologyproper
But what you did say was half true. Yes infinities can be different sizes, which is what I thought you were going to talk about, but they can not be different amounts.
I can have two different sized infinities that contain the same amounts.
This is why I can have an infinity of airplanes which are much larger in size than an infinity of babies.
This is why cantor was confused when he pared up a big circle outside of a small circle and extended lines between them
ecclesiologyproper 1 year ago
@ecclesiologyproper
a big circle will have the same amount of points on it's ring but the points will be bigger than a circle of the same amount of points in the middle.
In this way they contain the same infinite amount of points but one has larger points.
ecclesiologyproper 1 year ago
@ecclesiologyproper Cantor was using the wrong thinking tools; Galileo was closer to a practical answer except he probably had difficulty explaining Node density on the radial vectors.
AClarke2007 1 year ago
@ecclesiologyproper I don't reorder the list. The proof is basically "If we were able to construct a full list, then I'd be able to construct a number not on the list. This is logically impossible, therefore no such list can exist." No reordering of the list is necessary. (The reason the list appears to change in the video is to point out that we can find an extra number regardless of what candidate list we consider.)
rasmusfribble 1 year ago
@rasmusfribble
Yes you did reorder the list.
You made a list of fractions and paired up whole numbers with them and then you put a new fraction inside the list and forgot to put a new whole number in.
There are just as many fractions as there are whole numbers
ecclesiologyproper 1 year ago
@ecclesiologyproper First of all, I agree there are just as many fractions ("rational numbers") as whole numbers. The numbers in that list are "real numbers", and cannot all be expressed as fractions (some will be "irrational numbers").
Secondly, I did *not* add that number (0.44542856...) to the list. It would be impossible to assign a natural number to it, as any natural number you could think of would already have a real number assigned to it. Please reread my previous comment.
rasmusfribble 1 year ago
This has been flagged as spam show
@rasmusfribble
"any natural number you could think of would already have a real number assigned to it. "
ecclesiology: Wrong, because you can do the diagonal method with whole numbers and make a new whole number, same as you can do with rational numbers.
0.44542856 can be associated with 44542856 and so on.
ecclesiologyproper 1 year ago
@ecclesiologyproper You can't do the diagonal argument on whole numbers! If you do, you get an infinitely long decimal which is not a whole number. If you pair up the rationals correctly, you will get an IRRATIONAL number when applying the diagonal argument.
If you pair 0.44542856 with 44542856, you won't be able to pair 0.666... with anything but infinity. Cantor's diagonal argument is right and there's nothing you can do about it.
anticorncob6 11 months ago
This video is not natural... it's far too complex. Please be rational. Aleph-naught more of that behaviour from you young lad! [Enjoying the puns?]
nuodai 1 year ago
@nuodai Stop surjecting me to these puns. I can't function - for real!
rasmusfribble 1 year ago 4
This is actually a very good introduction to the idea of infinity (assuming the n00bs are not thrown by the whimsical yet perhaps distracting comments by the presenter - it made it more fun for me at least). Good job.
rohypnol55 1 year ago
Outstanding! You'll be famous one day lad!
redabdab 1 year ago
I remeber thinking about this topic for years. My Control systems teacher used to argue with me that infinite numbers had no tangible use, while I argued that they were like circles or squares, just an idealization (and a very useful one).
