Hilbert's Hotel Problem ... The Infinite Hotel Paradox (An EXPLANATION!)

[@instagram1572]

Excellent job of illustrating and explaining the subject in a slow and concise way that makes perfect sense, instead of trying to impress with fancy aesthetics and how fast you can spew out technical jargon. You are a natural-born teacher. Thank you

[@nijiheart6743]

You explained it better than my prof back when I was still studying. You can apply as a prof in my uni 😂🤣

[@catsarechingus4357]

This was incredibly well explained and I have thoroughly enjoyed watching. Thank you very much for this explanation, you bright young man.

[@HarrySurplus]

Thank you so much! Glad you enjoyed

[@elmothegod3314]

Thanks for explaining it fully like this, I’ve been confused for a while and i struggle to follow when I’m being taught things but you made it easy enough for me to follow and understand.

[@LEWISWATSONVLOGS1]

Man....really good video 👍 this is the last video I came across regarding this hotel and this video actually made sense haha . My brains have been working the past few days with this hotel 😄 the way I see it...the first person who arrives should be accommodated towards the end of the hotel and work them backwards haha that way no one will be moved all the time...absolutely crazy..amazing video dude 😄 ❤

[@gregheffley5745]

The infinite hotel would be the perfect place to run a pyramid scheme.

[@bae3m4n]

you made it pretty easy to grasp. i have a few questions though: 1. In the case of only one person wanting a room, why do we ask the person in room 1 to vacate: we could ask anyone. So for example if 10 people enter the hotel, we could just leave rooms 1 to 10 undisturbed and ask the people from room 11 onwards to move 10 rooms ahead. Like is that theoretically wrong for any finite number of guests? 2. If the Hilbert Hotel is infinite, how can it be considered full? I mean that there are an infinite number of hotel rooms, right? And those are all OCCUPIED. So since infinity doesn't end, the number of OCCUPIED hotel rooms doesn't either. So in a way the entire hotel is occupied and has no vacant room. No matter how hard I try to understand it, I can't put my head around it.

[@HarrySurplus]

We ask room 1 to move just for ease. This way the one new guest can go into the first room easily. Yes we could ask everyone from room 11 onwards to move up that would work too, but then the new guest would have to walk to room 11 instead of walking straight to room 1! & yes all rooms are full. Infinity never ends so we have an infinite number of occupied rooms, ie no vacant room. However, to accommodate one new guest, move room 1 to room 2, room 2 to room 3 ...... , room 1000 to room 1001, ..... room 90,000 to room 90,0001 etc (this will kind of never end as there are infinite rooms, hence why hilberts hotel is theoretical!) then because we started by moving room 1 to 2, nobody is in room 1. So the new guest can go into rooms one!

[@bae3m4n]

@@HarrySurplus thanks for the explanation! ☺️

[@IngvarLind]

To be absolutely correct, even infinite (transfinite) numbers can be precise. Which means, you can deal with a set of infinite objects as non-changing. If you do, you can't just add non-zero objects to an infinite collection and it will stay the same--then you would get a contradiction. For this to work, you'll need some other type of infinite arithmetic, like the hyperreal one, that lets you work with a mix of infinite and finite objects (plus infinitesimal ones) in much the same way as you deal with finite numbers.

[@keelanadams6948]

Really well explained and interesting, please do more like this!

[@HarrySurplus]

Thank you!

[@jannatulnaima4186]

is there any easy 3d video about this.I have seen some videos bt didnt get this yet

[@oneilclarke7563]

Thanks bro i was so confuse till i watch the video

[@Shivam-sb8df]

Let's say we have to accommodate one new guest. To do this, each existing customer moves to a higher room number by 1. Then there has to be an empty room in the end where the last existing customer can move into. So the infinite hotel does have one empty room already?!

[@pradyarsomaharsi6501]

If you put every guest in a room with Odd number you will “always” have +1 vacant (Even number) room. The empty Even number rooms will always be empty, but at the same time always available. Primarily because you assigned every guest into the Odd number rooms format. I think this prevent “fully-booked”, without moving any guests around (because it’s annoying 😆).

[@grassgrees6728]

please do more like this thank you

[@zoeevans946]

I find problems with infinity very confusing haha I remember going on a school trip which was for people who were interested in a level maths and it was so confusing haha

[@miladsayad2935]

at: 9:48 : 2^5 = 32 But wasn't room 32 already taken during 2N, when the person of room number "16" moved there? And 32 is an even number, wasn't all even numbers taken? You mentioned at 7:08

[@HarrySurplus]

No! These are two different cases. At 7:08 we are discussing what happens if we have an infinite number of new guest & at 9:48 we are discussing what to do if we have an infinite number of buses, with an infinite number of people on each bus! Two different cases

[@miladsayad2935]

​@@HarrySurplus I see, Suppose all even numbers are taken. And there is an infinite number of busses with an infinite number of guests. Then placing these guests in merely prime number rooms would also solve it right? Since Euclid proved that there an infinite numbers of prime.

