The Search for the Longest Infinite Chess Game

[@bruhmcbro2704]

I love how you start with something that resembles chess and end with massive structures designed only to prolong the inevitable.

[@Patashu]

The lower quadrants have been lined with bishop cannons of increasing size

[@Nzargnalphabet]

And yet, a math brain loves it

[@meks039]

thats such a good way of describing how my brain is reacting to this.

[@edgeman1135]

That awkward moment when you blunder into an infinite number of rook locks...

[@mongoose7857]

Just like real life

[@WolfgangShaffer-ey6ce]

Imagine being foolish enough to blunder checkmate in ω^3•10+ω^2•8+1001

[@Naviary]

🤦‍♂️ "You blundered! How could you not see that?"

[@godofnumbersakausername5226]

@@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

[@JaredJeyaretnam]

@@NaviaryYou fell for one of the classic blunders!

[@Maxyncydentz]

Hikaru be like:..... Yes that's checkmate in (imagine the amount of all real numbers)

[@mrt_pose]

Rookie error lululul

[@paperwhite3853]

It's kind of beautiful, that two kings are just having a staring contest, while all of this is happening

[@icantthinkofaname8139]

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

[@fntthesmth423]

Typical monarchy forcing everybody else to dedicate their lives to them lol

[@w花b]

The UK has no problem with that. Sucking money like leeches. Very impressive.​@@fntthesmth423

[@RFM__]

shout out to their patience because doing absolutely nothing all of that amount of time is huge‼️‼️

[@abdillahahmad7025]

"Now, now, there's no need to fight... why not settle this over a nice cup of tea?"

[@WillySalami]

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.

[@gianglai7346]

Omega-1 minus 1 is just Omega-1

[@barrianic4]

@@gianglai7346actually omega-1 minus 1 is ill-defined

[@aaravthediscoverer]

@@barrianic4 actually omega-1 minus 1 is e =m2

[@barrianic4]

@@aaravthediscoverer omega-1 does not have an imediate predessessor because it is a limit ordinal

[@aaravthediscoverer]

@@barrianic4 but does it have an immediate successor?

[@christopherearth9714]

The worst thing in Infinite Chess is probably the Bishop sniping you from 45 multiverses away.

[@Dexuz]

Biblically accurate

[@pmz-gaming6479]

@@Dexuz*why do I hear boss music*

[@ChoccoGlx]

Yeah, I hate that move.

[@Someone_adachi]

Average 5d chess game

[@manymanyatoms9327]

Oh crap the knight jumped over the interdimensional void and took my queen

[@tails183]

Of all things, I never thought I'd hear the phrase "bishop cannon" in my life.

[@DR-7]

BROTHERS! WE NEED TO CRUZADEEEE

[@glowstonelovepad9294]

I searched "bishop cannon" and apparently there is someone who is a bishop and named Cannon. en.wikipedia.org/wiki/William_Ragsdale_Cannon

[@Roovinggoove]

Pope cannon

[@elaundertale]

I mean... they ARE called "shooters" in my native language... I guess this is why

[@jacobortiz7087]

Bishop cannon is probably the most metal chess thing I've ever heard

[@gavinlol-lo1pd]

A higher game value does not necessarily mean the game will take longer, but instead it means your opponent can be more annoying

[@richardpike8748]

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

[@eeeee11235]

@@richardpike8748 2 move mate

[@eeeee11235]

​@@richardpike8748 ig depends on position

[@londegel]

​@@eeeee11235 not in infinite chess, since the king can just move backwards

[@Damianp00p]

E

[@vnXun]

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.

[@-minushyphen1two379]

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.

[@liam.28]

@@-minushyphen1two379the large veblen ordinal is also countable

[@kruje314]

bruh

[@thefastmeow]

that's what she said before she left me

[@MichaelDarrow-tr1mn]

@@-minushyphen1two379 the large one is also countable

[@John_Gillman]

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

[@tektek1100]

started laughing at omega³, checked how much of the video is remaining, oh boy

[@dazcarrr]

the moment terminology turned to "towers" and "cannons" i think this stopped being about chess

[@kruje314]

'What is this piece?' 'Tower.' 'ITS CALLED DA ROOK'

[@erka339]

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 !

[@Proto-EXX]

soon we’re gonna have nuclear warfare in chess if we keep goin higher with these ordinals

[@tigerghg7302]

Those massive structures weirdly remind me of the game of life

[@dazcarrr]

@@tigerghg7302 chess has become cellular automata

[@louisjagger2177]

The later stages begin to resemble Conway's Game of Life :)

[@Naviary]

In some ways it does resemble it doesn't it?

[@subscheme]

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

[@shauas4224]

@@Naviary now I'm thinking if infinite chess is Turing complete

[@T11235]

It's the game of life, except it's more complex

[@mathgeniuszach]

@@shauas4224 I wonder that too now

[@Steamworker_Evolair]

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!

