Countably infinite just means that you can match your list one-to-one with the positive integers. What the diagonal argument shows is that there are certain infinite collections that can't be matched one-to-one with the positive integers. In effect, some infinities are bigger than others. One concern you had was "how can an infinite list not contain everything?". That's easy, consider the list ABBBB..., BABBB..., BBABB..., etc. (each item has all Bs except for one A in different positions). Clearly this list doesn't contain any word with more than one A. So it's not difficult to write down an infinite list that doesn't contain everything. However, the diagonal argument is more special than this, it shows that there is no possible list that contains every word constructed in this way.
5:11 "there is no problem; just keep making more rooms". At no point in the thought experiment does the manager create any more rooms. No rooms are added to the hotel at any point.
'How do you have a complete infinite list?' How do you have an infinite hotel? You don't. You assume that you have. Similarly, he is assuming that even if he has a complete infinite list, there is a guest with a name which is not on the list.
@@KonstantinosII You can. Let's say you have a list of all natural numbers. That is an infinite list but you can assume which numbers will be included in that list and which will not. 754,652 definitely will be on that list but 1/2 will definitely not.
You're right. You can't literally have a complete infinite list. But you can assume you do. Why is it so hard to understand? I don't know how else to word this. I'm sorry.
I get where you're coming from. I'm thinking like this : If you can predict every item on a list, be it infinte, it's technically 'complete'. Like a list of all natural numbers as I said earlier. If I give you a number, you'll be able to tell whether it will be included or not, regardless of you not having that list.
Actually thats just the simplest way. To get it PERFECTLY RIGHT You would have to study for almost 10 years. Infinity does have an end. And some are bigger then others
Well... I'll take the one room move. But now imagine there is this infinite bus coming and you have to leave your current room to the room double your number- and you're in room 10 Million. Moving another ten million rooms would surely ruin my night! 😉
Flamin hell Connor stick with comedic reactions this just cabbaged my head in. Also when do the cleaners/room services get to go in and do a service if guests have to move a room for each new arrival 🤣🤣🤣😎
Countable infinity is a value that can be calculated, whereas uncountable infinity doesn’t have a value because it’s an endless value. It can pass countable infinity. Since countable infinity is a value, there’s always a limit. This is why you can’t fit a party bus that has uncountable infinite passengers. I hope this helps.
This hotel would need to display a sign that reads, _"No vacancies, guests welcome"._ This paradox demonstrates that an _"actual"_ infinite amount of _'things"_ is logically absurd, and is good evidence that the universe is finite... the past is finite... etc.
i mean it is a difficult idea to grasp for some people without any real context. infinity isnt really a simple concept. i think this video is a good video explaining the thought experiment specifically. what it doesnt really do is a great job of explaining exactly what "infinity" is and to not have any real context of infinity past "yeah its just all the numbers possible, seems obvious". then of course throwing in the idea of countable and uncountable infinities isnt going to be something most people will be able to grasp. stand up maths and vsauce have better videos on the explanation of concept of infinites than this video does. since this video doesnt really explain the concept at all.
It is so hard to look at the UA-cam comments on a subject like this after studying this kind of thing for a mathematics degree. So I won't. Just rest assured that the formal logic which applies to many aspects of computing and so many other areas of our lives would not be possible if it wasn't for people with "too much time on their hands" questioning the very nature of the physical world, the universe, biology, numbers, logic and the relationships between them all. Do you think you could make your own silicon chip? Do you think they actually work, though? Why? The vast amount of human effort spent on problems like this by very, very clever people has been exactly _because_ they weren't happy with "infinity is just infinity" or "air is just space without anything in it" etc. Be thankful for the work they have done. And be humble enough to accept that if you think something well established as interesting and true by scientific thought is wrong, trivial or pointless then you are most likely missing the point.
an infinite number of connors with an infinite number of vlogs with an infinite number of mcjibbins with an infinite number of mr mcjibbins with a cup of coffee how many vlogs will infinately be infinite if you know meet me in the home economic room if you dont , ,, feel bad ...... you will feel better soon gentlemen ...........or ladies we don`t infinitel discriminate here ..............................(sorry long pause) its all good
Don’t worry Connor, it pains me to say it but I got lost on this too; stuck on my preconceived idea of infinite Love Veritasium btw I enjoy his content Not sure if you’ve ever noticed the atomic number on his “Ve” part of Veritasium 🤷♂️🙄🤣 (it’s on screen at the end of this video)
Am I stupid? If you have an infinite number of rooms it means you have an infinite number of keys. When someone turns up just give them the next available key and write it down ad infinitum. Have I missed the point?
