The Search for the Longest Infinite Chess Game
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- čas přidán 17. 05. 2024
- We explore extremely long Infinite Chess games, starting with Mate in Omega, and progressively climbing to higher and higher transfinite ordinals!
Play Infinite Chess at: www.InfiniteChess.org
Support me and its development on my Patreon, and get your name in the credits of my videos: / naviary
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Access the spreadsheet board editor: docs.google.com/spreadsheets/...
\/ Credits for the positions shown, along with links to learn more about them! \/
Omega, Omega^2, Omega^3 by Joel Hamkins and C. D. A. Evans: math.colgate.edu/~integers/og2...
Omega*2 by Noam. D. Elkies: mathoverflow.net/a/63841/514442
Omega^4 by C. D. A. Evans, Joel Hamkins, and Norman Lewis Perlmutter: math.colgate.edu/~integers/rg4...
Omega_1 Proof by Matthew Bolan: mathoverflow.net/questions/63...
Subscribe to his channel! / @matthewbolan8154
Some clips are from Joel Hamkins presentation, “The theory of infinite games, including infinite chess”, watch it here! • TMWYF: The theory of i...
See more presentations on The Infinite Games Workshop: / @infinitegamesworkshop...
Note for the chapter “Climbing the Ordinal Tower”: I often mention “this to the power of this infinitely…”, or “infinite epsilons above this…”. When I word things like this, what I really mean is to take the limit, or supremum, as we approach doing that operation infinitely. I have chosen this wording to make things more intuitive to the viewer and those who may have zero ordinal arithmetic knowledge. This chapter is not meant to be a detailed explanation of ordinal arithmetic.
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Chapters:
0:00 Introduction
1:18 The Game Tree
3:21 Omega
5:17 Why Ordinals?
8:04 Omega+1
8:58 Omega*2
10:41 Omega^2
12:27 Omega^3
14:48 Omega^4
20:20 Matthew’s Proof
24:09 Climbing the Ordinal Tower
26:53 Conclusion
Other stock footage is from Pixabay.com
If I missed, or incorrectly gave credit, please don’t hesitate to contact me using my email! Located in the "About" section of my channel, click "View email address". - Hry
Imagine being foolish enough to blunder checkmate in ω^3•10+ω^2•8+1001
🤦♂️ "You blundered! How could you not see that?"
@@Naviary Imagine being foolish enough to blunder checkmate in phi(psi(W^W^W^W^eZw^w^(w^2)*2),0,n0,23,193)
PS: W is capital omega, w is lowercase omega, e is epsilon, Z is zeta, n is eta
@@NaviaryYou fell for one of the classic blunders!
Hikaru be like:..... Yes that's checkmate in (imagine the amount of all real numbers)
Rookie error lululul
I love how you start with something that resembles chess and end with massive structures designed only to prolong the inevitable.
The lower quadrants have been lined with bishop cannons of increasing size
And yet, a math brain loves it
thats such a good way of describing how my brain is reacting to this.
That awkward moment when you blunder into an infinite number of rook locks...
Just like real life
the moment terminology turned to "towers" and "cannons" i think this stopped being about chess
'What is this piece?'
'Tower.'
'ITS CALLED DA ROOK'
Tower is also a valid terminology, it is the name of the rook in many languages, also cannon is the name for the rook equivalent in Chinese chess although it moves differently, in this video's case the cannon name is appropriate to me as the rooks go as fast as a cannonball !
soon we’re gonna have nuclear warfare in chess if we keep goin higher with these ordinals
Those massive structures weirdly remind me of the game of life
@@tigerghg7302 chess has become cellular automata
A higher game value does not necessarily mean the game will take longer, but instead it means your opponent can be more annoying
It does beg the question, how long could a game last if black tried to play _as badly_ as possible? I.e. trying to get itself checkmated
@@richardpike8748 2 move mate
@@richardpike8748 ig depends on position
@@eeeee11235 not in infinite chess, since the king can just move backwards
E
It's kind of beautiful, that two kings are just having a staring contest, while all of this is happening
Bet after the 198,298,171,372,287,394,291th move they would wish they could just lunge at each other and have a fist fight
Typical monarchy forcing everybody else to dedicate their lives to them lol
The UK has no problem with that. Sucking money like leeches. Very impressive.@@fntthesmth423
shout out to their patience because doing absolutely nothing all of that amount of time is huge‼️‼️
"Now, now, there's no need to fight... why not settle this over a nice cup of tea?"
