Berry's Paradox - An Algorithm For Truth
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Sources and Further Reading
An introduction to Kolmogorov complexity and its applications - Ming Li
Infinity and The Mind - Rudy Rucker
www.lesswrong.com/posts/Kyc5d...
en.wikipedia.org/wiki/Kolmogo... - วิทยาศาสตร์และเทคโนโลยี
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You came *this close* to making a link between the halting problem and berry's paradox, which I've never thought of before. Would be cool to link this to Godel's Incompleteness theorem too. Great video!
Lets not forget von neumann
Oh
Or Church and Turing
"Describe yourself"
"Too complex for five words."
I really like this.
@@General12th:-)
Too good👏
I think the most information-efficient way to describe myself is:
"I"
@@vigilantcosmicpenguin8721 LOL!
"So it runs the program and returns the result..."
Ooohhh it's time for the Halting Problem!
It's time for Zeno Machine!
It’s amazing that Kolmogorov is mentioned only rarely, if at all, in online discussions of the greatest mathematicians. His name appears so often at such a high level in so many fields that he’s nearly unique. Among these are probability, logic, topology, fluid mechanics, and others.
Wow
He even gets fashion respect
I only discovered this amazing lady this year & she is amazing. Absolutly love all her videos. I Thank You
Thank you!!
I have followed her a bit longer but I always enjoy what she presents, especially a little min-bending like this one. The argument seems a little like some from Turing or Russell's Paradox.
I only found her this hour.
This is only her second video that I have seen.
Though I am not sure which of the two she posted first.
If a TH-cam creater posts 100 videos and I watch all of then in the opposite order that they were posted, which one can be said to be their first video that I watched? Video 1, or video 100? Couldn't both videos be said to be able to be described that way?
Yea she’s smart and cool.
I bet the other students in her math classes are jealous of her deep understanding of the material.
Well. Time to throw away my solomonoff induction script...
nah it's a cool topic and i'm sure you can cover it differently
Please don't! The world definitely deserves move videos presenting this facinating topic in a digestible way!
@@upandatom Thanks!
But yeah I have so many topics to cover, and it takes me so long to make a video, that if I don't expect my video to be the best one about that topic, I'm better off covering something else.
The plus side is I can now skip ahead and make videos that rely on these ideas, using this one as a prerequisite :D
@@RobertMilesAI AIXI is the obvious follow up ;)
@@RobertMilesAI How can it be the best if you cannot prove it’s the best?
You have a positive vibe gift of explaining things that I have no clue about.. I don't understand half of it but a lot of it sticks too!! Your way of presenting in your videos is so natural.!
This is my first time watching your videos, and this might be my favorite educational video among thousands of amazing ones I watched on TH-cam. I've recently been trying to produce these kind of videos in my leisure time, but there are so much technical difficulties it frustrated me a bit. I just hope one day I can make something good like this. Thanks for the inspiration.
Seems like that FindShortestString program wouldn’t work because of the halting problem. It would eventually come across a program string that it couldn’t tell if it halts or not
The FindShortestString program may never terminate itself. The main problem of the function is that it's making an assumption that it somewhat knows about all possible forms of computation. Let's assume for a moment that I create a new programming language where the number Pi can be generated by feeding it only one 1 bit of information. This way I can basically guarantee that FindShortestString will always return 1 for any input I throw at it because it completely sidesteps any complexity inherent to the process of decomposing that 1 bit of information into the original value.
@@dwightk.schrute8696 Ahh, the Codegolf paradox lmao
@@dwightk.schrute8696 Indeed, the choice of language determines the 'simplicity' of each string. However, regardless of which language we choose, we'll eventually reach programs which are so complex that they *implement other programming languages*, at which point we can use the shortest program in *that* language (plus the size of the implementation). For example, let's imagine we're looking for the Kolmogorov complexity of a string S by searching through machine code programs, but it just-so-happens that the string S only has a short description in Python; we'll call that description P. If we keep trying more and more machine code programs, we will *eventually* start to see programs which just-so-happen to contain Python interpreters; and programs which contain Python interpreters applied to Python programs; and eventually we'll find a machine code program which contains a Python interpreter applied to the program P. This program will be longer than P, but only by a constant amount (the size of the Python interpreter): the complexity of *any* string, measured using machine code, cannot be more than its complexity measured using Python, plus the size of a Python interpreter.
