One caveat: the hotel cannot be an infinitely large shape that connects with itself, such as a circle, ellipse, or any irregular shape that connects with itself, because all the rooms would then still be occupied.
i don't think that's true. if it's some loop, but we index the rooms against the continuum, we would have plenty of room. remember against a continuous interval, all this cute countable stuff has a Lebesgue measure of 0
well the power of a prime is composite. it's just unique but i assume you mean composites that are not powers of primes. seems in this example they are vacant. but peyam left out the secret caveat that we use those rooms for "special" guests
Usually people point out their hotel has an infinite amount of room before they pull the first trick. Then again, your enthusiasm is such that I basically accept everything you say.
6:00 When you think of uncountable sets, you don't necessarily have to imagine them as being continuous. Sure, the real numbers are uncountable and they're continuous. But the power set of integers is also uncountable-and AFAIK-its elements are _not_ continuous. (Correct me if I'm wrong.)
Or, for the infinite # buses with infinite # passengers, once you take our the existing tenants, add them as a new row to the infinite rows of buses and start to select them in a zigzag manner like (1,1) to (1,2) then (2,1),etc. Similar to how we can prove that Q is countable.
The last few things you said, in my opinion is the real mind bender. The fact that the C.H can't be neither proven nor disproved with our usual axioms, and the thought of how would a set with cardinality between א0 and א might look like 😊
@@sharpnova2 I'm not sure what you mean by that, we simply don't have a good short and useful (and hopefully consistent even if it can't be prove to be) list of axioms that can resolve the C.H in a convincing enough way. Currently, the debate is mostly whether א = א2 or א = א1 as if it was a dichotomy (which it isn't).
Maybe I should just avoid hotels. I was thinking about the Order of Operations video ... where math is made needlessly ambiguous or confusing. Another video idea (albeit simple) might be to discuss what it means to say something "costs 400% more" or is a "savings of 50%" ... where for example - the simple idea of something costing twice as much means it is a 100% increase in price or 200% the original value. I've noticed advertisements often confuse shoppers by doing this (e.g. 50% off). Such "encoding" seems to confuse people (and if you make a video, I don't have to explain it again to business associates!!! haha). I expect it sometimes is meant to confuse, instead of simply stating math in a way everyone would understand. Cheers ...
Dr. Peyam, your third example is much less convoluted than other solutions I have seen. Still, if an infinite number of busses showed up at my hotel each with infinitely many customers, I would still be worried.
alternate way to fill up the hilbert hotel in the infinite buses situation, move the hotels location to gensokyo. there are no buses in gensokyo after all according to cirno.
5:44 No, no, no dear Peyam the hotel is not almost empty. There are much more passengers in rooms with a number that's a power of a prime than passengers with a room number that ist not a power of a prime. Guess you numerate all room numbers which are not a power of a prime: a1 a2 a3 ... In the first floor (with infinite number of rooms) the first room number is a1 (1) the second room number is the first prime (2), the third room number is a2 (6), the fourth is the second prime (3) and alternately so on and on. In the second floor (with infinite number of rooms) are all room numbers with the second power of all primes, in the third floor are all room numbers with the third power of primes and so on. There is only one first floor but infinitiv floors above the first floor. Conclusion: There are much more rooms with a number that's a power of a prime than rooms with numbers that are both a power of a prime. Greetings from Germany the home of Hilbert ;-)
Before a new guest can be accommodated, at some point an existing guest has to be moved into an empty room contrary to the hypothesis that all of the rooms are occupied. There's no paradox because the premise that every room is occupied is false. Right?
And I would think the rooms will have different sizes (first room will have the width W, and the n room will be W/2^(n-1) for example). Otherwise this hotel would not fit on the earth planet.