This hotel analogy makes it harder to understand imo. It's basically a Cantor's diagonalization proof in a fancy dress. Veritasium actually explains it a lot better in another video: ua-cam.com/video/HeQX2HjkcNo/v-deo.html (at 4:30). It comes down to the difference between countable (for example, all integer numbers) and uncountable (for example, all real numbers) infinities.
To understand what this video is about is very simple. The problem is that we tend to confuse some similar concept as the same and that is where things get complicated. At 5:02 you said something that is the essence of the video: "...there's no number infinity you know and so you're just gonna keep going up". Infinity is something that appear when you do some process (like in this case putting people in some rooms). You lost it when later you talked about the infinite universes (contrarily to popular idea that is only a fiction idea. Interesting but certainly not science. But whatever, let us continue). You talked about it in a correct way (every possible way that things in the universe could be) but (you probably weren't aware) you implied that it was just like a certain defined quantity and there is no other infinity. You implied that there was a certain defined infinity and that it was the only one possible. The fact is that in modern maths (depending on the field and here just simplifying), infinity is a concept of process that hasn't a definite end for us "mortals" (mortals point of view being the point of view of the process). So, for example, if i have an algorithm that gives me all the even numbers, i can talk about the infinite even numbers. If it gives me the odd numbers, i will have a different infinite. Depending in the "never-ending" process, you will have different infinite. The important part here is to understand that, in this definition, infinite is not a quantity (that may change completely in another context). When you ask about what is "countably infinite" as mentioned in the video, It is just the idea that one can compare different infinite (infinite processes) with others. Let us say that we have a process that we call counting numbers, then we can use it to count all the even or odd numbers. That is the same as saying that their "infinity" are comparable. That also means that there might be some infinite process where we cannot use the "counting algorithm" to get all the individual element of it. We cannot count all of them. Given all this, later one can go further and begin to try to "quantize" the infinites and say that there are some that are more than others. A classic example is saying that the real numbers (all the point in an infinite line) are "more" than all the integers (integers numbers).
Vsauce has a much better video on this concept. he has a clearer explanation to the concept of infinites. this video is specifically more about the infinite hotel thought experiment specifically, but doesnt really dive as deep into the concept of infinities and larger infinities as the "how to count past infinity" vsauce video.
There is not one infinity, but rather more infinities than even any specific infinity. The term "infinity" is vague, mathematically. The first infinity is the only countable one and it is very limited. A complete matching between sets with differently infinite sizes is not possible.
There are various kinds of infinity. Not one. Infinity of natural numbers (aleph 0) is much "smaller" than infinity of rational numbers.. Our imagination functions well when we deal with finite sets, but is easly fooled by those infinites. But yes - all is OK - they are different. But - watch out ! 😉 natural and even numbers are - equally numerous, this is - the same kind of infinity 🙂🤷♂️
Actually, the infinity of naturals and rationals are the same infinity (Aleph null). The only subset of the real numbers that is uncountable (Aleph one) is the set of irrational numbers. in fact, the sets of naturals, intergers, rationals, algebraics are proovably countable.
They’re messing around with potential and actual infinities. This is why Aristotle said there’s no actual infinities, only potential ones. Otherwise, it leads to paradox (like this video).
It is not a "paradox" here. An infinite set is just a set that is not finite. What the video is saying is that some infinite sets are larger than other infinite sets
My interpretation of this problem is that there is an infinite amount of infinites. Unfortunately I think the narrators interpretation falls apart when he starts to question how many individual numbers constitute 1 particular infinite. I think he is completely overlooking the fact that it is impossible to know how large 1 particular infinity is compared to another infinity. It’s a ridiculous conundrum. I think it’s more of an exercise of discovery into oneself (where do you start and where do you end). What scientists seem to not understand is that we are the infinity, “I am in you and you are in me”, “love thy neighbour as you love thyself”. This maybe why scripture is so misunderstood and mostly by those that follow it and are devout to it. It cannot be understood by words, rather it is a knowing! In the same way you instinctively know that the conundrum makes no sense…
Hi, first time viewer. I was looking at the gabriel's horn paradox and saw this. And you're right, it's not possible to fill an infinite hotel. This paradox is just smoke and mirrors to distract from the contradictions that make it appear paradoxical. If your interested, I can explain my view. Just don't want to bore you otherwise. Take care.