[@deviantshade]

Not exactly a hotel dilemma, but more of how a memory controller organizes data from a memory bus in memory such as RAM. Not mentioned in the video is how the data "vacates" the memory cell with the controller knowing about it.

[@Mevely_]

Underrated

[@h1gh3xp_baby92]

so for example, the people in bus 1 will go to the first prime number 3 to the power of n, n is their seat number I got that, so for example, he is in seat 3, 3 to the power 3 is 27, what happens to the person already living in room number 27??

[@HarrySurplus]

Hello! People in bus 1 will go to room 3^n (where n is there seat number), but 3 is the second prime number. The first prime number is 2. So everyone that is always in a room will move to 2^n, where n is there original room number. Because the base are prime numbers, the new rooms will never overlap! So in your example, the personal already living in room 27 will move to room 2^27! 😌

[@h1gh3xp_baby92]

@@HarrySurplus oh tysm i got it tyyy

[@mee834]

The premise was that every room is full, which means the number of rooms (n) is the same as the number of people. Both the numer of rooms and the number of people are infinite. Same cardinality. If the one in room 1 has to move to room 2 and so on, the last one will move to room n+1. So that would mean there is n+1 rooms, which is ok since we're talking about infinity. But that also means that there never was the same number of people as rooms. The hotel wasnt actually full. What am I missing?

[@thetaomegatheta]

'But that also means that there never was the same number of people as rooms' Incorrect. Infinite sets have possess the property of having infinite proper subsets. This means that the set of natural numbers can be bijectively mapped to a proper subset of its own. f(n)=n+1 is one such bijective mapping.

[@mee834]

@@thetaomegatheta If every room is full, that means the number of rooms is the same as the number of people. Correct? This is true no matter if the number continues forever or if it is finite. If that is the premie, and if we could move people from room 1 to room 2 and so on, then the hotel wasn't full. I know what you mean by f(n)=n+1, but this violates the premise. Another way to look at the problem is to say that the number of rooms divided by the number of people is 1 (every room is full). When we are dealing with infinity, we are essentially entering the field of magic, since we can prove or disprove anything if we are allowed to divide by 0. Mathematicians sometimes suggest that 0/0=1, or at least I have seen some who do. But with 0 and infinity, we don't get real (in the sense of meaningful) numbers. If we start by 1000/1000, then go to 999/999 and so on down to 1/1, we will always get the ratio 1. It seems to make sense that 0/0 should also be 1, but it is undefined - meaning it can be anything or nothing. Only if the numerator and denominator have the same rate of change can we say the answer is 1. I can easily prove that 5=1 if I'm allowed to divide by 0. 5*0=1*0 -> 5=1. That's what it means, it's not math, it's magic. How many slices can you divide that invisible apple in front of you, using the invisible knife in your hand? It's a meaningless question.

[@thetaomegatheta]

@@mee834 'If every room is full, that means the number of rooms is the same as the number of people. Correct?' Not 'the same number', as there are infinitely many of both. There is one guest for every room. The sets map to each other one-to-one. 'If that is the premie, and if we could move people from room 1 to room 2 and so on, then the hotel wasn't full' I mean, this is demonstrably not true, but what is your argument that the guests can't just leave their rooms at the same time and then teleport to their new rooms? 'I know what you mean by f(n)=n+1, but this violates the premise' How does it violate the premise? 'Another way to look at the problem is to say that the number of rooms divided by the number of people is 1 (every room is full)' Incorrect. The operation of division is undefined in this context. You CAN define it, but that won't help you solve anything. 'When we are dealing with infinity, we are essentially entering the field of magic, since we can prove or disprove anything if we are allowed to divide by 0' We aren't 'allowed' to divide by zero, as there is no standard definition for it. You can produce such a definition, actually, but that would actually break some nice properties of division, no matter how you define it, and, in the end, you aren't actually getting any closer to a solution. Why did you even assume that we can just divide by infinity or by zero in this context? 'If we start by 1000/1000, then go to 999/999 and so on down to 1/1, we will always get the ratio 1. It seems to make sense that 0/0 should also be 1' And if we start with 1001/1000, go to 1000/999, and so on, down to 2/1, we will be approaching a different ratio. The point of division by zero being undefined is that no matter how you define it, it breaks some properties of division, and doesn't actually help with anything. What's the relevance of division by zero here, anyway? 'but it is undefined - meaning it can be anything or nothing' It is undefined. It doesn't magically become 'anything or nothing' based on your whim. Again, you can define it yourself, but I assure you, that's not going to be helpful. 'I can easily prove that 5=1 if I'm allowed to divide by 0' Well, you aren't allowed to do that. Case closed, I guess. So, what was this whole wall of text about division by zero when we don't use division by zero to solve any of the problems shown in the video?

