Hi everyone, I'm here to express my extreme gratitude for the amazing comments. We're clearly blessed with a cool and wholesome community. You all rock! I can tell you that my dad was extremely moved by all of your wishes. He asked me to convey this message, straight from him to all of you: " I want to thank you all for the many birthday wishes. It is an exceptional and unexpected gift. Also, my heart warms up when I see your appreciation for the work of my son. I would like to complement his motto 'keep learning' with mine: 'keep teaching'. Knowledge has this peculiar property that the more of it you give away, the more of it you have left. Have a goof life. "
BTW, we won't be able to solve *all* maths problems with Omega because Omega has no information about programs that have access to omega. This is the hierarchy of hyper computation or something like that, and it's really neat. But yes realistically all problems we care about are problems about programs that don't have access to omega
Well that is why omega is uncomputable, if it could be computed, we’d have that problem, we also actually know it’s irrational, because if it was rational it would be computable
Happy Birthday, All Angles's Dad! I too am a software engineer. I have an 18 month old little boy. I think there are many, ways in which I could succeed as a parent, but if in two decades time, my boy is making RUclips videos (or whatever has replaced them) which challenge me to think and learn in the same way that your boy has done for me today, I will feel super proud of what he has achieved. And if he traces any part of his love of learning and his willingness to challenge himself back to me, then I in turn will trace it back to videos like this one that have helped and inspired me. Thank you so much for helping to create a world filled with the passion and curiosity
Doesn't omega depend on the system being used? So it's not really a constant like pi, unless you specify the system you are working in (some specific lambda calculus, for example)
You're right, the exact value will depend heavily on all the choices we make along the way. I didn't want to get bogged down in the details, so I skipped over it. But well spotted.
Belated happy birthday to your Dad from France. Good work getting your (now grown) child interested enough in these topics that now they are making videos for the engineers like me who didn't get to learn the spicy bits of the math.
The halting problem is decidable for finite deterministic systems, so it's theoretically possible to calculate omega for some systems. Unfortunately, any problem worth solving with omega would require massive amounts of computational power
I just turned 75 too, and I'm also a Dad, so happy birthday to both us Dads! I found an interesting pattern for the composites of Euler's quadratic. Perhaps you can find it too.
Happy birthday to your dad from Poland! By the way, I believe even if we knew the value of omega, running all possible programs at once could be tricky - at a trillion in, we would still be looking at programs that print random constants
I think that to compute the number to a certain accuracy would require you to know the answers to all the problems it could solve, so it's less of an oracle and more of a compression algorithm. And an optimal one too, since it's not compressible further!
Oh, it's my dad's birthday today as well. I think he's 76, I always forget whether it's my mum or dad was born in 48 or 49. What a nice coincidence. :) Happy b'day to both our dads.
Great video, very well explained; the topic choice is perfect and carries a nice philosophical thought at the end, "I compress; therefore, I understand." is my favorite line. Btw I think there is a mistake at 10:36 since a perfect number equals HALF the sum of its divisors, not twice (actually, quite a bummer that it's not twice the sum, as that would make the problem of proving that an odd one doesn't exist pretty easy lol). Also, happy birthday to your dad. Maybe he can wish for the first 2^75 digits of omega as he blows out the candles on the birthday cake.