It is possible if you have infinitely many guests. It isn't really a paradox at all, just a rather simple exercise in basic set theory. "Filling" the hotel is equivalent to finding a one-one and onto function (bijection) between two infinite sets. There is nothing paradoxical or mysterious about this.
@@charlieharris4881 Hey, thanks for the response. Yea, I studied a lot of set theory. My objection for this point is that the hotel cannot be both full and infinite to start. In the veritasium video, he remarks that the hotel is full and also infinitely countable. That's a false statement. If the hotel is full, the hotel is countable. The number of guests and rooms is known. Infinity is defined as greater than any real or natural number. If the hotel is infinitely countable, it would never be known if the hotel is full. So, that's my point. Oh, and happy new year.
@@michael2974 I think there was something lost in translation. The set of rooms being "filled" means that there exists a bijection between the set of rooms and the set of guests. That's all. It doesn't matter if these sets are countable or not, as long as they have the same cardinality then the set of rooms can always be "filled". An uncountable set A can be "filled" by another uncountable set B as long as A and B have the same cardinality.
@QuixoticCarrot You know what, I think you're right. Something was lost, but I also think it's because veritasium took a great teaching tool and turned it into crap. A strict distinction must be kept between 'countable', 'infinitely countable', and 'infinitely uncountable'. A hotel that is full is 'countable', from beginning to end. An infinite number of rooms and guests is 'infinitely countable'. The other glaring error is the last bus arriving and it's stated that the last bus is 'infinitely uncountable'. It is definitely not because a one-to-one correspondence can be shown between the passengers and the hotel. I come down pretty hard on this video, because these are either deliberate falsifications to make mathematics look like voodoo, or a complete lack of knowledge of set theory, binary sequences, and infinity. Anyway, this was cool. That was a clear, easily understood comment you made. Aloha.
man stuff like this bugs me. here are three basic points why this video and the concept is wrong. 1 you can't add infinites together 2 mathematicians only think that you can because they manipulated the meaning of the word infinite in order to fit it into mathematics. 3 and I honestly mean this with the utmost respect for mathematicians because they are so insanely good at what they do, they're also the world's greatest violators of confirmational bias. (sorry hardcore Christians you lose!) let me give you an example of what I'm talking about. look up the word infinite it will appear as a noun. now look up synonyms for infinite. I looked at a list of around 30 of them before writing this, and every single one of them are very good synonyms and they're all adjectives. you can't add two adjectives together can you? you can add two nouns though! two elephants two Bridges to universes two infinites! but you can't add two adjectives. which brings us back around to that confirmational bias they're so good at. what they do they've convinced themselves of this. the problem is if you're rewatch the video and notice they're not talking about truly infinite things they're talking about infinite patterns then infinite matching numbers then uncountable infinites. let me put it properly they're talking about non repeating patterns matching numbers (which could be as simple as odds on one side even on the other) and uncountable sums. (the closest thing to something that is truly infinite in this problem but in truth more like all the sand on all the beaches in the world which is a whole whole lot but still not infinite). in layman's terms they compared apple juice orange juice and some other mystery sweet juice and called them all fruit juice but because the last one is beyond their understanding it could have been antifreeze.