Loved this video, to be honest it made me feel dumb which is good because that means it made me think. I would really like to hear more about this or other similar topics.
cachoengineering 1 year ago
let's divide an infinate number by zero
CueCappa 1 year ago
why would u give chainsaws to babies
Benandbjshow 1 year ago
the answer is simpler than the diagonal proof, real numbers can be infinitely long, therefore you can never define a starting point
BigGordonLips 1 year ago
@BigGordonLips Can, but don't have to be. 2.0000... is a perfectly good starting point.
rasmusfribble 1 year ago
@BigGordonLips Besides, pi is a well-defined infinite number, as is 2/3 = 0.6666... . Just because you can't write out its decimal expansion doesn't mean you can't use that number.
rasmusfribble 1 year ago
192/219ths is absolutely my new favorite number ever.
falstaffswims 1 year ago
you have not tried to pair it up in the V^V^V^V^V^V^pattern lol
death2opposeme 1 year ago
yey i figured out the dots at 6:00 :D ..(i know that u cold not care less.. but still)
Roffe192 1 year ago
Good show! .... and I actually could follow it too! Bravo!
doloppost 1 year ago
Infinity is a state not a value.
Infinity is infinity until defined.
MRDraak 1 year ago
Too bad people do not understand one simple concept:
Infinity is a state not a value. Infinity is infinity until defined.
MRDraak 1 year ago
thanks for making the diagonal argument clear
jlmknight 1 year ago 2
thanks for helping me almost answer calon's question of if infinity is a number.....
calon's decided that infinity is a number, but i'm not so sure yet.......
i guess it sorta is in a wat.........a proper amazing number if so.......
creuynni 1 year ago
@creuynni Well, I did say that infinity isn't a number, so now I am indeed forced to come round to Calon's house and eat all of his biscuits one by one!
rasmusfribble 1 year ago
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creuynni 1 year ago
come over here and eat all my biscuits!
from calon (age 6)
creuynni 1 year ago
@creuynni Excellent, do you have custard creams?
rasmusfribble 1 year ago
That was funny! I like it!
Charizard632 1 year ago
I wonder if Cantor would be able to pair all girls and boys on earth using bijection. Or would the diagonal argument apply?
rufiao 1 year ago
Good video but STOP WITH THE JOKES. they realy arent funny at all.
XANDERXXZ 1 year ago
Please do another that was brilliant !
Along with your ability to join wit and interesting maths.
badger1184 1 year ago
god, thanks for really taking the long away around to explain something interesting that your antics really made uninteresting with ease.
death2opposeme 1 year ago
that was awesome! make another!
ssslickguy 1 year ago
Brilliant video. I've just watched Horizon, which explained the same concept and you did it better. Well, more entertainingly. Claps.
tomspencer155 1 year ago
Great video. Loved it.
I would show it to my A Level students but they'd think I was a nutter. They don't like it when their teachers like "young people" things.
AlanKey86 2 years ago
Well done! You should make more videos.
Eiwalker 2 years ago
Hi very good video many thanks.
Philosphical point here but are the counting numbers something distinct from the decimals. i.e is 1 the sum of the bits in between.
My point is like Zeno's tortoise and the hare. spltting things samller and smaller
so by moving from 1 to 2 have you pasted through an infinite numer of points already as they are just notations for their contingent decimals.
like 2 circles of 1cm circumfrence and 2cm circumfrance. Seems trite to count numbers (not things).
39grotto 2 years ago
Numbers are things, surely?
rasmusfribble 2 years ago
good response, guess they are abstract things think your video thou proves there not normal things by any stretch, variously unknowable, unlimited and contradictory but never the less the essence of natural logic
39grotto 2 years ago
adamkyler 2 years ago
Great, but which natural number is paired up with 1/3 = 0.333...?
rasmusfribble 2 years ago
333=0.333
33333333333333333333333333=0.33333333333333333333333333
I do agree with adamkyler on this one, sounds the most logical and would allow pairing to give the decimal an infinity status, which it clearly has.