Hence Kolmogorov complexitites can only change by at most a constant amount when switching from one language to another.
By the way, you might like the 'minimalist languages' that John Tromp uses to calculate Kolmogorov complexities, like Binary Lambda Calculus tromp.github.io/cl/Binary_lambda_calculus.html
@@dwightk.schrute8696 This is wrong. Kolmogorov Complexity is defined per programming language. Usually it's left ambiguous with the assumption "for most programs most results are similar, so no need to worry about details", but any concrete number depends on the language or encoding
If it runs all programs, then at some point it will run it self.
So it will be infinite loop.
As a mathematician working in algorithmic information theory, I will definitely be saving this video for the next time I teach an intro course!
Just got my BS and planning to go on to do graph theory - but information theory is definitely one of the areas that calls to me pretty hard. Hopefully I'll have time to study it as well. Plus, who knows, algebraic topology or something. There's too much maths I want to do and I'm only mortal.
@@hughcaldwell1034 A good approach to information theory might be via ergodic theory and dynamical systems.
What kind of "work" are you doing, if you need to be inspired by a video made by an amateur?
@@aaaab384 "Amateur" content gets used a fair bit in teaching courses, because some people have a gift for educating, for explaining clearly and making things click for students, even if their interest in a particular topic is purely a passion for learning thing. These are not always the same people who necessarily specialise in researching a given topic. In fact, it is usually people who specialise in teaching, not in research, that make the best teachers. Go figure. So my question to you is: what kind of work are you doing that you think that showing a video to your class that clearly introduces a topic reflects at all on the quality of someone's research?
@@aaaab384 to be fair I think this when ever I see someone start their Conment saying their in some crazy post at their job Not this person necessarily but so many people just happen to be in the best position possible with all this insight and “humbleness” when they just wanna state their position and that they can relate. I can’t believe I just wrote all this……I feel you lol.
Occam's Razor is best expressed in words as "an explanation must be as complex as is necessary to explain a given phenomenon, but not any more complex than that." In other words, it's not "the simplest idea is the best" but rather "the idea that has no unnecessary elaborations or assumptions is most likely to be the best description of how the phenomenon is generated."
These videos are insanely valuable! This is only the second one I've seen, and I've already subscribed. Please never stop!!!
I have class in 4 hours and I haven't slept yet but I will watch this video because Jade teaches physics like no other.
haha thanks! i hope you enjoy math and computer science too :)
Dude it's math and computer science
She really has so positive and happy vibes, it makes learning fun.
Well so do some sociopaths...
@@orthocoinbitzantium1002 Like politicians . . .
The realization you express at 11:05 is really amazing. Thanks!
Simply Amazing!
Really what else is there to say except you presented it very well; logically linking quite complex ideas ! Wishing you all the success.
You: **looking for some universal algorithm or some complete axiom system**
Paradox: "Oh hello there!"
You: "Oh no!"
Kurt Godel: told you so
Turing: you'll never know when to stop looking...
I bet that one of the dislikes is from David Hilbert
At least Godel and Turing liked it.
Nunez and Lakoff can't decide whether to dislike or like it.
Its from people trying to find pattern in pi for 20 mins.
@@vigilantcosmicpenguin8721 also Tarski !
I do not know who David Hilbert is but I dislike this post because it assumes that everything is ones and zeros which is digital thinking. About 20 years ago I was discussing this with electrical engineers and ask them why we didn't go to trigital, positive one, zero and negative one.
Turns out they've actually started making some devices experimental e that use trigital.