You obviously know very little about mathematics and are not qualified to talk about this problem at all. This video is talking about Hilbert's paradox of the Grand Hotel which is a a thought experiment which illustrates a counterintuitive property of infinite sets, like concepts of uncountable and countable infinity. Noone is talking aboout something like adding infinities (while that is possible and you dont understand that obviously as well, because it has nothing to do with linguistic meaning of a word infinity which came long long after the concept of infinity). Simply put there are diferent types and layers of infinity and one infinity CAN be bigger then other, the fact that they are bot infinite doesnt matter. I will explain (but it is simplified a lot). Lets start with the set of whole positive numbers lets say. (1,2,3,4....58.,59....124,125....) This set is clearly infinite but if you specify concrete nuber(like 10 953 158) you can count to it easily. That is not the case with another infinity. Lets say set of real numbers (meaning numbers with decimals). If i say count to 5 you cant not even to 0,1 or any other. Why? Because between ANY two nubers in this set is infinite nuber of numbers. Between 1 and 2 we have 1.1 , 1.2 , 1.3 ... 1.9, but also 1.11 , 1.12.... 1,99 and also 1.111 , 1.112 .... 1.999 etc. Between 0.13 and 0.12 same thing infinite number of numbers. So in one case you can clearly count to given number but in other you can never reach any given number. For this reason any countable infinite set will be always smaller then uncountable infinite set. So yes you can very much say that the set of real nubers in interval [0,1] is bigger than set of all intigers. This is just a small part of the Hilbert's paradox of the Grand Hotel and there is much more like a Triangular number method or Prime factorization method and other tings. Also the paradox is formulated as: The statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms ---> and it is provable true. Let me end by saying that i am really sad that someone who has no knowledge in given area has the audacity saying something like "this concept is wrong" just because he/she doesnt understeand it. Imagine thinking you are so great that all experts in a given field are wrong and you are the only one right.
@@jakubfanta1494 in 2017 two mathematicians showed that two infinity's are actually the same as one another. the next logical conclusion to this would be that you can't add two infinity's together or that at least it's useless because infinity plus infinity equals infinity and all three of those infinities have the same value. yes give it 5 years and not only will you catch up on your reading but the rest of the world will be starting to come around to this as well. I'm not the first guy to say this but you're the only one arrogant enough to think he's in on the ground floor when your information is a decade old. and by the way of course the word is newer than what it describes. we're using modern English you clown. (not on the debate team either were you) however the word to explain what we're trying to say when we say infinite traces back to words of old French and Latin origin that we would most closely call uncountable or innumerable. thank you for opening up that doorway for me by the way. yes words that strongluly insinuate a mathematician should have nothing to do with them because they kinda say NOT NUMBERS. I really debated over whether or not I should even respond to you your response was childish insulting and I didn't want to lower myself to that and then I thought why not this guys ego is so big it's just going to bounce off of him anyway. look being completely Fair and if I didn't say it my original comment I meant to I admire mathematicians the formulas they use in the infinite hotel paradox I could never figure that stuff out. but whether you're talking about countable infinite or uncountable infinites the principal idea of what infinite is remains the same and that is what you can't multiply by or add together. look in basic math I can't just add to my volume right I can't just double it for the heck of it to make my math problem easier to do. and yet in the infinite hotel paradox that is what is happening. we need to fit more people so we simply double the number of rooms and since the number of rooms goes up infinitely supposedly that's doable. except it would be the same as doubling our square footage of rooms. think of it this way what if the hotel manager had everyone in room one and two tear down their center dividing wall and then room three and four did the same thing and then five and six and then assuming they all kept their doors intact they go ahead and double their room number and go to that room so one goes to two two goes to four three goes to six etc etc all the rooms are occupied everyone has their own room the square footage is just suddenly doubled I followed the same rules as the original paradox and yet I broke a major and basic fundamental rule of mathematics. you can't just change your numbers because you don't know where they end and that's the philosophical part that I don't think common mathematicians get. so no it's not audacity it's not even me. if you want to blame somebody blame Malliaris and Shelah. they're the mathematicians that proved two infinities are equal. and argue all you want in another couple years that's the work that's going to prove you're wrong here today. until then by the way you don't have any actual mathematic proof classic egomaniac methodology here but you can't bring up the very thing we're arguing as proof that you're right you're just proving your pretentiousness and close-mindedness again and again . PS these are all signs of a lower IQ than you want to admit you have
Countably infinite just means that you can match your list one-to-one with the positive integers. What the diagonal argument shows is that there are certain infinite collections that can't be matched one-to-one with the positive integers. In effect, some infinities are bigger than others.
One concern you had was "how can an infinite list not contain everything?". That's easy, consider the list ABBBB..., BABBB..., BBABB..., etc. (each item has all Bs except for one A in different positions). Clearly this list doesn't contain any word with more than one A. So it's not difficult to write down an infinite list that doesn't contain everything. However, the diagonal argument is more special than this, it shows that there is no possible list that contains every word constructed in this way.