Our universe _could_ be infinite-we don't know yet. Depends on how parallel lines behave in the real world. - So if our universe is indeed infinite, a hotel 'room' could just be a region of space. Kinda like the parking lots in my apartment complex. They were presumably demarcated by painted lines at some point, but today it's just contiguous space behind the buildings. Yet we residents somehow know which is our allotted parking space. 🤣 - Or the rooms are infinitesimal (why not), so the hotel is finite overall. - Or maybe the hotel uses the next generation Time Lord technology: Small on the outside, infinitely large on the inside.
I'd never go to that hotel since I'd have to change room every 5 minutes🤣
The hotel would get a countably infinite downvotes.
You'd have to change your room countably many times.
No time to relax...
I would be pretty cranky if a hotel clerk woke me up and asked me to walk an arbitrarily large distance to another room.
All guests with prime room numbers, enjoy the company from guest "1" or themselves.
In the prime rooms are lonely persons :(.
guest 1 is just comfortable with their divisibility
One caveat: the hotel cannot be an infinitely large shape that connects with itself, such as a circle, ellipse, or any irregular shape that connects with itself, because all the rooms would then still be occupied.
i don't think that's true. if it's some loop, but we index the rooms against the continuum, we would have plenty of room.
remember against a continuous interval, all this cute countable stuff has a Lebesgue measure of 0
Love Hilbert's Paradox - question: what happens to the ith hotel rooms labelled as composite numbers? Are they filled or left blank?
well the power of a prime is composite. it's just unique
but i assume you mean composites that are not powers of primes. seems in this example they are vacant.
but peyam left out the secret caveat that we use those rooms for "special" guests
Super spicy situation!
Usually people point out their hotel has an infinite amount of room before they pull the first trick. Then again, your enthusiasm is such that I basically accept everything you say.
welcome to the hotel mr. peyam
such a mathy place.. such a mathy place
I love that song!!!
Infinite rooms at the hotel dr peyam.....
I enjoyed both the explanation and the smiley faces Dr Peyam was drawing. :)
I dare you to not love this man ❤️
Proud of you Sir.
OMG, Dr. Peyam... Couldn't you do the in-between? Pleeeease??? It would make my life so much easier. 😉😉
Hahaha
What is paradoxical about this?
One runs out of time satisfying doubters before realizing a conclusion.
I was gonna ask the same question. All I heard here was an explanation of a Hilbert hotel. Where is the paradox?
That hotel would be receive very poor ratings online
Probably hahaha
6:00 When you think of uncountable sets, you don't necessarily have to imagine them as being continuous. Sure, the real numbers are uncountable and they're continuous. But the power set of integers is also uncountable-and AFAIK-its elements are _not_ continuous. (Correct me if I'm wrong.)
Yes. Thanks.
Many youtubers like veritasium also did videos on this but I liked your video the best
when a huge continuous mass of customer arrives and it speaks "no, no, no, we are all different"
Or, for the infinite # buses with infinite # passengers, once you take our the existing tenants, add them as a new row to the infinite rows of buses and start to select them in a zigzag manner like (1,1) to (1,2) then (2,1),etc. Similar to how we can prove that Q is countable.
The last few things you said, in my opinion is the real mind bender. The fact that the C.H can't be neither proven nor disproved with our usual axioms, and the thought of how would a set with cardinality between א0 and א might look like 😊
uhm.. i don't think anyone has shown that the c.h. is undecidable or anything like that yet
@@sharpnova2 I'm not sure what you mean by that, we simply don't have a good short and useful (and hopefully consistent even if it can't be prove to be) list of axioms that can resolve the C.H in a convincing enough way.
Currently, the debate is mostly whether א = א2 or א = א1 as if it was a dichotomy (which it isn't).
Diagonalization is another solution for putting ctb inf busses of ctb inf people.
A1 A2 A3 A4 A5 ...
But in my way we’re making more space haha
@@drpeyam You're right. We could put the guests in odd-numbered rooms.
That is exactly what I came up with...
If I would be the 100-th passenger of the first bus, I would be to tired to go in the room 3^100.