Plus the claim that this includes philosophy is the biggest bunch of crap! Philosophy also includes human behavior and emotions. These items cannot be summed up with one's, zeroes and negative one's.
You need to stop acting like everybody else is a jerk when you are the real jerk!
This is a fantastic video. Finally someone giving credit to Solomonoff's original work! Worth mentioning Levin's Kt complexity (Levin Search) which is a time-bounded version of Algorithmic/Kolmogorov complexity to go around un-computability. Please continue making such great videos :)
Yay^^
She even gives him fashion respect!
A minute in and I guess existential dread is what I’m doing this morning, lol!
Always a pleasure to see your videos.
hehe good morning :)
My psychiatrist: Jade with a shaving cream on her face doesn't exist and can't hurt you
Jade with a shaving cream on her face: 6:52
I didn't catch that day first, was looking away. WTAF?! 😂
still can't hurt you
Just found your channel. You are a brilliant performer. Keep these videos coming. They are fascinating.
You are my favourite amongst youtube educators and your way of explanations and your energy always amaze me
I really love these computer science videos, it's surprising as a computer science student just how much of computer science I'm not exposed to. This video has made me very interested in learning Algorithmic information theory! thank you!!
Yay! You’ve been working on this one for a while!
yes it took me a while to wrap my head around!
@@upandatom You did a tremendous job. Thank you!
Had to write that down, ‘information is the resolution of uncertainty’. Great video, really nice studio / background. I couldn’t follow on a few points (my limited comprehension, not Jade’s presentation) but a lot was covered here.
I hit like at the start of your videos in case I forget to do it later, and I've never came close to regret it. In my eyes you're unique in regards to content presented, I love how you're able to fit philosophy beautifully in your math, physics videos.
Just I thought of watching your videos on Paradox. And here came a New one. Thank God. Thank you. ☺
Occam's Razor is not merely "simpler is more likely" it is specifically, "plurality should not be posited without necessity". Or in less obtuse language: "all other things being equal, the simpler answer is preferred".
Or, the explanation which requires the least assumption is the prefered.
Yes, m00t, but she's explaining why this is the case.
Duh, she's literally explaining why it is preferred instead of just saying that it's preferred. I love how you reduced the substance of a statement while simultaneously saying that there's more to it. Oh the irony.
I usually translate it as, "all other things being equal, the fewer assumptions, the less likelihood for error."
Yes, but Jade chose the simplest way to explain it.
Such interesting topics, and so clear and concise. We all benefit so much from your videos. Thank you.
I feel like the paradox / contradictions are only present when we narrow our focus to partial descriptions.
Synchronicities seem special when we choose to become inspired by individual facets :)
Choosing to become amazed when observations are more/less like a maze.
Ur vids are more inspiring than morning coffee
This was so good! I learned loads and yeah, really fascinating journey, darned uncomputableness. Also, I love your animations, the characters are brilliant.
Hello there
Hello there
Thanks Dom :)
Lol just crap for other shills to copy and make VPN commercials, SHE LITERALLY SHOWED YOU NOTHING
JUST CLICKBAIT.
@@upandatom JADE! I really hope you can please respond to my last message when you can finally.
4:55 The string of 9's gave it away for me
Shame on me, I should've recognized the Feynman point!
@@T3sl4 Shame on me for not knowing it has a name!
It's been a year and a half since I last watched any video of yours. Glad to be back
Wow, what a journey! You've earned my subscription. Kolomogorov Complexity runs into the Halting Problem because it treats data compression like a recusive program. I'm not sure that would happen in a limited non-Turing Complete DSL though. That approach would get you a workable gauge theory if not the TRUE compression. I'm impresssed you managed to turn a math proof that we'll never know absolute truth into a positive and uplifting video. Informtation theory is very transforming and applicable everywhere. Great work!
4:40 obviously I suck at lip-reading. Surely you didn't say what I thought you said.
never ;)
"Only I didn't say 'Fudge'..."