@Tech Overkill We do?
I had no idea. That's pretty cool news, thanks!
5:11 "there is no problem; just keep making more rooms". At no point in the thought experiment does the manager create any more rooms. No rooms are added to the hotel at any point.
'How do you have a complete infinite list?'
How do you have an infinite hotel? You don't. You assume that you have. Similarly, he is assuming that even if he has a complete infinite list, there is a guest with a name which is not on the list.
@@KonstantinosII You can. Let's say you have a list of all natural numbers. That is an infinite list but you can assume which numbers will be included in that list and which will not. 754,652 definitely will be on that list but 1/2 will definitely not.
You're right. You can't literally have a complete infinite list. But you can assume you do. Why is it so hard to understand? I don't know how else to word this. I'm sorry.
@@KonstantinosII Well, if you can't assume, you miss out on some fun experiments like this video since you also can't assume an infinite hotel exists.
I get where you're coming from. I'm thinking like this : If you can predict every item on a list, be it infinte, it's technically 'complete'. Like a list of all natural numbers as I said earlier. If I give you a number, you'll be able to tell whether it will be included or not, regardless of you not having that list.
00:07:30 because infinity is endless. there is no end to the names.
Actually thats just the simplest way. To get it PERFECTLY RIGHT You would have to study for almost 10 years. Infinity does have an end. And some are bigger then others
Imagine making an infinite number of people move down a room for one person 💀
Well... I'll take the one room move. But now imagine there is this infinite bus coming and you have to leave your current room to the room double your number- and you're in room 10 Million. Moving another ten million rooms would surely ruin my night! 😉
An infinite number of infinite busses is the same amount as a single bus with infinite people
Flamin hell Connor stick with comedic reactions this just cabbaged my head in. Also when do the cleaners/room services get to go in and do a service if guests have to move a room for each new arrival 🤣🤣🤣😎
Countable infinity is a value that can be calculated, whereas uncountable infinity doesn’t have a value because it’s an endless value. It can pass countable infinity. Since countable infinity is a value, there’s always a limit. This is why you can’t fit a party bus that has uncountable infinite passengers. I hope this helps.
This hotel would need to display a sign that reads, _"No vacancies, guests welcome"._ This paradox demonstrates that an _"actual"_ infinite amount of _'things"_ is logically absurd, and is good evidence that the universe is finite... the past is finite... etc.
Imagine the crush trying to get a drink at the bar .....
or the toilets
As someone who understands this video, his reaction just annoys me
I know
i mean it is a difficult idea to grasp for some people without any real context. infinity isnt really a simple concept. i think this video is a good video explaining the thought experiment specifically. what it doesnt really do is a great job of explaining exactly what "infinity" is and to not have any real context of infinity past "yeah its just all the numbers possible, seems obvious". then of course throwing in the idea of countable and uncountable infinities isnt going to be something most people will be able to grasp. stand up maths and vsauce have better videos on the explanation of concept of infinites than this video does. since this video doesnt really explain the concept at all.
It is so hard to look at the UA-cam comments on a subject like this after studying this kind of thing for a mathematics degree.
So I won't.
Just rest assured that the formal logic which applies to many aspects of computing and so many other areas of our lives would not be possible if it wasn't for people with "too much time on their hands" questioning the very nature of the physical world, the universe, biology, numbers, logic and the relationships between them all.
Do you think you could make your own silicon chip?
Do you think they actually work, though?
Why?
The vast amount of human effort spent on problems like this by very, very clever people has been exactly _because_ they weren't happy with "infinity is just infinity" or "air is just space without anything in it" etc.
Be thankful for the work they have done.
And be humble enough to accept that if you think something well established as interesting and true by scientific thought is wrong, trivial or pointless then you are most likely missing the point.
Him :How is there an affinity list
Me : how is there an infinite hotel and infinite busses 🐐
an infinite number of connors with an infinite number of vlogs with an infinite number of mcjibbins with an infinite number of mr mcjibbins with a cup of coffee how many vlogs will infinately be infinite if you know meet me in the home economic room if you dont , ,, feel bad ...... you will feel better soon gentlemen ...........or ladies we don`t infinitel discriminate here ..............................(sorry long pause) its all good
Don’t worry Connor, it pains me to say it but I got lost on this too; stuck on my preconceived idea of infinite
Love Veritasium btw I enjoy his content
Not sure if you’ve ever noticed the atomic number on his “Ve” part of Veritasium 🤷♂️🙄🤣 (it’s on screen at the end of this video)
00:09:15 that doesn't obey the laws of physics though.