[@Shivoham2243]

Same😂😂

[@skittybug6937]

Legitimately made me understand that topic better than Vsauce if only because I was able to visualise it through _checks notes_ Rook Towers and Bishop Cannons

[@Pnod666]

And here I thought VSauce's video was absolutely insane! He doesn't even get into as many Greek Letters!

[@ChrisAsHell]

you know what? let’s call it a draw

[@Thepreacher_1]

*black* "Nah, id make you surrounder" *Moves first bishop to a Google Plex*

[@LeonardoPereiradaFonseca-y2r]

​​​@@Thepreacher_1*white* Hah! Silly you!! You just moved inside my bishop's range!!

[@wumaster1]

I love how you played a nuclear alarm in the background when talking about omega^4.

[@Error422win]

For good reason

[@Error422win]

For good reason

[@Naviary]

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

[@treelol]

u have my respect

[@wesleystoltz8421]

Would it be possible to reach a higher checkmate clock with custom made pieces?

[@Naviary]

@@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!

[@Patashu]

@@Naviary Ok now I need to see the video on infinite checkers 👀

[@arcaltoby5772]

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.

[@가시]

I've finally found an area where transfinite ordinals are useful

[@maldoror-13]

"useful"

[@rykehuss3435]

@@maldoror-13to the gods

[@nathanJensen-or3sv]

*semiusefull

[@fluffly3606]

Now to create a starting position where reaching said useful situation is actually plausible

[@coocato]

geometry dash theoretically possible levels as well

[@arandomdiamond2]

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.

[@NathanSimonGottemer]

@@Nomasunpibeboludothe imaginary unit i is finite in the 5-adic integers, among others. Or, really that’s the square root of -1: there are two separate numbers whose square plus 1 equals zero in that set

[@macchiato_1881]

​@@Nomasunpibeboludotransfinite ordinals are no less fictional numbers than the natual numbers.

[@SimoneBellomonte]

@@NathanSimonGottemerWhat did the coward say? 🗿

[@SimoneBellomonte]

@@macchiato_1881^

[@NathanSimonGottemer]

@@SimoneBellomonte …ngl I forgot 😅

[@GenTheFurredArtist]

This is basically just what happens when you're really determined to NOT lose.

[@Jakitz]

0:28, GothamChess when you miss mate in 549 in gte "AHHHHHHHH"

[@Mika-lt6lr]

Man i’ve not once left a comment on a KZbin 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.

[@Naviary]

Thank you, I really appreciate it!

[@objectshowfan362]

We've seen how far we can go with infinitely many pieces, but how far can we go with only finitely many pieces?

[@Naviary]

@@objectshowfan362 This is still an open question! We know so far that omega^2 is possible.

[@TymexComputing]

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 :)

[@Unknown-vg2mf]

@@objectshowfan362 It is bounded by Church-Kleene ordinal (the first nonrecursive ordinal, also the supremum of the recursive ordinals)

[@eliascastillojerez6778]

10:58 Tier 2 announcements were a plot twist I wasn't expecting. Great video and great narration!

[@Henry3.1415]

The intro with the infinite chess game zooming out to show the text is so cool

[@findystonerush9339]

3.1415? Pi? 3.14159 this is pi followed by!

[@TacoChess]

@@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

[@lightningfirst689]

That mate-in-Omega⁴ position is oddly terrifying.

[@Dondoki_]

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."

[@TheFlame_Hawk]

Nah, why play a game that has a preset winner

[@Dondoki_]

@@TheFlame_Hawk for fun

[@Drippyglaze]

You sound dyslexmic

[@abellematheux7632]

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. )

[@matthewbolan8154]

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.

[@neopalm2050]

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).

[@abellematheux7632]

I will edit my comment tomorrow if I don't forget (it's night here).

[@TymexComputing]

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!

[@zaringers]

Hmmm grave interessant ça aussi

[@neopalm2050]

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.

[@neopalm2050]

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.

[@danielyuan9862]

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.

[@danielyuan9862]

​@@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.

[@neopalm2050]

@@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).

[@angelmendez-rivera351]

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.

[@AGamerNamedSky]

"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

[@wesleystoltz8421]

I like that if you set all these numbers to one mathematically these extremely big numbers all equal 1

[@Thepreacher_1]

​@@wesleystoltz8421not if you have a trillion of them stacked on top of each other

[@Bestmotion2389]

26:40 Imagine if after a few seconds of silence you hear: "Or is it...?"

[@TheArtOfBeingANerd]

All the ordinal numbers were just blowing my mind. Not the size of them, but the fact that we have notation for it

[@cobble616]

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

[@steeevealbright]

I had this exact thought about Jon Bois

[@beyondobscure]

Never heard of that before. It's great!

[@tsevasa]

Insanely well made! This truly is a game for the gods and we have only begun to scratch the surface :)

[@Naviary]

Thank you!