[@mee834]

@@thetaomegatheta I would argue that one guest for every room is the same as every room is full. That’s what one-to-one means. What did it have to do with the wall of text, divide by zero and all that? As you know, infinity can be written as 1/0. 5 times the number of guest (or rooms) could be written as 5/0. Meaning: it’s just arbitrary and meaningless juggling with numbers. The point of infinity and multiplying by 0 is that you can get any value you want. And if we allow those types of errors, we can say that the hotel is never full. But the same number trick can be used on both the number of people as the number of rooms. It’s not that it’s wrong, it’s that it’s about as meaningsful as dividing the invisible apple with the invisible knife.

[@thetaomegatheta]

@@mee834 'I would argue that one guest for every room is the same as every room is full' Well, that is incorrect as soon as we relocate the guests, as there is also, for example, one room for every even-numbered room. 'That’s what one-to-one means' I mean, again, there is one room for every even-numbered room. There is also one room for every tenth room. One room for every room which has a number that equals a power of a prime number. And so on. The guests in those rooms, even though they can, after relocating, do not automatically fill the entirety of the hotel. 'As you know, infinity can be written as 1/0' I know that it can't be written like that. '5 times the number of guest (or rooms) could be written as 5/0' No, it couldn't. 'The point of infinity and multiplying by 0 is that you can get any value you want' No, you can't. As soon as you start treating 1/0 as evaluating to infinity, you lose a bunch of important properties of division. You can't, for example, get an inverse of that division anymore, which ruins your entire argument even further. 'And if we allow those types of errors' We don't, so I'm just dismissing the rest of your post as based on a false premise.

[@christinapeterson8210]

Super helpful, thank you

[@HarrySurplus]

😁😁

[@shreya6704]

Enough study tips, now give some skincare tips, bruh how clear af your skin is 🤨🤨😂😂 Jk😛 much support & Blessing to you for helping us & guiding us the right way 😄✨🤗

[@HarrySurplus]

I do use a charcoal soap lol but that about it haha thank you

[@JudeKennedyATCL]

Guess who's early for once!! Also I love Hilbert's hotel! (Numberphile and ViHart memories)

[@HarrySurplus]

yaaay

[@user-gi3yr1gl7j]

I love this content!!!

[@HarrySurplus]

Thank you !!

[@kadenvanciel9335]

Say, I wonder if anyone interested actually worked out the ships problem for Hilbert's considered folklore.

[@krystelcarmelo3791]

Hilbert’s Hotel will always have rooms for countably infinite guests. The set of real numbers, however, is uncountable. What issues can come up for a Real Infinite Hotel? Reply real quick

[@JudeKennedyATCL]

How to add an infinite number of guests to the already full infinite hotel... Build a second infinite hotel on top of the original. Job done😂

[@shreya6704]

Hahaha omg 😂😂 well it's a right idea indeed 😂😂😂😂

[@JudeKennedyATCL]

@@shreya6704 I did think about using omega infinities too. 1+W =\= W+1

[@shreya6704]

@@JudeKennedyATCL haha bro ur thinking is on another level 😂😂😂😂😂😂

[@shreya6704]

@@JudeKennedyATCL how are u feeling about a level math so far? 😀 I'm finding it moderate 🤨

[@JudeKennedyATCL]

@@shreya6704 I'm doing A2 Maths and A2 further maths this year, Maths is fine, further maths is amazing!! It's so interesting and I actually love going to the classes haha

[@Username-ww2cd]

Spoiler: move each person to the next room along

[@dudicorn6503]

That's the big issue with potential infinity in this case which is that it is smaller than that which it is being compared to by necessity. Let's take an infinite line within an infinite space. This line is presumably infinite yet we are at an end of this line, however we are not at an end of this space. This means that the line has to be smaller because although it continues infinitely, the fact that we can perceive its end gives it a size w.r.t. the space which it occupies. To make more sense of this, we may never be able to get to the other end because there is no other end, but there is still one end nonetheless

[@jamestagge3429]