-Ralphie
An AI that asks itself AI Paradox?
Earth: It'll call themselves human.
Wow you really blew up now. I remember watching you at only a couple thousand subs I'm really happy to see your channel grow :)
I love the way you talk, gestures and all! It's incredible!
Why can't my profs be as simple as you and as interesting as your goofy animations
Great video!
Really appreciate it clearing so many of concepts!
4:56 I can't believe I actually knew this. That's knowledge that's been waiting 15 years for an application!
Brilliant video! It's truly grand seeing how philosophy, math and everything comes together!!
Excellent video. Very clear on a tricky topic.
Jade: What' the biggest number you can think of?
Me: 6
I think you don’t even need to go as deep as you did to find it impossible. Wouldn’t the halting problem make FindShortestString impossible before even getting to Berry’s Paradox? Once FindShortestString reaches code that loops infinitely, it could never proceed, since it would be impossible to tell if the code would be looping forever (so neither a right or wrong output string, just no output) or if it would just be running for a really really long time.
Wow... How delicate the concept is! Interesting!
13:46 . so, the way you helped me to truly comprehend this...thank you for your videos. you and your entire team
Great video! This is quite similar to-and in my understanding what Alan Turing postulated through-the halting problem paradox.
I didn't like math in high school but you make the subject so entertaining and understandable!
Amazingly written, explained. Really great video to explain a mind blowing stuff 🔥
Thank you, glad you liked it!
This channel is amazing, well presented, simple explanations, very entertaining.
This discussion makes me think of the Mandelbrot set and fractal equations . Great Video .
Science is important to spread and i often offer Recommendations,
but today i wanna do it a bit differently and try to bring-in People to watch 'some More News',
a Satire-Version of corrupt and biased News-Channel.
Just like Scientists bring Attention to Climate-Change and such Issues,
that TH-camr bringts Attention to Homelessness and various other Issues.
Unbiased and informed, his Talks about 'Obvious Solutions to Obvious Problems' are a Masterpiece.
It's always a better day when there's a new Up and Atom video.
That statement is universally true.
A fascinating presentation, Jade! As I was watching it, I was reminded of a poem I wrote. I think I should dedicate it to Ray Solomonoff! :)
*Majestic sunrise*
It’s sad but true that one can be
adept at counting numbers,
yet at the same time fail to see
which ones make up the wonders.
All things in life may be defined
as strings of ones and zeroes.
Discernment of a finer kind
transforms someones to heroes.
I love it!
@@djpete2009 Thanks :)
This is great. ^-^
@@momom6197 Thank you for that :)
I love how technical yet newbie friendly this channel is
Deeply interesting and wonderfully presented!
You know the video is good if it has an undefined Like-Dislike Ratio
infinite
"I took out the first 99, so it wouldn't be obvious for the mathematicians out there"
That was a sucker punch!
She actually took out the first 100 decimal places (101 digits), and I recognised it almost immediately because I memorised it to 100 decimal places a while back, and then decided to keep going about a year later, but stopped at about 120. If she'd cut off a few more digits, I would have had no idea.
I'm not a mathematician.
Another problem I have with the Find Shortest String algorithm (even if ignoring Barry's paradox), is that given a large enough number of bits, it'll eventually run into programs that do not halt, and since in general it's impossible to compute if a program halts (as proven by Alan Turing), it's possible for Find Shortest String to run a program that doesn't halt, and runs forever on that particular number of bits.
Very nicely presented! Well done!
Awesome video! I love your work, Jade. Question: around minute 8-9 are we risking undercounting complexity when we convert complex objects to strings since each programming language or 'context' contains information that we overlook or undercount in that conversion?
Practically yes, but mathematically these proofs are done with Turing machines which are defined by their ability to mimic any other program, which in this case requires the simplest way possible.
POV: you’re a computer, but they won’t stop asking you philosophy questions.