Think about it. If you call something infinity, wouldn't it be logical to assume it is infinite and thus paireble with the normal count?
jackenco 2 years ago
I can assure you that I have thought about it. You cannot pair every non-terminating decimal with "infinity", firstly because "infinity" is NOT a natural number, and secondly because even if it were a natural number, you cannot pair it up with more than one decimal - that's not a bijection!
rasmusfribble 2 years ago
"You cannot pair every non-terminating decimal with "infinity", firstly because "infinity" is NOT a natural number"
Just because infinity is not a natural number doesn't mean that you can't pair natural numbers leading up to it. There are an infinite amount of natural numbers, so you could pair them trying to get to infinity until you lost your mind. But what if 0 was paired with infinity?
adamkyler 2 years ago
Also, what is a bijection?
adamkyler 2 years ago
@adamkyler
A bijection from one set to another means that:
1)you can pair up the numbers one to one liek in the video
2)every number in both sets is used
affablegiraffable 1 year ago
@adamkyler You may as well have asked "what if 0 was paired with a croissant?" as neither infinity nor a croissant is in the set of natural numbers.
Infinity. Is. Not. A. Natural. Number.
(A bijection, as I defined in the video, is a pairing up of two sets such that each element is paired with precisely one element in the other set. The rigorous definition is that it's a function which is both surjective and injective.)
rasmusfribble 2 years ago
I love this kind of comedy. And I learned something. Thanks, man!
felenoral 2 years ago 10
I don't know. It seems to me that for every number you can find between 1 and 2, there will always be an integer available to match up with it. Which is to say: you can keep counting these numbers between 1 and 2 without limit, infinitely. So it seems meaningless to say there are more of them than there are integers. There would be more of them only if you would run out of integers and still have some numbers left between 1 and 2. But you will never run out of integers.
FranzFafner 2 years ago
It intuitively seems meaningless, yes, but the diagonal argument shows that it's true!
You will never run out of natural numbers if you pair things up one by one. But if you do all of the pairings at the same time then you will use all of the natural numbers, and there will be real numbers left over.
rasmusfribble 2 years ago
@rasmusfribble Is there a proof of the diagonal argument somewhere? I don't consider it obvious enough to be an axiom, and every demonstration I've seen of it lacked the rigor I'd prefer.
theboombody 1 year ago
@theboombody Which step in the proof do you disagree with? There is a minor point about decimal expansions which I skip over in the video (namely that some numbers have two decimal representations, e.g. 0.1 = 0.0999... - this can be fixed by forbidding change of a digit to 0 or 9.) - other than that it's pretty watertight.
rasmusfribble 1 year ago
@rasmusfribble I'm not disagreeing with the diagonal argument, I just think that since I never see it presented in a sentential logic format, I don't consider it to be a pure proof.
theboombody 1 year ago
@theboombody
The point is for each number in one of the sets there's a corresponding number in the other set.
From a mathematic perspective this video is poorly made, since by applying definitions its a lot easier to see that the following operators (+)a, (*)a, (^)a for any complex number a doesn't apply, we can further prove that 2^x (x |-> y)for any infinite set x is grander than x.
FundamentalPhysicist 1 year ago
@FundamentalPhysicist I was more concerned with the diagonal argument itself than the size of the sets. I just posted a response to this video about set size. It's poorly made too. I show that there's a 2 to 1 ratio and a 1 to 1 ratio between the sets (0,2] and (0,1], if that means anything.
theboombody 1 year ago
@FundamentalPhysicist Huh? Are you saying that the video is poorly made because it doesn't go on to describe power sets?
rasmusfribble 1 year ago
@theboombody The logic is sound, it's just reductio ad absurdam. (i.e. suppose a bijection f from N to R exists, now construct a real number that cannot be in the domain of f - this is a contradiction with f being bijective, so our initial assumption is false.)
Do you insist that any proof be entirely in sentential logic format before accepting it? The whole point of a language of mathematical proof is that we don't have to lug hundreds of sentential logic statements around the whole time.
rasmusfribble 1 year ago
@rasmusfribble I figured a proof of the diagonalization argument could be done with the few symbols that already exist, and I think there are less than ten. There's a symbol for "and" "or" "not" "for every" "for some" and I forget what else, but there aren't very many. And you can just use plain old letters or Greek letters to represent particular statements.
theboombody 1 year ago
i have a theory i thought of how many number are between 1-2 (yes i made this)
so if decimals go on for infinity the nubers would be 1/i (i= infinity)but if you divided any number (only using digits) infinity times like 1/infinity and 40/1 it would be reduced to nothing so in a sense there are 0 number between any 2 numbers
pk4eva2 2 years ago
You made it look very easy for sure great video
schatzen121 2 years ago
so is 1 plus 1 still 2? or is it 1.9999999999999999999999...?