The paradox of the Hilbert Hotel is nothing but cheap verbal acrobatics to try to impart some relevance to the concept of infinity to be used within various aspects of materiality which it does not have. If one wishes to question what I have just claimed, he should tell us all when counting, what the very last number would be just before the next one was infinity. The Hilbert hotel is supposed to be some brilliant intellectual construction when in fact it’s a sophomoric word game which tells us nothing about anything. Consider…first, the hotel could not have infinite rooms or infinite guests because each of these is a quantifiable entity and by trying to accommodate them in a context of infinity, which is not, the entire proposition fails on its face, though I know that most who embrace this paradox as genius understand that. The two concepts are mutually exclusive. The concept of infinity is by definition non-delineable, non-quantifiable. There is no infinite line or volume which can ever be filled by quantifiable entities, EVER, as I suggested above. Secondly, if for the sake of the discussion we assume the premise that there are infinite rooms then that is a consideration of the concept which is, to be sure, intellectually manageable. However, once we claim that there are also infinite guests we cannot but understand that each room, though infinite in number is by the definition of the proposition itself, occupied or full. Here the two “floating concepts” of infinite rooms and infinite guests are in a kind of conceptual equilibrium, necessarily so or the proposition contradicts itself. If then, we were to say that we will move this infinite number of guests by one to make room for one more, we are forced to consider that the shift of customers cannot be made for each room is again, by definition full and also by definition, always full. The very outline of the structure of the proposition shows it is necessarily so and is that by which it also collapses. To make this clearer, imagine traveling down the line of infinite rooms and looking in. You would find each room occupied as you traveled, forever. That “is” how the proposition is defined/stated. One can’t have it both ways. To consider that room could be made for new guests, especially an infinite number by clever manipulation of the numbers used as identifiers and the manner in which they are listed is childishly transparent as means of proposing a solution but only in the immediate sense, i.e., for the sake of a visible or quantifiable number of guests relative to rooms (outside of the context of infinity). As for the “uncountable infinity” of those with the names composed of various combinations of the letters “A” and “B”, he claims that there would be an infinite number of the combinations of these letters forming the names of these infinite individuals but that because of a clever manipulation of the material list made of them, there would be one that was not on it and thus the infinity of these guests could not be given rooms. But if there are an infinite number of combinations of these letters, how could there be one that is not accommodated? Consider…he confines the compilation of this name by beginning with the first letter of the beginning of an infinite list, open ended in this first entry, leaving a choice to be made as to who is first. Below in the first list you see the list used in the video but with ABBA…at the top, the one making the arrangements. The constrains to some measure what can follow. But what if he made the arrangements and was not placed first on the list? Then this idea of one who will not appear on the list fails. ABBA….then becomes the one not to be accommodated and yet there he stands. All others are accommodated by the definition employed in this video. ABBAAAAAAAAAAAAAAA ABABABABABABABABAB BBAABBBBBBBBBBBBB BBABAAAABAAAABAAA = BABA…. ======================== BABABABABABABABAB BAAABBBBBBBBBBBBB BBABAAAABAAAABAAA BBABAAAABBAA = ABBA…. This is simply a manipulation of the definition of a concept which cannot be defined as such. So, if we consider the moving of each guest down a room to make room for one more, for example, this works for those rooms within our immediate sight or understanding of limited rooms, but not of the rooms and guests down the line. If the infinite rooms have infinite guests then by that very statement which is the very definition of the proposition, all are full and no room can be made, for by extension of the proposition, we are proposing that we never consider the actual infinity of the rooms or the guests in doing so. This would be the only means of making that room for the new guest, let alone and infinite number of new guests. There are two aspects of this proposition in the video. One is the material understanding of the number of guests and of rooms. The other is the manipulation of numbers and names listed to give the appearance by the mathematics that the accommodations of new guests can be made. They are “not” one and the same and are in fact disconnected. It is simply put...that infinity cannot exist within or with regard to any aspect of materiality makes the coupling of concepts of the finite, i.e., rooms and guests and the infinite impossible and by that, the proposition as well. Infinity might be a useful place holder in advanced math or the like. But as a means of defining anything in, for example, physics or cosmology, it is pure sophistry. I might be accused of employing conventional considerations in materiality in my judgment of the proposition, but it was by them exactly that it was defined in the first place.

[@thetaomegatheta]