Fabulous job Jade.... I love a nice paradox for Sunday lunch.
Came back to watch this again. Your videos look so much better by the way
Hey Jade, good to see you after so long!
thanks :)
@@upandatom :)
What are the paintings/pictures you have in the background? And where can you get them? :)
displate.com
So cool, yet so simply explained. Thanks!
The first problem with Find Shortest String is the Halting Problem. No way to compute if the program bring tested will ever stop and give an answer.
I feel like the context (compiler) used to interpret information needs to be taken into account. For a specific compiler the kolmogorov complexity can be computed using the technique in the video. But if you can pick the ideal compiler for an input string then the complexity of any string is 0 as the compiler could just hard code that string when the input is empty.
"For a specific compiler the kolmogorov complexity can be computed using the technique in the video." Not for a language that is Turing complete, because the program could run forever and so the algorithm wouldn't go on to the next string.
@@b43xoit ah, right
Actually I think it’s taken into account mathematically, because yeah the shortest program to print string x is ./printx. So you can either shove the complexity into the compiler / language or into the algorithm, either way it’s a constant complexity.
Something disproved is equal to something proved! So I am pretty sure that the old version would not tell his young version to stop. He would have appreciated him, just like we did!
Thanks for the great content.
Very well done. Thank you for the great information and great laughs. :)
Efficiency and realistic computability can always be problems with very abstract math. Machine learning has an efficient algorithm for computing something like this it’s called and encoder decoder. The encoder, in a sense, creates the program and the decoder runs it. The real problem to make this general is the have a language that describes every possible encoder decoder architecture.
Great Vid as Always!!!
Your channel is pure gold. Your teaching skills are unique and you don't hesitate to introduce viewers to topics that are seldom approached by other channels.
Many thanks to you and your team. A TH-cam gem.
Thank you so much :)
This is a really well made video. Thank you it!
Love the way you explain everything ❤️
It’s the truth there’s no algorithm for truth!
As always a great video; your presentation was elegant and clean👍🏽😊
thank you I appreciate that :)
Hmmm, Fakebook and Twitter seem to think they have truth algorithms, no?
You know Jade, it's unkind to make jokes about shaving. Some of us have to do it four times a day.
Joke's aside, this video *looked* gorgeous, loved the little text effects, colours and all the graphics. Worth the wait!
For surgery, I suppose.
Thanks Rohin :)
Maybe try a new blade?
Cringe
One of the most brilliant videos I've seen. Tears in my eyes, literally. Big thanks! Keep on...
Science is important to spread and i often offer Recommendations,
but today i wanna do it a bit differently and try to bring-in People to watch 'some More News',
a Satire-Version of corrupt and biased News-Channel.
Just like Scientists bring Attention to Climate-Change and such Issues,
that TH-camr bringts Attention to Homelessness and various other Issues.
Unbiased and informed, his Talks about 'Obvious Solutions to Obvious Problems' are a Masterpiece.
Ahhh, my favorite YT math teacher. Always ready to learn. Thanks Jade. And thank you
Wait, wasn't the same thing proven earlier in Gödel's incompleteness theorem, and Turing's halting problem?
Yes it is. A shorter proof. If I have an algorithm for truth, that algorithm can definitely say whether a program can halt or not. So an algorithm for truth should be able to solve the halting problem. But Church-Turing thesis implies that halting problem can't be solved. Yaah and Church Turing Thesis I guess was solved in 1930s
They're definitely related. The reason FindShortestProgram doesn't work is that it will have to look at some programs which loop forever, but (due to the Halting Problem) it has no way tell those apart from very-long-running programs, and hence it can't choose which programs to 'skip'. Levin Complexity is a computable alternative to Kolmogorov Complexity, which takes (the logarithm of) the running time into account; hence it's able to 'skip' programs which take an exponentially long time (whether or not they happen to loop forever or not)
Algorithmic Information Theory actually takes things a bit deeper than Gödel and Turing. For example, Chaitin's incompleteness theorem proves that (a) the amount of 'algorithmic information' in a system of mathematics limits the theorems it is able to prove, (b) the algorithmic information of a system of mathematics comes entirely from its axioms, (c) the information in those axioms are essentially random bits, which we're free to choose either way. In other words, theorems are just re-statements of the axioms; and are hence assumptions.