Of all things, I never thought I'd hear the phrase "bishop cannon" in my life.
BROTHERS! WE NEED TO CRUZADEEEE
I searched "bishop cannon" and apparently there is someone who is a bishop and named Cannon.
en.wikipedia.org/wiki/William_Ragsdale_Cannon
Oh man, I really hate when I'm just casually playing and suddendly Stockfish tells me I have a mate-in-Omega-1 minus 1 position.
Omega-1 minus 1 is just Omega-1
@@gianglai7346actually omega-1 minus 1 is ill-defined
@@barrianic4 actually omega-1 minus 1 is e =m2
@@aaravthediscoverer omega-1 does not have an imediate predessessor because it is a limit ordinal
@@barrianic4 but does it have an immediate successor?
26:13 After infinitely many levels of incomprehensible infinity, we finally reach SMALL Veblen ordinal, what a journey to reach something that's literally called small.
That’s because it is still countable (there is a one-to-one correspondence between its elements and the natural numbers). ε_0, mentioned in the video, is also countable, and ε means small in maths. So it is also “small”, in the sense of still being countable.
The real numbers are uncountable, so there are more real numbers than announcements and types of announcements in Infinite Chess.
@@-minushyphen1two379the large veblen ordinal is also countable
bruh
that's what she said before she left me
@@-minushyphen1two379 the large one is also countable
The later stages begin to resemble Conway's Game of Life :)
In some ways it does resemble it doesn't it?
Well yes because of the nature of forced mates restricting the moves when assuming optimal play, choice is lost and it becomes a cellular automata which is very interesting
@@Naviary now I'm thinking if infinite chess is Turing complete
It's the game of life, except it's more complex
@@shauas4224 I wonder that too now
I went in expecting a relatively standard chess video, i was not expecting you to basically recreate Vsauce's "how to count past infinity" video within the language of chess. That was an absolutely incredible watch, i applaud you on your efforts!
Same😂😂
I've finally found an area where transfinite ordinals are useful
"useful"
@@maldoror-13to the gods
*semiusefull
Now to create a starting position where reaching said useful situation is actually plausible
geometry dash theoretically possible levels as well
The intro with the infinite chess game zooming out to show the text is so cool
3.1415? Pi? 3.14159 this is pi followed by!
@@findystonerush9339fun fact pi is TINY it has a lot of digits but it is small because it starts a with 3 so if you round down in 3
It is possible to get mate in ω₁. Just not in any actual chess board. All you have to do is give the board to the opponent and tell them they have to set up a board in which you have the winning move. They have access to every countable ordinal move count, and so the move counter when you give them the board is ω₁.
However, you have to give them infinite time to set up such a board. Unfortunately, if you restrict yourself to the boards that can be represented by a bounded amount of information, this is suddenly a countable ordinal again. You must afford them a literal eternity to make this particular announcement for it to truly be mate in ω₁. They have to _actually be able to spend this eternity_ in order for it to work. If you just afford them an unbounded amount of time, you force them to make an announcement that decides between a countable set of countable ordinals (each being the best they can do if given n years), which is just not good enough.
Ordinals are so trippy sometimes. I suppose if you let the opponent to set up the chessboard, it would be "mate in omega_1", since omega 1 would be the smallest ordinal greater than all the others.
@@neopalm2050If you give someone time to set up a board, is each moment an announcement because you have to go at a finite speed, but there's no limit to how fast you can be, unless you account for the speed of light. But there are probably ways to set it up so the same thing happens but without that nuance.
@@danielyuan9862 I was imagining a situation where the only real announcement would be the actual board state. Anything done up to that point, they could take back. I was also assuming there was an upper bound on how often information can be set (information that determines the board state).