Good stuff Sir. I appreciate the simplistic explanations. I think problems like these will be easily solved by next generations.
Ones ceiling will be a floor for another...
verkaserduch 2 years ago
A challenge: find an explicit formula that maps N onto Q. It's easy to find one from Q onto N, just map p/q to Max(p,1) or some such.
at1with0 2 years ago
Please tell me you want to become a math teacher!
lunchlady55 2 years ago
Interesting. Its one of those things, you either get it or you don't. Ive done some A level maths but not too much, however I do understand most of the principles behind what you have said. The way I would pair up the final numbers (on the vid) would be simply to remove the zero and the decimal point, and reverse the numbers. So 1 would be infinity, 0.00000-----(infinate zeros) followed by one would be 1... etc... however the only difficulty is, labeling 0.
What do you think, or am i stupid?
askmagus 2 years ago
Good idea - unfortunately you again run into trouble with numbers such as two-thirds, i.e. 0.666... - how are you going to reverse an infinite number of 6es?
Also, 0.000...001 doesn't make any sense... how can we have an infinite number of zeros if they eventually end?
You're not stupid. Don't ever feel stupid for asking questions, even ones that may seem basic. It is only by asking questions that we can make progress in mathematics. :)
rasmusfribble 2 years ago
you should have covered the the size of the irrational numbers versus the size of the rationals. a bijection can be formed between the rational numbers and the natural numbers, therefore rationals have the same cardinality as naturals. then notice that irrationals unioned with rationals = real numbers, which has greater cardinality than natural numbers. thus irrationals have same cardinality as the reals, and thus there are more irrationals than rationals(even though they're both dense sets)
ibreakkidslegs 2 years ago
Could have done, but I think there's enough to think about in this video for most people. Besides, I'd have gone over 10 minutes. :P
rasmusfribble 2 years ago
Brilliant!!!
When I first read Cantor's Diagonal argument I couldn't believe it. It seem amazing that there are more irrational numbers between 0 and 1 than there are natural numbers!
I'm off to have a cheese toasty now!
36trooper 2 years ago
:D I hope you showed that cheese toasty who's boss!
rasmusfribble 2 years ago
two things what is the implications of not being able to pair up decimals, you said it would take forever to do so which it would to with integers too. also if you assume that the infinite number of zeros before a decimal are accounted for surely you can treat decimals as integers for the purpose of pairing them up, as integers also have an infinite number of zeros but the base starting point is different..
fremanofkol 2 years ago
Yes, it would take an infinite amount of time to write out a mapping from integers to naturals, for example, but the point is that a mapping does exist even if it can't be done physically. No mapping exists for reals to naturals.
Are you suggesting e.g. 0.002 pairs up with ...000002? Problem there is that 0.02, 0.2 also would pair up with ...00002 which means it's not a pairing (it's not "injective").
rasmusfribble 2 years ago
if infanites can be bigger than infanites
then are than an infanite number of infanites?
would that be infinity to the infinith?
and if you looked at infinity to the infinith, sideways, would it look like the number 88?
i think it would, just a little bit
kionay 2 years ago
That observation should definitely be considered for a Field's medal.
rasmusfribble 2 years ago
I almost wanted to say that an infinite number of babies or chainsaws would take up the entire universe (and all of nothingness as well), but then I reminded myself that it was a gag for comedic effect and used solely to explain how things can be paired up.
KurosenvsGrither 2 years ago
Haha!