There isn't too much for differences between The History of Maths and the works of HP Lovecraft
To the point where comparing famous mathematicians to Lovecraftian protagonists is typically considered not funny for the reason of "that is an old joke--heard it too often--now it is lame"
@@chriswarburton4296 Church-Turing thesis relies on the idea of "computable" functions. Sure, in the classical model of computation, which either humans, mechanical machines and digital systems can do, that is the case. Although a bit recently, there was a breakthrough in theory of math and physics: th-cam.com/video/HL7DEkXV_60/w-d-xo.html
So maybe if (in the near future) we define "computation" as quantum computation, this thesis no longer holds.
This is just my own understanding of the topic. Would love to hear your opinion on it too!
6:14 That's protein folding and the riemann zeta function, both known to be involved in hard to solve problems. But can anyone figure out what chemical that is on the right or explain which problem it represents?
All I know is that it's an acid.
Chucking the structure into a Google image search narrowed it down to something in the fluroquinolone family of antibiotics, that may have been cheating though
What are the equations on the pictures in the background?
I can only identify Entropy, the cool equation, and Schrödinger Equation. The third pictures I can't place. Maybe something with gravity because of the big G?
edit: nevermind, it's general relativity en.wikipedia.org/wiki/General_relativity
Now I'm curious too!
I've always been quite fascinated with kolmogorov complexity and one thing that intrigues me is the fact that I cannot entirely state the difference between this and the halting problem. Is it a different form of the same thing?
Thanks for your videos
Love your videos Jade!
2:58 "As you can't describe all numbers, it logically follows that there must be a biggest number that you can describe"
That's incorrect. Just suppose I can only describe every other number. In that case I cannot describe all numbers but there still is no biggest number I can describe since I can always jump to the next one I can describe by adding 2 to the last one I could describe.
Unless your life ends the moment you just added 2 to the last one and the description ends abnormally. Which definitely will happen at some point on your quest by ALWAYS jumping to the next one.
@@nickdsp8089 That only works if we are talking about the numbers I actually count. What I took Jade to mean when she said "The highest number you can describe" is not the highest number I have or will ever describe in my lifetime, but rather the highest number I could in theory describe (without having to count up to it or whatever). Even if there is a highest number I will have described at the end of my life, it is not necessarily the highest number I could in theory have describe.
@@austinpowers7670 You don't have to count up to it, but you do have to describe it; and that takes time (and symbols, if writing it down).
> it is not necessarily the highest number I could in theory have describe.
Yes it is! You say "in theory", but don't say *which* theory. The relevant theory in this case is Algorithmic Information Theory, which says there *is* a highest number you can describe. In fact, Algorithmic Information Theory says there is a highest number that *mathematics* can describe! For example, see risingentropy.com/are-the-busy-beaver-numbers-independent-of-mathematics for a number which is easy to describe (the 7918th BusyBeaver number), but whose actual value is impossible to calculate in ZFC (i.e. the "usual" axioms of mathematics). The same BusyBeaver trick can be used to defeat any system of mathematics :)
Nice catch!
@Chris, algorithmic information theory isn't invoked in the syllogism Austin is criticising, so it's outside the scope of the conversation. Chris isn't claiming that there is no largest describable number, only that the given statement doesn't prove that there is
8:28
Me, an intellectual: ah yes, string theory.
Don't think these are the same kind of strings, though.
@@benurm2390 yes, that was the joke.
I'm sure I can find a shorter string if I just cut a spool of thread.
Most of it is over my head. But your presentation is so ordered, that i glean something interesting from every video!!