What you are describing is not a mate-in-ω(1) at all. Indeed, what you are not describing is not even infinite chess to begin with.
It's important to understand that there is an ordinal (omega^CK, Church-Kleen ordinal) lower than omega_1 (noted as capital omega in the video) that is no longer recursive (i.e. it can't be reached by the construction shown in the video).
As a result, the plateau required for an omega^CK mate is strictly incalculable. In other words, there's no way to describe the position of the pieces unambiguously ("describe" in an algorithmic sense).
As a result, mankind will never know a mate in omega^CK (however, all smaller mates are feasible).
Edit : In the replies to this comment, there are some very pertinent remarks for you to read, including a reply from Matthiew.
( PS: Incidentally, no program would be capable of calculating a sequence of mats whose ordinals tend towards omega^CK (otherwise, we could use them to construct omega^CK mats). In other words, it can be shown that there is a mate shorter than omega^CK that mankind will never be able to achieve because of the computability of the Universe.
The meaning of a non-recursive ordinal is very difficult to grasp in this context, since we're dealing with such large infinities that the consequence of what I'm saying is not perceptible. )
( I'm using a translator to express myself, as I'm French and the terms are getting a bit technical, so I hope it's still intelligible. )
Among computable positions (once you precisely define computable positions) omega^CK is an easy upper bound, and my construction suffices to show you cannot do better. If you do not ask the position to be computable, then for any countably branching well-founded tree my construction gives a position with game value equal to the rank of the tree, so all countable ordinals indeed occur as game values of some (not necessarily computable) position.
In fact, we can say a little more. Given as an oracle a function f:N -> N such that the image of f is well ordered under the Kleene-Brouwer ordering and of order type alpha, my construction shows that there is a position, computable relative to f, with game value alpha. This shows that restricting to positions of any level of the lightface Borel hierarchy (e.g. computable, Sigma_2, arithmetic, hyperarithmetic, etc), the correct upper bound is the supremum of all ordinals belonging to that level.
You can describe it. You just need to reach beyond anything equivalent to the standard turing machine operations to do so. Non-computability doesn't stop the busy beaver function from being expressible. You just can't write a program that generates them (or even prove what numbers it outputs past a certain point).
I will edit my comment tomorrow if I don't forget (it's night here).
Thank you - its a very informative comment :) - heard about cardinals and ordinals , little and big omega notation but really missed the Calvin-Klein definitions (intentional typo). I am really not sure what do you mean by "computable" - someone referred to turing machine idea but i have no issue with having a power of ω wide computer register or just write a sentence (function) that states i can browse the whole board in an instant and calculate the formula on it :). The universe has ONLY got 10^80 atoms, but the quantum deterministic wave function has been immersed in a Hilbert space i see no issue with saying that everything is achievable just by creating and idea and truly believe it :). Thank you!
Hmmm grave interessant ça aussi
This is what I’ve been working on the past several months! I couldn’t split the topic into more cliff-hangers on you guys, so here’s covering it all! By far my biggest project. I hope you enjoyed!
Consider subscribing 😉 And come join the discord! discord.gg/NFWFGZeNh5
u have my respect
Would it be possible to reach a higher checkmate clock with custom made pieces?
@@wesleystoltz8421 Unfortunately not, with only countably many infinite squares on the board, you can never create a piece that can move to uncountably many squares, which would be required to reach Omega_1. The exception is you would have to create a piece that can make infinitely complex moves (like, chain infinitely many moves into a single move). Infinite Checkers has this property, and can reach uncountable ordinals!
@@Naviary Ok now I need to see the video on infinite checkers 👀
In Infinite Chess, you could get a position Mate-in-ω_1 if there is infinite pieces on the board. This has been already proven. Although, with only a finite piece, you can't make a position with Mate-in-ω_1. Keep in mind that some Mate-in-x position have the value of x greater than ω_1^CK.