Am I stupid? If you have an infinite number of rooms it means you have an infinite number of keys. When someone turns up just give them the next available key and write it down ad infinitum. Have I missed the point?
This hotel analogy makes it harder to understand imo. It's basically a Cantor's diagonalization proof in a fancy dress.
Veritasium actually explains it a lot better in another video: ua-cam.com/video/HeQX2HjkcNo/v-deo.html (at 4:30).
It comes down to the difference between countable (for example, all integer numbers) and uncountable (for example, all real numbers) infinities.
Just forget the definition of infinite in lingustics.
To understand what this video is about is very simple. The problem is that we tend to confuse some similar concept as the same and that is where things get complicated.
At 5:02 you said something that is the essence of the video: "...there's no number infinity you know and so you're just gonna keep going up".
Infinity is something that appear when you do some process (like in this case putting people in some rooms).
You lost it when later you talked about the infinite universes (contrarily to popular idea that is only a fiction idea. Interesting but certainly not science. But whatever, let us continue). You talked about it in a correct way (every possible way that things in the universe could be) but (you probably weren't aware) you implied that it was just like a certain defined quantity and there is no other infinity. You implied that there was a certain defined infinity and that it was the only one possible.
The fact is that in modern maths (depending on the field and here just simplifying), infinity is a concept of process that hasn't a definite end for us "mortals" (mortals point of view being the point of view of the process). So, for example, if i have an algorithm that gives me all the even numbers, i can talk about the infinite even numbers. If it gives me the odd numbers, i will have a different infinite. Depending in the "never-ending" process, you will have different infinite. The important part here is to understand that, in this definition, infinite is not a quantity (that may change completely in another context).
When you ask about what is "countably infinite" as mentioned in the video, It is just the idea that one can compare different infinite (infinite processes) with others. Let us say that we have a process that we call counting numbers, then we can use it to count all the even or odd numbers. That is the same as saying that their "infinity" are comparable.
That also means that there might be some infinite process where we cannot use the "counting algorithm" to get all the individual element of it. We cannot count all of them.
Given all this, later one can go further and begin to try to "quantize" the infinites and say that there are some that are more than others. A classic example is saying that the real numbers (all the point in an infinite line) are "more" than all the integers (integers numbers).
Vsauce has a much better video on this concept. he has a clearer explanation to the concept of infinites. this video is specifically more about the infinite hotel thought experiment specifically, but doesnt really dive as deep into the concept of infinities and larger infinities as the "how to count past infinity" vsauce video.
There is not one infinity, but rather more infinities than even any specific infinity. The term "infinity" is vague, mathematically. The first infinity is the only countable one and it is very limited. A complete matching between sets with differently infinite sizes is not possible.
There are various kinds of infinity. Not one. Infinity of natural numbers (aleph 0) is much "smaller" than infinity of rational numbers..
Our imagination functions well when we deal with finite sets, but is easly fooled by those infinites. But yes - all is OK - they are different. But - watch out ! 😉 natural and even numbers are - equally numerous, this is - the same kind of infinity 🙂🤷♂️
Actually, the infinity of naturals and rationals are the same infinity (Aleph null). The only subset of the real numbers that is uncountable (Aleph one) is the set of irrational numbers. in fact, the sets of naturals, intergers, rationals, algebraics are proovably countable.
How can you have an infinite number of people plus one ?
🤯 x Infinity
I'm infinitely confused
🤣🤣🤣
They’re messing around with potential and actual infinities. This is why Aristotle said there’s no actual infinities, only potential ones. Otherwise, it leads to paradox (like this video).