Hi, nice explanation video ! I'm currently working on a video on the exact same subject and I have some remarks, especially on the part "how the oracle works": The Omega that you describe in your video is not defined on a prefix computing model which mean that it is possible that your Omega can be greater than 1 for example if the 2 programs of length 1 (the ones encoded by 0 and 1) both halts they will both add a wheight of 0.5 which will make the total already equals to 1. And by your defenition that would mean that even if there are only thoses two programs which halts the probability of any programs to halts will be 1 (Omegas can be seen as a probability but not in a direct way) . At 17:48 you said that you work only on the shorts programs which means that you can only compute a lower bound of that omega because there are programs on size greater than n that can have an impact of the first bits. To do so you have to not work only on the programs of size n but all programs of your computing model. Since there is an infinity of programs you can't do the trick of "I do one step of each programs and I start again" because that would mean that you can only performs at most one step of each program, to get over this you can use what we call a "Dovetailling" which works like the bijection between N and N^2. I know i am being really pedantic about theses little details but as I said, this is a nice explanation video that probably make the whole concept understandable for a person that don't know about it already and all of thoses details can take a while to explain and might hurt your audience retention so keep it that way 😄
A key step in our journey to figuring out if a program halts or not is using omega-n, where n is the "length" of our program and then used to take the first n digits of omega. A few questions I've been thinking about as a result of this and my thoughts (feel free to chime in): Questions - Is it possible that omega has less than n digits? I.e. does omega have infinitely many digits? Does a random number have to have infinitely many digits? Why is omega "random"? Thoughts - From the video, a key point is that a number is random if (and only if?) it is incompressible. Thus, if a number is not random, we could write a program to write out its digits. If it is random, we could not do that. So, if a number has finitely many digits, we should be able to write a program with finitely many steps to write out the digits of that number. So a random number must have infinitely many digits. Secondly, Turing proved through the halting problem that we can't have a program determine whether all programs will halt or not. Thus, we can't compress the probability that a random program halts, and so omega must be random. Since omega is random, it must have infinitely many digits, meaning we could always take the first n digits for arbitrary n. I'm little shaky on that second jump. Let me know if I'm missing something or can think about it in a different way. Great video and happy birthday to dad!
If you Like this video and topic, you should DEFINETLY go and buy yourself a copy of Gregory Chaitin's book - "Meta Math". It is an amazing book for math and computer nerds in general, but covers the story behind the exploration of Omega coming straight from the man himself, and gives insight into his thought process on discovery and knowledge.
Late happy birthday to your dad! I'm israeli but I recently moved to the netherlands for a master's degree in logic in UvA. Working hard on giving you some future content :)
Good video! I'm seeing quite a few people questioning the reasoning and results of the last section of the video. While I'm certainly no expert I wanted to give some notes on where this video is skipping over details (some I think are a bit crucial). This isn't to discredit the video! I understand that some simplifications must occur so the video is accessible: 1. If we actually allow all binary sequences to define a program and add 1/2^k whenever that program halts, notice that this number can be as large as n, when considering just the sequences of length n. This number therefore isn't a probability. I would urge watchers to think about how you might actually describe the probability over an infinite sequence of options. The answer is to define a measure, which is where the 1/2^k thing comes from. 2. The only reason the 1/2^k thing works is because the constant is defined with respect to a *prefix-free*, *universal* turing machine. You might think of this as the 'programming language', which leads to the different values of omega, but this language has restrictions, namely that if one program p halts, then any program which has p as a prefix (is just p with some extra characters added on) then this program *cannot* halt. With this restriction, and this restriction only, the 1/2^k computation step makes sense, (what would that mean if these values summed more than 1, how would we know a prefix has contributed?) 3. The way the 'simulate all programs' step is animated wouldnt work, because we would have to execute a countably infinite set of '1st steps' before returning to the 2nd step of the first program. The solution here is to use a diagonal approach, only executing the xth step of the yth program once all previous programs have computed at least the x+1th step. (Search dovetailing on wikipedia and the subsection on infinite sequences if confused) I found this set of slides helpful for further reading: Search for 'Computation and Thermodynamics - UCR Math' Again to reiterate - I liked the video! But just wanted to add more detail for eager viewers looking to better understand the concept.
![Seungmin "그렇게, 천천히, 우리(As we are)" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/kAzmhLHePqU/mqdefault.jpg)
Hi everyone,
I'm here to express my extreme gratitude for the amazing comments. We're clearly blessed with a cool and wholesome community. You all rock!
I can tell you that my dad was extremely moved by all of your wishes. He asked me to convey this message, straight from him to all of you:
"
I want to thank you all for the many birthday wishes. It is an exceptional and unexpected gift.
Also, my heart warms up when I see your appreciation for the work of my son.
I would like to complement his motto 'keep learning' with mine: 'keep teaching'.
Knowledge has this peculiar property that the more of it you give away, the more of it you have left.
Have a goof life.
"
Many Happy Birthday wishes from the North Carolinas.🎉
BTW, we won't be able to solve *all* maths problems with Omega because Omega has no information about programs that have access to omega. This is the hierarchy of hyper computation or something like that, and it's really neat. But yes realistically all problems we care about are problems about programs that don't have access to omega
Well that is why omega is uncomputable, if it could be computed, we’d have that problem, we also actually know it’s irrational, because if it was rational it would be computable
@@NzargnalphabetIt’s even transcendental, as all algebraic numbers are computable.