Man i’ve not once left a comment on a CZcams video ever, but this vid was actually just too incredible too not praise you for. This almost makes me want to try and construct an omega^5 (and higher) position. Great video as always man.
Thank you, I really appreciate it!
We've seen how far we can go with infinitely many pieces, but how far can we go with only finitely many pieces?
@@objectshowfan362 This is still an open question! We know so far that omega^2 is possible.
Yeah - i am sure that many of the 4000 Grandmasters wouldnt like to deliberately analyze this knowledge - still i am sure that if i played infinite correspondence chess and i trained on ω tactics there would be someone who learned the whole set of ω^2 and would beat me i wouldnt know how :)
@@objectshowfan362 It is bounded by Church-Kleene ordinal (the first nonrecursive ordinal, also the supremum of the recursive ordinals)
the moment of history, the third naviary's video
Only countably many videos until the ω-th video
Unlike **some company**, he actually know what comes after 2
This is a great way to explain infinity. Most people don't understand it but starting from something that we can feel and showing that it can get beyond anything we can imagine yet still never reach true infinity is very satisfying.
People misunderstand than the "infinity" that shows up in most conversations about infinity isnt actually infinity its actually absolute infinity wich is the last number before we enter the realm of imaginary numbers
Insanely well made! This truly is a game for the gods and we have only begun to scratch the surface :)
Thank you!
@@Naviary, I tried joining the Discord, but it said that I'm unable to accept the invite.
@@RickMattison314 That's weird. It should work! Maybe try a different link? discord.gg/bWbgYqX7Re
@@Naviary, still nothing.
Edit: NVM. It worked on my phone.
@@RickMattison314 Great!
At 15:05 Hikaru be like: pf that's a simple forced mate in ω⁴ position.
Joke aside: that ω⁴ was very entertaining to watch!
"we can have a game length of any number we desire, even exceeding the time in seconds until the heat death of the universe. we just have the move the rook that many spaces away" is such a crazy and hard ass sentence
The worst thing in Infinite Chess is probably the Bishop sniping you from 45 multiverses away.
This was such a good video, the ending sort of reminded me of 17776, and how the people in that story play thousand years games of football. I could also imagine them playing those really long infinite chess games
I had this exact thought about Jon Bois
Never heard of that before. It's great!
Thanks for the tutorial, now I know how to deal with this when it comes up in my games!
Why does this feel evil
10:58 Tier 2 announcements were a plot twist I wasn't expecting. Great video and great narration!
The real question is why you'd sit through omega^4
Even if I’ve watched Vsauce’s video on infinities that talks about larger ones, this video still blew my mind. To say this is well made is an understatement. Omega/10 video.
omega is a ordinal, not a cardinal
@@godofnumbersakausername5226 lol, good point
This is honestly one of the best mathematics videos I've seen on youtube. The only thing that could be considered missing, in my opinion, is mention of the difference between cardinal exponentiation and ordinal exponentiation; it'd call back to the "least ordinal greater than all finite ordinals" from before, while giving some context to why omega^omega is still countable while 2^aleph_0 is no less than the cardinality of omega_1. I don't think there's any good place this could fit within the video (because you went on to very concisely describe all countable ordinals), and seen as you did a great job with the script, I don't think adding it would make the video better than it currently is, but it did come to my mind. Can't wait to hear more come from this project in the future!
Thank you
If there's one thing I would have included more, it honestly probably would have been greater explanation of ordinal arithmetic! You are correct with the script being a little tight, not sure where I could have paused the story to explain arithmetic. More videos will come!
2^aleph_0 is the cardinal of a set of applications from a set of cardinal 2 to a set of cardinal aleph_0, such as bit sequences. A sequence of bits contains an infinite amount of information.
You'll notice that all the elements of omega^omega are written with a finite amount of information. So it's more analogous to the set of finite bit sequences (wich is countable).
@@abellematheux7632 This is inaccurate. A sequence can be encoded entirely with finite information only, using a recursion. In fact, trying to think of cardinality as being about information begin with is incorrect.
@@angelmendez-rivera351
I denote F^E the set of applications from a set E to a set F.