Oh this was delightful. Right up my alley. I enjoyed this.
Yeah. All of science comes down to running 7zip.
Or maybe not.
Oh, 7zip doesn't do compression that way, it is statistical compression with a rolling window applied on parts of the total stream (and the output, yes, it mixes input and output, that's the cleverness of data-compression in the real world), think of it as a clever Morse code generator, it just does a bit of ordering (Huffman trees) so it takes less bits than before.
Its basically "cheating", because, of course you want 7zip to finish running before the heat death of the Universe.
When ya first posted the picture of you shaving, I thought it was a [INSERRT RAZOR BRAND HERE] sponsorship.
haha how would that sponsorship have come to be?
I loved the discordance... Explaining simplifying things while a quick unexpected clip raises a bunch questions. O_o
@@upandatom Dollar Shave Club?
Wonderful and informative, as always! Love your enthusiasm and excitement for the field that you are in! And, I would probably lose to you at rock, paper, scissors 🤣
Very well explained. Thanks Jade.
Didn’t expect to see Devin (Legal Eagle) to make an appearance 🤯
Where tho?
@@fajrulramdhan2005 In the bit with everyone on a call
Edit: 8:14
3:21 Liberryan... I see what you did there.
:D
Hi!
- Can you take a picture of anything in an infinite resolution? Like every single atom in one go, and more?
- It would produce countless contradictions. (15:29)
- You must find a suitable resolution first.
As a programmer, I denote series of 0|1 rather as *streams* (in this case, a bitstream) to avoid confusion with strings.
Because "String" in almost every program language is used for an array of characters (usually one or two bytes) representing some kind of text containing all letters of the alphabet (and some more).
Outro music slaps tho!
You forgot to mention how this is related to the Russell's paradox and co.
And here is question. Is the Occam's razor a "truth" that shaves all those truths, and those only, who do not shave themselves?
And by the way,. all of this comment is a pure lie.
this is actually a work of genius
Ah you barber you :)
I left two comment for her, check those out, involves Russel too.
There is actually no paradox in the folliwing
I mention that a certain town has a barber who shaves every man who does not shave himself.
...
Clue: think woke
...
Clearly the town has a lady barber.
Next attempted paradox please...
@@trueriver1950 Actually yes there is no paradox even when the barber is a man. This issue is more serious than you imagine and there is no place making jokes about it. It cost 3 times world war and we are in third one. Principles, value of information, certainty of human difference are replaced with practicality.
First we see this approach in history with Greek philosopher Parmenides; then the Roman empire followed the same rule with power and cruelty. Russell applied it to knowledge and science by using modern mathematics.
We are in the same cruelty but the specialty of human difference and solving problems with certainty do not exist anymore.
Roman Empire was demolished Istanbul with another type of culture difference; now Russell’s modern politics will be demolished from the same type of culture with certainty.
This is the main reason why compression algorithms must do tradeoffs, because they can only approximate the output of find_shortest_string.
This is also similar to the reason why compilers can only partially solve the halting problem, usually by doing Control-Flow-Graph analysis
I was (in my head) essentially *screaming* "halting problem" shortly after 12:00. But that led me to another thing: The halting problem *is* solvable under some constraints: Specifically, you can simulate a finite state machine without inputs until it loops or halts; and a Turing machine with a finite tape is just a finite state machine with no inputs while running again. So in the worlds of computers, "all you need" is enough memory to detect loops, which funnily enough only requires double the amount of memory (plus some constant offset for the cost of simulation itself), not counting the description of the simulated machine - if all we need is the algorithm to terminate, we only have to run the FSM for as many steps as it has states; since if it's still running after, it has to have visited a state more than once, which, since it doesn't get any input besides the initial state, would mean it got into an infinite loop.
So now I'm wondering: *Could* you define such a complexity not just by program size, but also memory and/or computation time size? That seems as though it ought to be computable, though still not computationally feasible. Or did I make a mistake somewhere in my logic?