Happy birthday to your Dad from Boston! 75 is a big one, congratulations!
Happy Birthday, All Angles's Dad!
I too am a software engineer. I have an 18 month old little boy. I think there are many, ways in which I could succeed as a parent, but if in two decades time, my boy is making RUclips videos (or whatever has replaced them) which challenge me to think and learn in the same way that your boy has done for me today, I will feel super proud of what he has achieved. And if he traces any part of his love of learning and his willingness to challenge himself back to me, then I in turn will trace it back to videos like this one that have helped and inspired me. Thank you so much for helping to create a world filled with the passion and curiosity
Thank you so much for that uplifting comment. Let's keep spreading knowledge!
Happy anniversary from Spain, and thank you for inspiring your son!! We absolutely love his incredible videos.
Muchas gracias!
Creía totalmente que sería el único español aquí. Me alegra saber que no.
I'm a little bit late to this, but greetings from Kazakhstan 🇰🇿, I hope your dad is doing great and thanks for such good quality videos
Thank you!
Happy birthday to your dad from the east of Spain!
Gracias!
Doesn't omega depend on the system being used? So it's not really a constant like pi, unless you specify the system you are working in (some specific lambda calculus, for example)
You're right, the exact value will depend heavily on all the choices we make along the way. I didn't want to get bogged down in the details, so I skipped over it. But well spotted.
I was wondering about this, neat to hear.
It'd be fun to try and calculate a few trivial bits of omega :)
Happy birthday from France !! Take care of yourself, hoping everything's going great !
Happy birthday to your dad from Oregon!
Thank you sir for making my day! This video was absolutely incredible. Happy birthday to your Dad from Düsseldorf, Germany!
Danke schön!
Happy cake day for you dad 🎉
found the redditor
Happy birthday from Duluth Minnesota!
I like the Idee of omega. Happy birthday from Switzerland to your father.
Thank you! Switzerland is one of our favorite hiking countries.
Happy birthday from Canada!
Hey! This is also my mother's birthday
Congratulations to your mother!
This was one of the best videos i have watched in the couple of weeks on RUclips, thank you!
Happy Birthday to your Dad from Tennessee
Belated happy birthday to your Dad from France. Good work getting your (now grown) child interested enough in these topics that now they are making videos for the engineers like me who didn't get to learn the spicy bits of the math.
Happy birthday padre! Blessings from Colorado!
Muchas thanks!
Happy birthday to your dad from canada :D
Fijne verjaardag vake! (Best wishes from Belgium)
Happy birthday to your dad from France :)
happy birthday dad, from koper, slovenia
Happy birthday wishes from Melbourne in Australia!
Happy birthday!
Happy birthday to your dad from Texas! Congratulations on the big 75!
Happy birthday from Germany ❤
Happy Birthday to your dad from Brazil!
Happy Birthday from South Florida!
The halting problem is decidable for finite deterministic systems, so it's theoretically possible to calculate omega for some systems. Unfortunately, any problem worth solving with omega would require massive amounts of computational power
Happy birthday from Marseilles, France, from a fellow software engineer. :))
Happy Birthday to your dad!
Happy birthday to your Dad from Quebec, Canada!
Happy birthday to your dad from Italy!
Happy B birthday from Antibes in France
I have fond memories of walking along the Cap d'Antibes. Thanks for the wishes!
Happy Birthday from New Mexico.
Happy birthday from Houston, TX!
I just turned 75 too, and I'm also a Dad, so happy birthday to both us Dads!
I found an interesting pattern for the composites of Euler's quadratic. Perhaps you can find it too.
Congratulations on your recent birthday!
Happy birthday to your dad from Poland!
By the way, I believe even if we knew the value of omega, running all possible programs at once could be tricky - at a trillion in, we would still be looking at programs that print random constants
You're right, the entire idea of omega is mostly theoretical, not very practical.
Happy 75th birthday! Thank you for having such a great son! From Brazil!
happy birthday from Canada!
Happy birthday mr. Dad, from central Michigan.
Happy birthday to your dad!!
Happy 75th birthday from the Blue ridge Mountains of North Carolina! May it and the days that follow be wonderful 💕
Happy Birthday from Chicago...
Happy birthday to papa from England
Happy birthday from France!
Happiness and many more years of life for your dad!