Let beth_n be the (ordinal) sequence of cardinals such that beth_0=alef_0 and beth_{n+1}=2^alef_{n}. Let X be a set of cardinal beth_{n}, and the set 2^X={0,1}^X is of cardinal beth_{n+1}. More generally, let E be a set of finite cardinal, E^X is of cardinal beth_{n+1} like 2^X.
Finally, if X is not in bijection with a set of the form 2^Y, then X is in bijection with a union of sets all of lower cardinal than X and all of different cardinalities. For example, the union of sets X_n of cardinal n has cardinal beth_0.
All elements of omega^omega can be written with a finite amount of information, i.e. with a finite number of characters in a finite alphabet. However, the number of characters per element is not bounded.
If there is no way to represent the elements of omega^omega by sequences of characters in a finite alphabet such that the number of characters is bounded, then omega^omega is not finite (obvious). However, omega^omega is in bijection with a set included in the set of finite sequences of possible characters in this alphabet. By denoting this alphabet E, X is therefore in bijection with a subset of the union of E^n, making it a set of cardinal beth_0.
2^beth_0, in turn, is in bijection with a set of the form 2^X.
In E^X with finite E, I like to call E the alphabet and its elements characters when I'm vulgarizing. So, to compare infinite sets that look like E^X, just compare the cardinal of E. I like to call the cardinal of E the amount of information needed to write the elements of E^X. It's as if, for f belonging to E^X, we wrote, for each x belonging to X, f(x).
Of course, this is a vulgarization procedure. In reality, we don't really write down this amount of information. But it does help to recognize the size of a set: the elements of R are written with beth_0 decimals, those of R^R with beth_1 reals (which themselves are written with beth_0 "information"), and those of Q with a finite number of digits.
I really hope I've made myself understood. It's probably just a misunderstanding of my intention and the way I use the words "sequence" and "information".
I don't blame you for criticizing me, of course, and you can tell me if I wasn't clear.
I'd like to point out once again that I'm very bad at English and that I use a translator, which can be a big source of misunderstanding.
8:04
Oh hey, it's the Code Bullet song!
8 by 8 chess is just 2 groups of children fighting (with a leader) and that GM's there is just super smart leaders.
But infinite chess is THE REAL BATTLEFIELD between empires in the multiverse
Several universe died during 1 turn of Infinite Chess. The only epic chess battle you couldn't not miss.
@@xaf15001 you couldn't not miss?
@@DoNotSin Yeah, you had to miss it, you'd be dead long before turn 1 finished 🤣
I love how you played a nuclear alarm in the background when talking about omega^4.
started laughing at omega³, checked how much of the video is remaining, oh boy
OOOOOOOH now I see how my confusion about omega + 1 from the last video gets resolved. It's very elegant!
Glad I could clear your understanding!
Please tell me why the big boss number at the end that can never be reached (omega 1) is also called OMEGA AND absolute infinity @@Naviary
This Omega principal is actually relavent to the game Magic The Gathering, and is inbuilt into the rules. Essentially, in that game, it's very possible to generate infinite loops and combos. In that scenario, the way the rules work is that once you demonstrate an infinite loop, you are then allowed to shortcut actually doing that loop N number of times, where N is a number of any size of your choosing. The "priority" is then given to your opponent, who can agree, or name a smaller number that they will choose to interrupt and intervene at if they have an action that can do so, which is rare after the first loop.
Because of this, it's not extremely rare for monsters to end up with a billion power, or to give yourself a googol health, etc etc
I haven't played Magic, but that is actually quite interesting!! For certain actions it allows you to pick an arbitrary amount of steps to repeat that action?
The same is true for yugioh as well! They have the same rule of “demonstrate a loop once to show that it’s infinite and then declare how many times you are going to perform it”
so this is what the memes mean when they say that magic the gathering is turing complete
yeah you need to demonstrate both that you can create a loop, but also that you can choose to stop the loop, otherwise you either win, lose or draw game depending on the loop's effect on both players' life total
@@StriiderEclipse I thought they just banned cards that cause infinite loops? (Freaking Pole Position, man.)