Maybe I should just avoid hotels. I was thinking about the Order of Operations video ... where math is made needlessly ambiguous or confusing. Another video idea (albeit simple) might be to discuss what it means to say something "costs 400% more" or is a "savings of 50%" ... where for example - the simple idea of something costing twice as much means it is a 100% increase in price or 200% the original value. I've noticed advertisements often confuse shoppers by doing this (e.g. 50% off). Such "encoding" seems to confuse people (and if you make a video, I don't have to explain it again to business associates!!! haha). I expect it sometimes is meant to confuse, instead of simply stating math in a way everyone would understand. Cheers ...
Agreeeed
Dr. Peyam, your third example is much less convoluted than other solutions I have seen. Still, if an infinite number of busses showed up at my hotel each with infinitely many customers, I would still be worried.
Thank you!!
Can you make an uncountably infinite hotel please
Hahahaha
@@drpeyam thanks
alternate way to fill up the hilbert hotel in the infinite buses situation, move the hotels location to gensokyo. there are no buses in gensokyo after all according to cirno.
Funny
Can you recommend a book that deeply explains the modern physics mathematics ?
Not that I know of
I think I don't want to be a porter in that hotel... :-)
Probably it is situated in Las Vegas.....
Same hahaha
Musical rooms/chairs, just don't go to bed.
5:44 No, no, no dear Peyam the hotel is not almost empty. There are much more passengers in rooms with a number that's a power of a prime than passengers with a room number that ist not a power of a prime.
Guess you numerate all room numbers which are not a power of a prime: a1 a2 a3 ...
In the first floor (with infinite number of rooms) the first room number is a1 (1) the second room number is the first prime (2), the third room number is a2 (6), the fourth is the second prime (3) and alternately so on and on.
In the second floor (with infinite number of rooms) are all room numbers with the second power of all primes, in the third floor are all room numbers with the third power of primes and so on.
There is only one first floor but infinitiv floors above the first floor. Conclusion: There are much more rooms with a number that's a power of a prime than rooms with numbers that are both a power of a prime.
Greetings from Germany the home of Hilbert ;-)
Sorry ... "not" instead of "both" in the last sentence
Kinda funny how room #1 is left empty in scenario 3. Reserved for the owner of the hotel, I guess lol
Or vip guests haha
Video is 6:28 long- Coincidence? I think not.
I love this paradox.
wat
Mint
Ordinary math students: trying to understand the paradox .
Me : wondering how they gonna build such an infinitely long hotel 🤣
For putting countably infinitely many people in countably infinitely many busses, I would have gone with square numbers.
But then 4^2 = 2^4 for example
Before a new guest can be accommodated, at some point an existing guest has to be moved into an empty room contrary to the hypothesis that all of the rooms are occupied. There's no paradox because the premise that every room is occupied is false. Right?
Not a paradox, you can always switch/move rooms. Just tell an existing customer to move out
A countably infinite hotel?!? What universe is Hilbert coming from? How do you make a countably infinite hotel out of brick and mortar?
And I would think the rooms will have different sizes (first room will have the width W, and the n room will be W/2^(n-1) for example). Otherwise this hotel would not fit on the earth planet.
Put the hotel inside gabriel's horn
Our universe _could_ be infinite-we don't know yet. Depends on how parallel lines behave in the real world.
- So if our universe is indeed infinite, a hotel 'room' could just be a region of space. Kinda like the parking lots in my apartment complex. They were presumably demarcated by painted lines at some point, but today it's just contiguous space behind the buildings. Yet we residents somehow know which is our allotted parking space. 🤣
- Or the rooms are infinitesimal (why not), so the hotel is finite overall.
- Or maybe the hotel uses the next generation Time Lord technology: Small on the outside, infinitely large on the inside.
What about room 1 after all the rearranging in the last scenario? You somehow made a full hotel have an empty room by adding more clients.
RJ
when a new guest come, why do not just put him in the N+1 room rather shifting other guests???
paradox do not existe but on the mind. thank you so much.
This Hotel exists only in Hilbert's imagination....