I was about to say that, actually, if the universe is infinite, then you could fit the babies and chainsaws into one "half" of the universe. But then I realised your comment was posted for comedic effect...
rasmusfribble 2 years ago
Fantastic video. I will send it to my students.
resselgmada 2 years ago
mhmm just stating that there are an infinite amount of numbers says that there is always an infinite amount of any type of number and all about in between. no matter how small or big
yup
BigPortu2 2 years ago
My friend, i am from Portugal,and i speak genuine Portuguese and i have a funny Portuguese accent....i think that for you to listen to an american to speak would be something like me, listening to a brazillian?? (and it is funny) ohww! by the way...i am not at mathematics...but i know alot about fractal imaging...and i know that in a Mendelbrot set for instance...you always find a paired sequence until the never coming infinity...??? i know i am stupid!! but what i say it is true :)
Icneumone7 2 years ago
I always wonder how different accents in foreign languages sound to a native speaker. Hurts my head to be honest! :)
Are you saying the Mandelbrot set is countable (i.e. can be paired with the naturals)? I don't think that's true.
rasmusfribble 2 years ago
I feel like i'm watching an episode of Charlie Brooker's screenwipe on maths. Excellent video!
drsponk630 2 years ago 5
wow that was charming and smart
danielmetz 2 years ago
wow, that was smart and charming
danielmetz 2 years ago
You are right. Both Double(n) and Prime(n) are functions. But very different functions for our purpose.
We immediately know Double(infinity)=infinityX2=infinity.
On the other hand Prime(infinity) we are lost. We need to do more than one iteration.
Another interesting thing we could do is to call these infinites (remmember different Prime() may have different algorithums) negative alphies.
Israe5l 2 years ago
No no no!! We NEVER evaluate double(infinity) or prime(infinity). There is no such thing.
For example, when bijecting 0,1,2,3,... with 1,2,3,4,..., I used the function f(x) = x+1. Nowhere did I evaluate f(infinity). Instead I note that this is defined for all x, and it is bijective (pairs up numbers uniquely). Give me a number and I'll show you what its partner is.
I repeat, in this field we never evaluate a function at infinity!
rasmusfribble 2 years ago
Comment removed
Israe5l 2 years ago
So, is the prime() a bijection even if it is defined it with an algorithum? For prime() the range is the all of primes. So, can you still prove the bijection. Because, that seems to put the cart before the horse. How do you know the set of all primes, which is the range of the function?
Israe5l 2 years ago
Again, please please please don't worry about algorithms - it makes no difference if a complicated algorithm rather than a closed form is required or not. Besides, "multiply by 2" is an algorithm as well. Existance is all we care about.
The prime() function works because there is always a "next" prime number - e.g. 7 follows 5, there are no primes in between. This allows the function to be defined. There is no such thing with the reals, for example - what is the next number after pi?
rasmusfribble 2 years ago
what if the "next" prime number is infinite far? (different type of infinity to think about, too.)
maybe I will just become a math snob and stick to my algorithums. Being a math person and not a compu sci person algorithum is an uncharted world.
I guess, I got hooked by the mission of meaningless calculations.
Israe5l 2 years ago
If the "next" prime number is "infinitely far" away, then that means there are no more prime numbers. Take a look at the Wikipedia article "Euclid's Theorem" for a proof that there are an infinite number of prime numbers. :)
rasmusfribble 2 years ago
Maybe math depends on locality. What do I mean by that? That means if the calculation is made in a different space-time-situations it will produce different results. For example in the blackhole 2x2=5. Maybe these meaningless calculations will affect the way we think (I can't tell you how). Anyways, just the process of setting up the project will require a lot of interesting work. It might be more interesting than hogging the computer screen for visuals.
Israe5l 2 years ago
This was quite an interesting video. It got me thinking =]
xGamer911x 2 years ago
You are awesome math lord!
Digikong 2 years ago