Imagine getting skewered by a bishop on the square b925836
Or if after a is aa and then ab,ac,ac… until zz then aaa,aab… you can be skewers at guem204593
This is basically just what happens when you're really determined to NOT lose.
20:26, pitch perfect delivery lolol
i love how with this configuration you could hide a bishop extremely far away, get it onto position and then snipe the queen from 3 kilometers away
That's why you never resign, kids.
Hey look that's me.
Thank you for your contributions!
Minecraft youtubers are going crazy these days.
I am your new subscriber who found your channel from this video
Right as I was rewatching your previous videos you drop yet another banger, great work as always.
The way you explained this made it so simple to understand and still very interesting, this is a "5-tier" video 🙂
Mate-in-Omega^4 reminds me of hostage situations in movies.
"I hate math"
"I hate physic"
"I hate chess"
"Is non-exist"
I just had this idea: If we start with the normal arrangement of pieces, we have two rooks. And since the board is infinite, we can't promote our pawns! Well, most (almost all) of the positions shown here are impossible anyway lol.
Great video! I loved it! I have always loved the concept of infinity. You got a sub!
Thanks! In competitive play, the current rules allow promotion at the normal ranks 1 & 8. But yes, for the positions I showed, there was no promotion, and pawns never have the opportunity to queen...
But what about infinite chess 2, with infinite amount of pieces?
Really looking forward to the in-game board editor. Thanks for the amazing vid! Mind is beyond blown with how complex these positions have to be
Now make chess but pieces can move to decimal values of spaces instead of just whole numbered spaces, allowing for an infinite amount of spaces between each space.
Eg inbetween the numbers 0 and 1 there are an infinite amount of decimals, so one could conceivably create an infinite amount of ordinal number checkmates between them
This seriously has to be one of the best videos I've ever watched. I normally don't ever leave comments but this deserves it. I was drafting a lot longer of a comment talking about all the little details between the script and the editing I noticed that made it great, but it was getting too long so I'll just say that I noticed them and leave it at that. Great video, ω/10 :)
Thank you. I tried to make it the best I could!! All the little details count.
the best video!!! i am looking forward for w^w position
I imagine if there is an after which you are immortal in you just calling up your buddy and saying "Yo, are you down for a quick game of chess? ill set it up in the Omega to the power of 3 position."
very well made! haven't seen anything that makes me so invested in quite a while. props to u!
A couple years ago, I heard about a chess engine's analysis of a game showing in a certain position had a forced mate in 256 and was amazed at this concept. It's so far beyond that now?! All I can say is Wow!
This video shattered my brain but every second was worth it. Great explanation of ordinals!
The really funny thing though, is that all of these different ordinal counts are pointless. What I mean is that if there is a mate in omega-squared game, once it's actually played, a mate in omega game can be played that is longer than it. In one sense, the games are getting longer and longer, but in another, arguably more real sense, none of them are any longer than the first rook tower.
What an amazing video 😭❤ I love how it gradually went from normal-looking chess to the never-ending nesting of countable infinities using nodes and towers
Amazing video! As a member of the intersection in the venn diagram between chess and math I've really enjoyed it! Can't wait to see what other wonderful concepts you will show us in the future!
Bruh the editing and explanation is just too good 😭 Bro is severely underrated
Get the camera Mom,
Naviary just uploaded
“Oh no, I blundered mate in Omega 1!!!”
“I resign”
I love how it seems like there's a countably infinite number of named ordinals. Transfinite mathematicians have too much time... transfinite time it seems. And they love naming numbers!
There is only a countably infinite amount of objects which can be described in any formal language with a finite alphabet of symbols.
hell yes, I was waiting for this video to come out and it's every bit as good as I hoped
Next things to explore:
Mate in w^2 with finite pieces (may be possible to prove it's impossible to setup? but the fact that you can get w*n with arbitrarily high n in constant pieces makes it tantalizing...)
Complexity class of/computers made in chess and chess variants
Mate in w^2 with finite pieces is known
Amazing video! I love the delving into the infinites of ordinals, and can't wait to see what you produce next!