From Brazil!
Obrigado!
Happy Birthday from Indonesia
happy birthday from sweden
Happy birthday from New Zealand
I think that to compute the number to a certain accuracy would require you to know the answers to all the problems it could solve, so it's less of an oracle and more of a compression algorithm. And an optimal one too, since it's not compressible further!
A big happy birthday to your Dad from another software engineer in the beautiful Wicklow hills of Ireland.
Happy birthday from Baltimore!
HB from Boulder, CO, USA! coder dads are important!
Oh, it's my dad's birthday today as well. I think he's 76, I always forget whether it's my mum or dad was born in 48 or 49.
What a nice coincidence. :) Happy b'day to both our dads.
Our best wishes to your dad as well!
@@AllAnglesMath He was 75 as well. :) Had a big bbq with the grandkids in the evening. Was really good!
HBD to your dad from Korea!
Happy birthday from Egypt 🎉
Happy Birthday from Hawaii!!🥳🥳🥳
Great video, very well explained; the topic choice is perfect and carries a nice philosophical thought at the end, "I compress; therefore, I understand." is my favorite line. Btw I think there is a mistake at 10:36 since a perfect number equals HALF the sum of its divisors, not twice (actually, quite a bummer that it's not twice the sum, as that would make the problem of proving that an odd one doesn't exist pretty easy lol). Also, happy birthday to your dad. Maybe he can wish for the first 2^75 digits of omega as he blows out the candles on the birthday cake.
You're absolutely right about the perfect numbers. My mistake.
HAPPY BIRTHDAY, MR. ANGLES! from Atlanta, USA
Thanks! Would be weird if that were really our last name. Like the Paul Simon song: You can call me All 😉
Happy birthday to your dad!
Best wishes from Poland - Cracow !
great video
Gelukkige verjaardag, papa! (Evergem, Belgie).
Bedankt voor de fijne wensen!
Happy birthday to your dad from india🎉
Happy Birthday from Columbus, Ohio, USA! ❤
Congrats dad!
Hi, nice explanation video !
I'm currently working on a video on the exact same subject and I have some remarks, especially on the part "how the oracle works":
The Omega that you describe in your video is not defined on a prefix computing model which mean that it is possible that your Omega can be greater than 1 for example if the 2 programs of length 1 (the ones encoded by 0 and 1) both halts they will both add a wheight of 0.5 which will make the total already equals to 1. And by your defenition that would mean that even if there are only thoses two programs which halts the probability of any programs to halts will be 1 (Omegas can be seen as a probability but not in a direct way) .
At 17:48 you said that you work only on the shorts programs which means that you can only compute a lower bound of that omega because there are programs on size greater than n that can have an impact of the first bits. To do so you have to not work only on the programs of size n but all programs of your computing model. Since there is an infinity of programs you can't do the trick of "I do one step of each programs and I start again" because that would mean that you can only performs at most one step of each program, to get over this you can use what we call a "Dovetailling" which works like the bijection between N and N^2.
I know i am being really pedantic about theses little details but as I said, this is a nice explanation video that probably make the whole concept understandable for a person that don't know about it already and all of thoses details can take a while to explain and might hurt your audience retention so keep it that way 😄
Thanks for the interesting feedback. As soon as your video is ready, feel free to post a link here. Looking forward to learning more!
Happy Birthday from Washington State.
Oh Ω, not ω. That's not so surprising, but I'll watch anyway, 😀. Chaitin's work is always worth revisiting!
Happy birthday to your dad! Mine will be 75 next week!
Thanks, and send your dad our best wishes in turn!
Happy birthday from South Africa!❤ 2:15
Happy bday for your dad, from 🇵🇭
Happy birthday from London and a 68 year old retired software engineer.
happy birthday for your dad from Turkey 🎉
Such a beautiful country. Thanks!
Happy b-day from belgium!
Dankjewel / merci bien!
@@AllAnglesMath geen probleem/no problem
A key step in our journey to figuring out if a program halts or not is using omega-n, where n is the "length" of our program and then used to take the first n digits of omega.
A few questions I've been thinking about as a result of this and my thoughts (feel free to chime in):
Questions -
Is it possible that omega has less than n digits? I.e. does omega have infinitely many digits?
Does a random number have to have infinitely many digits?
Why is omega "random"?