'first, the hotel could not have infinite rooms or infinite guests' It very much could. The hotel is just an analogy for the set of natural numbers, and, unless you can somehow prove that there are only finitely many natural numbers, you are yet to substantiate your claim. 'because each of these is a quantifiable entity and by trying to accommodate them in a context of infinity, which is not, the entire proposition fails on its face, though I know that most who embrace this paradox as genius understand that' There are infinitely many natural numbers, which, by your logic, are a 'quantifiable entity' (whatever that means), and, again, by your logic, there are only finitely many of them. Lol. 'Secondly, if for the sake of the discussion we assume the premise that there are infinite rooms then that is a consideration of the concept which is, to be sure, intellectually manageable. However, once we claim that there are also infinite guests we cannot but understand that each room, though infinite in number is by the definition of the proposition itself, occupied or full. Here the two “floating concepts” of infinite rooms and infinite guests are in a kind of conceptual equilibrium, necessarily so or the proposition contradicts itself' That's just a lot of words to say nothing of substance. Instead of thinking rigorously and using logic, you just make that word salad. 'If then, we were to say that we will move this infinite number of guests by one to make room for one more, we are forced to consider that the shift of customers cannot be made for each room is again, by definition full and also by definition, always full' You make this claim, but you provide no inference, no basis, no substantiation for it. And the claim is pretty wild. You are saying that the function f(n) for all natural n has a non-empty complement to its image within the set of natural numbers (N\Im(f)). That is, of course, demonstrably not true, and yet, you make this claim with such confidence. 'To consider that room could be made for new guests, especially an infinite number by clever manipulation of the numbers used as identifiers and the manner in which they are listed is childishly transparent as means of proposing a solution but only in the immediate sense, i.e., for the sake of a visible or quantifiable number of guests relative to rooms (outside of the context of infinity)' Again, more word salad. What is being done is that we use the property of infinite sets that they have infinite proper subsets, which allows us to map the set of rooms one-to-one to some of their proper subsets. 'But if there are an infinite number of combinations of these letters, how could there be one that is not accommodated?' Google 'Cantor's diagonalization argument' and 'Cantor's theorem'. The gist of it is that no matter how you assign the rooms to the named guests, we will always be able to construct a name (that one of the guests would have to have) that would differ from every name of guests who got their rooms by at least one letter. 'Consider…he confines the compilation of this name by beginning with the first letter of the beginning of an infinite list, open ended in this first entry, leaving a choice to be made as to who is first' Because it doesn't matter who the 'first' housed guest is. We literally have to find a way to construct the name regardless of who is the first one. And we do succeed in it. 'But what if he made the arrangements and was not placed first on the list?' If that person doesn't appear on the list, then they are not housed, which means that you have failed to house every guest. And we are, indeed, trying to prove that housing every guest is impossible. 'Then this idea of one who will not appear on the list fails' Lol no. Quite literally that very person whom you are removing from the list would not appear on the list. As per your own condition. 'ABBA….then becomes the one not to be accommodated and yet there he stands. All others are accommodated by the definition employed in this video' Well, yeah. So, then, they are not accommodated, and you haven't found a way to accommodate everybody. Which is kind of the point. 'So, if we consider the moving of each guest down a room to make room for one more, for example, this works for those rooms within our immediate sight or understanding of limited rooms, but not of the rooms and guests down the line' What the hell does that even mean? How did you come to this conclusion when relocating guests simply corresponds to us choosing a different function for their arrangement? We are literally going from using f(n)=n to arrange the guests to f(n)=n+1. It is done simultaneously and concurrently for every room and guest. 'If the infinite rooms have infinite guests then by that very statement which is the very definition of the proposition, all are full and no room can be made' Rooms can be freed by mapping the set of rooms one-to-one to its own proper subset. That is possible due to the properties that all infinite sets have. And which is demonstrated in this video. 'I might be accused of employing conventional considerations in materiality in my judgment of the proposition' I accuse you of having no grasp on logic and being bad at mathematics and language skills. You wrote a lot while ignoring the actual relevant definitions, and then made a bunch of unsubstantiated (and wrong) claims, but also some claims that don't support your point in the slightest, and then asserted that the author of the video is wrong.

[@jamestagge3429]