There can be many infinities in set theory, each infinity is enclosed in its own set
It is not a "paradox" here. An infinite set is just a set that is not finite. What the video is saying is that some infinite sets are larger than other infinite sets
find a better video that describes set theory and infinity
My interpretation of this problem is that there is an infinite amount of infinites. Unfortunately I think the narrators interpretation falls apart when he starts to question how many individual numbers constitute 1 particular infinite. I think he is completely overlooking the fact that it is impossible to know how large 1 particular infinity is compared to another infinity. It’s a ridiculous conundrum. I think it’s more of an exercise of discovery into oneself (where do you start and where do you end). What scientists seem to not understand is that we are the infinity, “I am in you and you are in me”, “love thy neighbour as you love thyself”. This maybe why scripture is so misunderstood and mostly by those that follow it and are devout to it. It cannot be understood by words, rather it is a knowing! In the same way you instinctively know that the conundrum makes no sense…
1st
Life’s to short for that crap
Since you're both american, you might as well say Veritasum.
I think the bottom line is, this is a mathematical problem, not a logical one.
Hi, first time viewer. I was looking at the gabriel's horn paradox and saw this. And you're right, it's not possible to fill an infinite hotel. This paradox is just smoke and mirrors to distract from the contradictions that make it appear paradoxical. If your interested, I can explain my view. Just don't want to bore you otherwise. Take care.
It is possible if you have infinitely many guests. It isn't really a paradox at all, just a rather simple exercise in basic set theory. "Filling" the hotel is equivalent to finding a one-one and onto function (bijection) between two infinite sets. There is nothing paradoxical or mysterious about this.
@@charlieharris4881 Hey, thanks for the response. Yea, I studied a lot of set theory. My objection for this point is that the hotel cannot be both full and infinite to start. In the veritasium video, he remarks that the hotel is full and also infinitely countable. That's a false statement. If the hotel is full, the hotel is countable. The number of guests and rooms is known. Infinity is defined as greater than any real or natural number. If the hotel is infinitely countable, it would never be known if the hotel is full. So, that's my point. Oh, and happy new year.
@@michael2974 I think there was something lost in translation. The set of rooms being "filled" means that there exists a bijection between the set of rooms and the set of guests. That's all. It doesn't matter if these sets are countable or not, as long as they have the same cardinality then the set of rooms can always be "filled". An uncountable set A can be "filled" by another uncountable set B as long as A and B have the same cardinality.
@QuixoticCarrot You know what, I think you're right. Something was lost, but I also think it's because veritasium took a great teaching tool and turned it into crap. A strict distinction must be kept between 'countable', 'infinitely countable', and 'infinitely uncountable'. A hotel that is full is 'countable', from beginning to end. An infinite number of rooms and guests is 'infinitely countable'. The other glaring error is the last bus arriving and it's stated that the last bus is 'infinitely uncountable'. It is definitely not because a one-to-one correspondence can be shown between the passengers and the hotel. I come down pretty hard on this video, because these are either deliberate falsifications to make mathematics look like voodoo, or a complete lack of knowledge of set theory, binary sequences, and infinity. Anyway, this was cool. That was a clear, easily understood comment you made. Aloha.
man stuff like this bugs me. here are three basic points why this video and the concept is wrong.
1 you can't add infinites together
2 mathematicians only think that you can because they manipulated the meaning of the word infinite in order to fit it into mathematics.
3 and I honestly mean this with the utmost respect for mathematicians because they are so insanely good at what they do, they're also the world's greatest violators of confirmational bias. (sorry hardcore Christians you lose!) let me give you an example of what I'm talking about. look up the word infinite it will appear as a noun. now look up synonyms for infinite. I looked at a list of around 30 of them before writing this, and every single one of them are very good synonyms and they're all adjectives. you can't add two adjectives together can you? you can add two nouns though! two elephants two Bridges to universes two infinites! but you can't add two adjectives. which brings us back around to that confirmational bias they're so good at. what they do they've convinced themselves of this. the problem is if you're rewatch the video and notice they're not talking about truly infinite things they're talking about infinite patterns then infinite matching numbers then uncountable infinites. let me put it properly they're talking about non repeating patterns matching numbers (which could be as simple as odds on one side even on the other) and uncountable sums. (the closest thing to something that is truly infinite in this problem but in truth more like all the sand on all the beaches in the world which is a whole whole lot but still not infinite). in layman's terms they compared apple juice orange juice and some other mystery sweet juice and called them all fruit juice but because the last one is beyond their understanding it could have been antifreeze.