Thoughts -
From the video, a key point is that a number is random if (and only if?) it is incompressible. Thus, if a number is not random, we could write a program to write out its digits. If it is random, we could not do that. So, if a number has finitely many digits, we should be able to write a program with finitely many steps to write out the digits of that number. So a random number must have infinitely many digits.
Secondly, Turing proved through the halting problem that we can't have a program determine whether all programs will halt or not. Thus, we can't compress the probability that a random program halts, and so omega must be random.
Since omega is random, it must have infinitely many digits, meaning we could always take the first n digits for arbitrary n.
I'm little shaky on that second jump. Let me know if I'm missing something or can think about it in a different way.
Great video and happy birthday to dad!
Happy birthday from Berlin!
Happy birthday from the Americas
Happy Birthday to your dad from India 🎊
Today is my mom's birthday too
Thank you, and happy birthday to your mother. Wish her all the best from us!
@@AllAnglesMath ❤️
If you Like this video and topic, you should DEFINETLY go and buy yourself a copy of Gregory Chaitin's book - "Meta Math".
It is an amazing book for math and computer nerds in general, but covers the story behind the exploration of Omega coming straight from the man himself, and gives insight into his thought process on discovery and knowledge.
Thanks for the tip!
Late happy birthday to your dad! I'm israeli but I recently moved to the netherlands for a master's degree in logic in UvA. Working hard on giving you some future content :)
Thanks! If you have a link to the research you're doing, maybe we can share it or even consider making a future video about it.
Shalom.
Happy Birthday to your Dad from the panhandle of Florida
Woah we are very close. I'm from the panhandle of Texas. Happy birthday to you, All Angle's dad!
My country has no panhandle, but we do have some funky bits flying off in the north.
Happy birthday to a lucky father, from the Sonoran Desert
Happy birthday to your dad from Taiwan🎉🎉🎉!
Happy birthday from USA!
Happy birthday to your father from Leuven!
Mijn geboortestad!
My late congratulations!
happy birthday to your dad from minnesota 🎉
Happy birthday from Finland, Angle Dad
Happy (3 * 5^2)-th birthday to your Dad from Poland!🎉
Yup, that seems to work out to 75. Thanks!
Good video!
I'm seeing quite a few people questioning the reasoning and results of the last section of the video.
While I'm certainly no expert I wanted to give some notes on where this video is skipping over details (some I think are a bit crucial). This isn't to discredit the video! I understand that some simplifications must occur so the video is accessible:
1. If we actually allow all binary sequences to define a program and add 1/2^k whenever that program halts, notice that this number can be as large as n, when considering just the sequences of length n.
This number therefore isn't a probability. I would urge watchers to think about how you might actually describe the probability over an infinite sequence of options. The answer is to define a measure, which is where the 1/2^k thing comes from.
2. The only reason the 1/2^k thing works is because the constant is defined with respect to a *prefix-free*, *universal* turing machine. You might think of this as the 'programming language', which leads to the different values of omega, but this language has restrictions, namely that if one program p halts, then any program which has p as a prefix (is just p with some extra characters added on) then this program *cannot* halt.
With this restriction, and this restriction only, the 1/2^k computation step makes sense, (what would that mean if these values summed more than 1, how would we know a prefix has contributed?)
3. The way the 'simulate all programs' step is animated wouldnt work, because we would have to execute a countably infinite set of '1st steps' before returning to the 2nd step of the first program. The solution here is to use a diagonal approach, only executing the xth step of the yth program once all previous programs have computed at least the x+1th step. (Search dovetailing on wikipedia and the subsection on infinite sequences if confused)
I found this set of slides helpful for further reading: Search for 'Computation and Thermodynamics - UCR Math'
Again to reiterate - I liked the video! But just wanted to add more detail for eager viewers looking to better understand the concept.
Thank you for clarifying! I must admit that I didn't catch all those details myself.
Bit late but HAPPY BIRTHDAY FROM THE NETHERLANDS!!
Dank je!
A bit late maybe, but happy birthday from the netherlands!
Dankjewel!
happy 75th from Luxembourg! 🎉
Happy birthday all algles dada from Brazil. Me and my girlfriend loves your videos
Best wishes to your dad from Tennessee! (And California)
Happy birthday to you dad - he must be very proud of you!
He is, and it's mutual. Thanks!
Happy birthday from Ukraine. I'm software engineer as well, love math