@@thetaomegatheta Part 1.....................Me - 'first, the hotel could not have infinite rooms or infinite guests'…in material terms. You - It very much could. The hotel is just an analogy for the set of natural numbers, and, unless you can somehow prove that there are only finitely many natural numbers, you are yet to substantiate your claim. Me - Here you prove my point. A “room” is a material entity and quantity. A natural number is not material but rather an intangible, a concept, etc. Within material, there an be no infinity. To be material, to exist materially an entity must be finite, delineable and quantifiable, period. Every composite entity which exists, each and every one is composed of molecules which are composed of atoms, which are composed of sub-atomic particles which are defined by physicists as “discrete packets of energy”, again, finite, delineable and quantifiable. From the last listed to the first, each a material construct. Even empty space, that composed of all of the energy fields in concert is a material construct which we know for it is quantifiably distorted or warped by the presence of a mass (entities which exist within it such as the earth). That should be a sufficient take care of your demand of me for a proof of my point. Me - 'because each of these is a quantifiable entity and by trying to accommodate them in a context of infinity, which is not, the entire proposition fails on its face, though I know that most who embrace this paradox as genius understand that' You - There are infinitely many natural numbers, which, by your logic, are a 'quantifiable entity' (whatever that means), and, again, by your logic, there are only finitely many of them. Lol. Me - Again you are way off base and your haughty attitude only embarrasses you. You make the wrong headed defense of this piffle your cause for some reason. As I stated above, the natural numbers ARE NOT MATERIAL. They are conceptual values and thus can be considered in an unending progression, but even then cannot be considered in any quantifiable instant in time to be infinite, but rather only a progression extending out into forever. This is not infinity for after any measure of time one would stop at a quantifiable value. Within the context of consideration within the material context, even this conceptual infinity would not be complete. LOL 'Secondly, if for the sake of the discussion we assume the premise that there are infinite rooms then that is a consideration of the concept which is, to be sure, intellectually manageable. However, once we claim that there are also infinite guests we cannot but understand that each room, though infinite in number is by the definition of the proposition itself, occupied or full. Here the two “floating concepts” of infinite rooms and infinite guests are in a kind of conceptual equilibrium, necessarily so or the proposition contradicts itself' You - That's just a lot of words to say nothing of substance. Instead of thinking rigorously and using logic, you just make that word salad. Me - My proposition here is precise and clear. That you cannot defeat it explains why you resort to childish tactics such as claiming “you are wrong” which is not an argument. If you had the answer, an argument which would defeat what I stated you would have offered it and made your point. You didn’t because you can’t. You are being a coward. Me - If then, we were to say that we will move this infinite number of guests by one to make room for one more, we are forced to consider that the shift of customers cannot be made for each room is again, by definition full and also by definition, always full'. You have demonstrated twice now that you are incapable of understanding my critique which is really quite simple and straightforward. Why not try? Minimally it is interesting. You continue to run away by accusing me of spewing “word salads” which if my propositions were that, you could certainly demonstrate how and why they are that. Consider…the rooms of the Hilbert hotel are the holes in one stanchion of a ladder, infinitely tall. The guests are the holes on the other stanchion. The rungs which extend form the holes on one stanchion to the corresponding holes on the other satisfy that “all the rooms are full”, Hilbert’s qualification, not mine. If we accept this material context for the sake of his proposition, we know then that his appeal to the unending number of rooms must be also made to the unending number of guests in a one to one relationship precisely because “all the rooms are full”. Here it is clearly demonstrated that the concept of infinity CANNOT be paired with that which is material. There can be no room made for additional guests because as a component of the progression of rooms and guests, they are all already, permanently matched in that all the rooms are full. Did he say that all the rooms were full up to this point but not beyond? No. Did he say that the infinity of the guests was smaller than the infinity of the rooms? No, for that would defy the qualifying principle or condition that “all the rooms were full” which would mean all the time and unendingly. What is it you don’t get? You - You make this claim, but you provide no inference, no basis, no substantiation for it. And the claim is pretty wild. You are saying that the function f(n) for all natural n has a non-empty complement to its image within the set of natural numbers (N\Im(f)). Me - Now there is some word salad. What you don’t understand is that my complaint is that Hilbert tried to pair infinity within a material context and that can only generate a conceptual contradiction. It is nonsense. He then defied his own paradox in its very definition. A line of infinite rooms would have a guest in each and every one for its entire unending progression because he claimed it to be so by stating the all the rooms were full. You keep returning to the math which is not part of this discussion. It is that the material concepts employed in the definition of the hotel CANNOT BE PAIRED WITH THE CONCEPT OF INFINITY! His hotel paradox is just…dumb. It makes absolutely no sense. Perhaps he authored it as a metaphor for his mathematical propositions. That’s fine but that does not mean that it is not clearly nonsensical. Don’t be so thick. You - That is, of course, demonstrably not true, and yet, you make this claim with such confidence. Me - 'To consider that room could be made for new guests, especially an infinite number by clever manipulation of the numbers used as identifiers and the manner in which they are listed is childishly transparent as means of proposing a solution but only in the immediate sense, i.e., for the sake of a visible or quantifiable number of guests relative to rooms (outside of the context of infinity)' You - Again, more word salad. Me - Your favourite phrase but also that which shows you are incapable of defeating my points or you would and you don’t. Again, saying “you’re wrong” is not an argument.