Stockfish: "1 missed win"
The missed win:
I didn't even notice that 29 mins had flown by. I wish you and your project all the best!
i knew it would get REALLY wild when the bishop cannons appeared
banger video! it's great to see how your video skills evolved with this infinite chess journey, and i'm all for it :D
my only issue is the part of the ordinals getting """bigger""" feeling a bit too fast without a lot of the aritmethic context (and my brain doesn't help :P) but i see why you would approach it that way (on the bright side it makes me want to learn more about it so yay)
anyways, Ω/10.
This is a great video and honestly a really good explanation of transfinite ordinals. Will recommend this to a fair few people, this is great. Best video i have watched on youtube in a while mate. Cheers and hope your day is going well whenever you read this.
That mate-in-Omega⁴ position is oddly terrifying.
This is a masterpiece! It may very well be the best video I have ever watched!
I'm very interested to see how you are going to make infinite structures work. Your chessboard is obviously finite (the need for coordinates for storing the piece positions ensure that), but having structures that extend to the edges without taking up unfathomably large amounts of memory (or do so after the first move) sounds like a interesting challenge, especially with the complicated patterns these boards have. It sounds very possible, though.
I will need a chunk-like system, where I only have a finite area loaded at a time. Definitely a challenge, and a challenge to optimize it too!
All the ordinal numbers were just blowing my mind. Not the size of them, but the fact that we have notation for it
The bishop really becoming sniper with this one
I've never seen so much dedication to a game of chess, good job (for leaving my brain in liquid form)
Unbelievably cool, I’m amazed by how much effort you put in - it was 100% worth it. Best of luck to you! Genuinely one of the best videos I’ve ever seen on this platform (as a chess player and set theorist I may be biased but still…)
Same here!
I love how it is incredibly logical, some deep thinking and you will understand. Pretty great video
I love the nuclear bomb warning siren when it changes to a higher omega lmao
Imagine white, after mate in Omega^4, all the hard work to get to the king, just statemates in the end
This was an extremely well done video.
Really liked this video! Very informative and educational without necessarily wanting to be that. Keep it up!
Screw Vsauce, this is the best video for understanding transfinite numbers.
BROO I DIDN'T EVEN REALISE THIS CHANNEL IS NEW BUT IT IS REALLY GOOD 🔥🔥🔥
FINALLY A VIDEO!!!! NAVIARY THE GOAT!!!!
Damn, this is amazing, I watched a few videos about infinite ordinals and omega before but never really understood it properly. This made it so much clearer.
i didnt die before this video ended, clickbait
I wonder what is the biggest mate-in-x we can reach with a finite amount of pieces. After all, you start a position with only 16 pieces, and you can only reach up to 10 of any given piece. How big does it get with this constraint?
That... is another story to tell! This one is actually still an open question. We don't know yet.... But we do know that at least Omega^2 is possible with finite pieces!
@@Naviary❤
Hey Naviary! This might be a stretch but I wonder if the mate-in-X could even be undecidable?
We already saw in the mate in omega-4 how close each side was to winning.
What if the game position represented the rules of a undecidable game?
This whole video reminded me a lot of Sylver's Coinage, which itself is unsolvable rn. So I guess if you simulated that game in infinite chess SOMEHOW, it'd be mate-in- implication of math problem 🎉
I think so! Pretty sure if we were just handed a mate-in-omega1CK position, it would be impossible for it to figure that out, as it's non recursive and uncomputable, so it must be impossible to create an algorithm that can calculate the clock for every single possible position!
I've been waiting for this video to continue the infinite chess saga, thanks!
I saw omega next to a chess board and felt like paul revere warning that the mathematicians are coming to take our safe space
Can you make a video explaining the Mate in Omega^Omega (theoretical) mate please? 🥺
😉👍
@@NaviaryI would like to know as well! I feel like you kind of glossed over how exactly the higher order mates work in the bishop zugzwang position.