You obviously know very little about mathematics and are not qualified to talk about this problem at all. This video is talking about Hilbert's paradox of the Grand Hotel which is a a thought experiment which illustrates a counterintuitive property of infinite sets, like concepts of uncountable and countable infinity. Noone is talking aboout something like adding infinities (while that is possible and you dont understand that obviously as well, because it has nothing to do with linguistic meaning of a word infinity which came long long after the concept of infinity). Simply put there are diferent types and layers of infinity and one infinity CAN be bigger then other, the fact that they are bot infinite doesnt matter. I will explain (but it is simplified a lot). Lets start with the set of whole positive numbers lets say. (1,2,3,4....58.,59....124,125....) This set is clearly infinite but if you specify concrete nuber(like 10 953 158) you can count to it easily. That is not the case with another infinity. Lets say set of real numbers (meaning numbers with decimals). If i say count to 5 you cant not even to 0,1 or any other. Why? Because between ANY two nubers in this set is infinite nuber of numbers. Between 1 and 2 we have 1.1 , 1.2 , 1.3 ... 1.9, but also 1.11 , 1.12.... 1,99 and also 1.111 , 1.112 .... 1.999 etc. Between 0.13 and 0.12 same thing infinite number of numbers. So in one case you can clearly count to given number but in other you can never reach any given number. For this reason any countable infinite set will be always smaller then uncountable infinite set. So yes you can very much say that the set of real nubers in interval [0,1] is bigger than set of all intigers. This is just a small part of the Hilbert's paradox of the Grand Hotel and there is much more like a Triangular number method or Prime factorization method and other tings.
Also the paradox is formulated as: The statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms ---> and it is provable true.
Let me end by saying that i am really sad that someone who has no knowledge in given area has the audacity saying something like "this concept is wrong" just because he/she doesnt understeand it. Imagine thinking you are so great that all experts in a given field are wrong and you are the only one right.
@@jakubfanta1494 in 2017 two mathematicians showed that two infinity's are actually the same as one another. the next logical conclusion to this would be that you can't add two infinity's together or that at least it's useless because infinity plus infinity equals infinity and all three of those infinities have the same value. yes give it 5 years and not only will you catch up on your reading but the rest of the world will be starting to come around to this as well. I'm not the first guy to say this but you're the only one arrogant enough to think he's in on the ground floor when your information is a decade old. and by the way of course the word is newer than what it describes. we're using modern English you clown. (not on the debate team either were you) however the word to explain what we're trying to say when we say infinite traces back to words of old French and Latin origin that we would most closely call uncountable or innumerable. thank you for opening up that doorway for me by the way. yes words that strongluly insinuate a mathematician should have nothing to do with them because they kinda say NOT NUMBERS. I really debated over whether or not I should even respond to you your response was childish insulting and I didn't want to lower myself to that and then I thought why not this guys ego is so big it's just going to bounce off of him anyway. look being completely Fair and if I didn't say it my original comment I meant to I admire mathematicians the formulas they use in the infinite hotel paradox I could never figure that stuff out. but whether you're talking about countable infinite or uncountable infinites the principal idea of what infinite is remains the same and that is what you can't multiply by or add together. look in basic math I can't just add to my volume right I can't just double it for the heck of it to make my math problem easier to do. and yet in the infinite hotel paradox that is what is happening. we need to fit more people so we simply double the number of rooms and since the number of rooms goes up infinitely supposedly that's doable. except it would be the same as doubling our square footage of rooms. think of it this way what if the hotel manager had everyone in room one and two tear down their center dividing wall and then room three and four did the same thing and then five and six and then assuming they all kept their doors intact they go ahead and double their room number and go to that room so one goes to two two goes to four three goes to six etc etc all the rooms are occupied everyone has their own room the square footage is just suddenly doubled I followed the same rules as the original paradox and yet I broke a major and basic fundamental rule of mathematics. you can't just change your numbers because you don't know where they end and that's the philosophical part that I don't think common mathematicians get. so no it's not audacity it's not even me. if you want to blame somebody blame Malliaris and Shelah. they're the mathematicians that proved two infinities are equal. and argue all you want in another couple years that's the work that's going to prove you're wrong here today. until then by the way you don't have any actual mathematic proof classic egomaniac methodology here but you can't bring up the very thing we're arguing as proof that you're right you're just proving your pretentiousness and close-mindedness again and again . PS these are all signs of a lower IQ than you want to admit you have