[@thetaomegatheta]

@@jamestagge3429 'Here you prove my point. A “room” is a material entity and quantity' No. It's a stand-in for an element in the set of natural numbers, i.e. a natural number, which are not material entities. It's like when, during a math class, a problem goes something like 'you have 5 apples in your pocket; you give one apple to your friend; how many apples are in your pocket now', and you protest, 'but I don't have any apples in any of my pockets, and I can't fit so many within any of them anyway'. 'A natural number is not material but rather an intangible, a concept, etc' Correct, natural numbers are non-material entities. And so are the rooms in this problem. It's like you have never attended any math classes in your life. I'm going to skip the rest of the paragraph, as that's just a lot of words to say nothing, based on a premise that has just been demonstrated to be false. 'As I stated above, the natural numbers ARE NOT MATERIAL' Yes, and neither are the rooms. The rooms are just an analogy for the set of natural numbers. You are basically arguing against using relatable analogies in the formulation of math problems, while using really dumb argumentation to boot. The problems could be restated along the following lines, without invoking the words 'room(s)', 'hotel', and 'guest(s)', but without changing anything: 1) Let's look at the set of natural numbers N. Find a bijective function f from N to N (so, f: N->N), such that its image has a non-empty complement in N (so, N\Im(f) must be non-empty). One appropriate answer to that problem would be f(n)=n+1, as N\Im(f) = {1}, because the inverse function f^-1(n)=n-1 doesn't map 1 to a natural number (it maps 1 to 0: f^-1(1)=1-1=0). 2) Let's look at the set of natural numbers N. Find a bijective function f: N->N, such that the complement to its image within the set of natural numbers, N\Im(f), is an infinite set. One appropriate answer to that problem would be f(n)=2*n, as N\Im(f) = {1, 3, 5,...} includes every odd natural number, which there are infinitely many of. 3) Let's look at the set of natural numbers N and a set S of infinitely many sets of the same cardinality as N, S={N_1, N_2, N_3,...}. Find a function f: N->N, and a family of functions g_i: N_i->N, such that they map their domains bijectively to subsets of N and their images do not intersect. One solution to that problem would be to find a family of infinitely many infinite non-intersecting proper subsets of N, and map the given sets to those subsets. One such family of infinite proper subsets is the family of sets of powers of prime numbers, so {{2^n: n is natural}, {3^n: n is natural}, {5^n: n is natural},...}. The appropriate functions could be f(n)=2^n, g_1(n)=3^n, g_2(n)=5^n,..., g_i=p_i^n,..., where p_i is the ith prime. Now that we have merely reworded our problems, without changing anything of substance, none of your shallow musings about rooms supposedly being material objects don't apply, and your entire line of criticism fails. And, again, we didn't change anything substantial in the wording of the problems, we just got rid of relatable analogies. 'My proposition here is precise and clear' Your 'proposition' is that you don't like the wording, and then try to invent BS reasons why this set of problems is somehow bad because infinitely many material rooms couldn't exist. It is also not particularly clear, as you do try to obfuscate your point behind a lot of text. 'That you cannot defeat it' I can. I have already done so. I have dismantled your point in multiple ways. And your only response thus far is that 'rooms are material, therefore the infinite hotel can't exist'. But hey, I can humour you.

[@thetaomegatheta]

@@jamestagge3429 'However, once we claim that there are also infinite guests we cannot but understand that each room, though infinite in number is by the definition of the proposition itself, occupied or full' 'Definition of proposition' is an undefined expression in this context. It doesn't refer to anything. You could have meant 'initial conditions laid out in the formulation of the problem', but then having infinitely many guests and infinitely many rooms doesn't automatically make every room occupied (for example, the guests could just be occupying only odd-numbered rooms), meaning that there is no actual implication, contrary to your claim. The condition 'every room is occupied' is required to be stated separately. 'Here the two “floating concepts” of infinite rooms and infinite guests are in a kind of conceptual equilibrium, necessarily so or the proposition contradicts itself' 'Conceptual equilibrium' is a meaningless expression in this context. You are inventing your own, not widely accepted terminology here, without actually providing any definitions for your terms. At best. At worst you are just using words at random. You have also, of course, not demonstrated that this 'conceptual equilibrium' is somehow necessary for the 'proposition' to not contradict yourself. The lack of definition alone is enough to dismiss your attempt at an argument. 'If then, we were to say that we will move this infinite number of guests by one to make room for one more, we are forced to consider that the shift of customers cannot be made for each room is again, by definition full and also by definition, always full' You haven't proven that the shift of guests cannot be made. You merely asserted that. When that is demonstrably false. Each room is also not full by definition, let alone always full. Every room is full INITIALLY, as per the INITIAL conditions set up in the problem. 'The very outline of the structure of the proposition shows it is necessarily so and is that by which it also collapses' You make another assertion without any sort of proof. 'To make this clearer, imagine traveling down the line of infinite rooms and looking in. You would find each room occupied as you traveled, forever. That “is” how the proposition is defined/stated' Okay? And? What is preventing every guest from leaving their rooms, concurrently, and then immediately teleporting to their new rooms? What are we supposed to consider here? 'One can’t have it both ways' What are these 'both ways'?