@@aav56 It's a little hard to understand. I would recommend reading up more on Matthew's proof himself. But basically there exists an algorithm that tells us exactly where to place the nodes to obtain the ordinal value we want. I briefly mention here that an w^w announcement would descend to an w^n position for any value n. An e_0 announcement would descend to w^w^w... for any height n. Basically any announcement of any size N can descend to any ordinal T that is included in the infinite sequence leading up to it.
In the bishop tree, if we want to make higher ordinal positions, we can always just take existing trees we have made, and repeatedly place them as choices in the first branch of the tree. This will always give us higher ordinals.
@@Naviaryis it possible for an Omega^^Omega checkmate?
@@ihateyoutubehandles444 That's just written as e_0 (epsilon zero), and yes!
This is all very interesting, but since most of these positions require an infinite amount of material, what I'd like to know is, what is the longest (currently known) infinite chess game position that can be achieved from the regular starting position
As in the normal setup? Or any position allowed with finitely many pieces?
@@Naviary From the normal starting position. The setup can be implausible, that is, it can require one or both players to have played moves that are suboptimal or even nonsensical. But it should be _theoretically_ possible to get to the position from the normal setup.
This is a fun gem of a channel. Really entertaining stuff keep it up :D
This feels like star wars. Just send out bigger cannons to counter bigger cannons and you can't lose; but of course you must build the cannon that counters the cannon which countered the counter cannon first. It just works. Trust me bro.
You know it's a good day when Naviary uploads
Interspecies probabilistic union agrees
The mate in omega^4 would actually be omega^5. Your opponent can choose how long they would like to take to move.
"The original bishop moving up is the tier 5 announcement, each rook tower is a tier 4 announcement. Each pawn movement in the rook towers allows the activating of a bishop cannon tier 3 announcement. Each bishop fired from that cannon is a tier 2 announcement. And each gateway door threat it makes is a tier 1 announcement, deciding how long it will take before it moves!" 🤣
by that logic normal chess is omega*n where n is the number of moves in the game
HOW DID I NOT KNOW ABOUT THESE ORDINALS!!!?!!?! This is one of the most amazing things I've seen in my life, thank you for making this!
this was fantastic! any math youtube fan who understands or enjoys even a bit of chess will love your video, Naviary
It feels like at some point you'd hit a problem of density, and higher ordinals would require more room than is afforded in just 2D space
It does feel like that though, right? More details on the proof can be found following a link in the description.
It feels like this a bit, but if you are about to run out of the room, I think you can double the spaces between each node. Since you can do it infinitely, you will never run out of the room
@@FancyHalcyonit seems like it'll only raise a tier instead of infinitely
If one were to develop continuous infinity chess, I believe a mate in (Big) omega 1 position would be possible. But likely the construction too psychotic for anyone to actually formulate
A way of making continuous chess would be to make each piece take a certain amount of area on the board, two pieces area can't overlap (Of course this would also correspond to captures), and a piece can move continuously on a line of its movement, excluding knights and kings. Blocking the movement of a bishop would happen on the point where if you move the bishop's area along its line it would overlap another piece. If this piece is the enemy's you can capture it on any point up to the point you leave its area. This also means you could make an arbitrarily small move, as well as an arbitrarily large move (In discrete infinite chess there is only the arbitrarily large)
How would continuous infinite chess work? Infinite checkers also has interesting properties, allowing game values larger than omega 1.
EDIT: That is interesting!
Pawns can move continuously up to one of their areas length forward. Or two of them on their first move. And capturing works like a jump move, making it discrete (Otherwise it would break the rules of normal chess). En passant can be made, but is a bit weird
@@zionfultz8495 I made a game like that as a school project. Granted, since it runs on a computer, it's not actually continuous, but it's close enough lol
@@NaviaryWill you make a video on infinite checkers? or any other infinite game variants
This video is making so many mathematical concepts make sense. What an incredibly, incredibly well made video. This is what youtube should be for.
great content, thank you. it's unfortunate it takes sooo much time to make, i love your videos :<
the world of infinity (or infinities probably) has always fascinated me.