[@thetaomegatheta]

@@jamestagge3429 'You have demonstrated twice now that you are incapable of understanding my critique which is really quite simple and straightforward' Your 'critique' is that you don't like the analogy, and, therefore, somehow, the problem itself is bad. The moment we strip the wording of the analogy, none of your criticisms apply, while the problem remais the same. If anybody has demonstrated anything, it's you who demonstrated their inability to understand formal logic and basic set theory, while refusing to think rigorously about the presented problems. 'Consider…the rooms of the Hilbert hotel are the holes in one stanchion of a ladder, infinitely tall. The guests are the holes on the other stanchion. The rungs which extend form the holes on one stanchion to the corresponding holes on the other satisfy that “all the rooms are full”, Hilbert’s qualification, not mine. If we accept this material context for the sake of his proposition, we know then that his appeal to the unending number of rooms must be also made to the unending number of guests in a one to one relationship precisely because “all the rooms are full”' That is correct. What's the problem? 'Here it is clearly demonstrated that the concept of infinity CANNOT be paired with that which is material' Okay? We aren't dealing with anything material here, yet you keep bringing up material entities here. And how does the reformulation of the analogy using a ladder instead of a hotel 'clearly demonstrates' what you say it does, anyway? 'There can be no room made for additional guests because as a component of the progression of rooms and guests, they are all already, permanently matched in that all the rooms are full' Good thing that we aren't creating any new rooms. We are just using the existing ones and the fact that infinite sets have infinite proper subsets, which allows us to map the set of rooms to one of its proper subsets one-to-one. They are also not permanently matched. They are initially matched that way. So, that's another error on your part. 'Did he say that all the rooms were full up to this point but not beyond? No' 'Beyond' what? Every room is initially occupied, as per the initial conditions set up in the problem. 'Did he say that the infinity of the guests was smaller than the infinity of the rooms? No, for that would defy the qualifying principle or condition that “all the rooms were full” which would mean all the time and unendingly' Okay? And? The set of rooms and the set of guests both share the cardinality (i.e. they both have the same 'size'/'amount of elements'). There is, indeed, one guest for every room. And, because there are infinitely many rooms, we can rearrange the guests and put them into a proper subset of the set of rooms. What's your point here? 'Now there is some word salad' If you don't understand fundamental terms of set theory and math in general, such as 'complement', 'image', 'empty set', or 'set of natural numbers', while also being so confident in your ignorance, that's on you. I'm using terms that have a very direct involvement in the problems shown in the video. 'What you don’t understand is that my complaint is that Hilbert tried to pair infinity within a material context' No. The hotel is just a relatable analogy. Your inability to grasp the context of analogies is rather baffling. Anyway, as I have shown, if we get rid of the analogy and just reword the problems in purely mathematical terms, we address your criticisms without changing the problems themselves. 'It is nonsense' Your inability to understand basic mathematics, including logic and set theory, doesn't make these really simple, introductory set theory course-level problems for freshman or, at most, sophomore math students, 'nonsense'. It's just you, your arrogance, and wilful ignorance. 'A line of infinite rooms would have a guest in each and every one for its entire unending progression because he claimed it to be so by stating the all the rooms were full' That is correct. So, what's the problem? What prevents us from rearranging the guests? 'You keep returning to the math which is not part of this discussion' 'Math is not a part of the discussion of basic math problems, introduced in lectures for math students to showcase some counterintuitive properties of sets, which are objects that are studied in math'. Haha. I'm afraid, today is not April Fools', though. 'It is that the material concepts employed in the definition of the hotel CANNOT BE PAIRED WITH THE CONCEPT OF INFINITY!' The rooms and the guests are just analogies for objects that you yourself recognise as non-material. Unless you want to explain how the set of natural numbers 'can't be paired with the concept of infinity' (whatever that means), you are only making a joke of yourself. 'His hotel paradox is just…dumb. It makes absolutely no sense' You not being able to understand basic math doesn't make this dumb or nonsense. You looking at a reformulation of the problem using only mathematical terms, and then calling it 'word salad' doesn't make this dumb, or nonsense, or a word salad. You being confident in your ignorance and refusing explanations doesn't make this dumb or nonsense. It just makes you a joke. 'Perhaps he authored it as a metaphor for his mathematical propositions' The fact that the thought clearly passed your mind and then immediately left it is just sad. Your intellectual laziness even prevented you from pursuing this thought further. 'Your favourite phrase but also that which shows you are incapable of defeating my points or you would and you don’t' I already have dismantled your points multiple times, but alright, I'll humour you again.

[@kadenvanciel9335]

The intro is a terrible-sounding version of ilmango's intro.

[@HarrySurplus]

turns out it’s the same song lol. thanks for telling me intro sounds terrible :))))