[@RickMattison314]

@@Naviary, I tried joining the Discord, but it said that I'm unable to accept the invite.

[@Naviary]

@@RickMattison314 That's weird. It should work! Maybe try a different link? discord.gg/bWbgYqX7Re

[@RickMattison314]

​@@Naviary, still nothing. Edit: NVM. It worked on my phone.

[@Naviary]

@@RickMattison314 Great!

[@triplebog]

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

[@Naviary]

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?

[@StriiderEclipse]

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”

[@harleyspeedthrust4013]

so this is what the memes mean when they say that magic the gathering is turing complete

[@Deh9o11en8or]

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

[@AndrewBlechinger]

@@StriiderEclipse I thought they just banned cards that cause infinite loops? (Freaking Pole Position, man.)

[@LordShrek420]

“You fool! You forgot my bishop 10 trillion light years away can stop your checkmate!”

[@caioxlive1318]

I had an idea to generate an ω^5 announcement. Remember that on the ω^4 announcement you fire the bishop first as the Tier-4 Announcement? you can make an tier 5 mirroring the Rook Towers and putting an piece with the movement restricted. This piece would be manouvered to protect a key square, where a pawn is going to be placed to fire the bishop. If the bishop takes the Key pawn under that circunstance, the piece would recapture and be free to checkmate the king on the throne room, or release a Bishop to do the job.

[@mrorcadood]

Thanks for the tutorial, now I know how to deal with this when it comes up in my games!

[@maximdegi]

the moment of history, the third naviary's video

[@BabayChannel]

Only countably many videos until the ω-th video

[@parapasarunbuenrato8773]

Unlike **some company**, he actually know what comes after 2

[@kevincsellak296]

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!

[@Naviary]

Thank you

[@Naviary]

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!

[@abellematheux7632]

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).

[@angelmendez-rivera351]

@@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.

[@abellematheux7632]

​@@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.

[@Patrick-gm3fb]

The scope of this video was just to explain super obscure chess problems but it turned out to be the best, most intuitive explanation of infinity (and different levels of infinity) I've ever seen.

[@Himsjdndms012]

Why does this feel evil

[@Roovinggoove]

Bc it is

[@jitendrasangwa7780]

Yea

[@stanimir5F]

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!

[@Jagrofes]

“Here here takes here takes here here and now you just win…”

[@EnerJetix]

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.

[@godofnumbersakausername5226]

omega is a ordinal, not a cardinal

[@EnerJetix]

@@godofnumbersakausername5226 lol, good point

[@thesenate1844]

​@@godofnumbersakausername5226Expanded chess should also come with new pieces, like Cardinals which are Bishops but better

[@godofnumbersakausername5226]

@@thesenate1844 Yeah, or else every game would be a draw.

[@raptorialmage]

Imagine getting skewered by a bishop on the square b925836

[@plasmakitten4261]

So basically, we won't know what the longest theoretical infinite chess game is for certain until someone figures out the continuum hypothesis

[@danielyuan9862]

It's already figured out. It can neither be proven nor disproven. So you can state it as either and the math still works completely fine.

[@plasmakitten4261]

@danielyuan9862 Yeah but which choice is "better" is an open debate in analytic philosophy.

[@noahhuguenin]

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!

[@willlllllliam]

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 :)

[@Naviary]

Thank you. I tried to make it the best I could!! All the little details count.

[@_Epidemic_]

Right as I was rewatching your previous videos you drop yet another banger, great work as always.

[@aloysiuskurnia7643]

OOOOOOOH now I see how my confusion about omega + 1 from the last video gets resolved. It's very elegant!

[@Naviary]

Glad I could clear your understanding!

[@Blue_FirewalI]

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

[@liam.28]

@@Blue_FirewalIi have never heard it called OMEGA but my guess for that would be that the symbol is capital omega (Ω) instead of lowercase omega (ω) and absolute infinity is a different ordinal

[@TSANOOvlogs]

19:38 I like the subtle difference of the Omega^2 announcement here compared to the other tiers. Instead of the announcement being a move played by a piece, this announcement is the act of selecting which bishop tower you decide to unload. The move that follows is the tier 1 announcement, choosing where to place the selected bishop. The "distance" of the announcement is proportional to the number of bishops in the selected bishop tower, which feels pleasing. Neat stuff!

[@aze4308]

please make a behind the scenes of this video! it’s so cool how the 3d graphics work

[@truckjumperdude]

The way you explained this made it so simple to understand and still very interesting, this is a "5-tier" video 🙂

[@janaki3829]

8:04 Oh hey, it's the Code Bullet song!

[@archysimpson2273]

Thanks, I was going nuts trying to figure out where I heard that from

[@That-One-Frog]

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!

[@Naviary]

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...

[@crowreligion]

But what about infinite chess 2, with infinite amount of pieces?

[@JacobGruzewski]

This is honestly one of the best videos I have ever watched and you deserve way more subscribers

[@numberblocksfiftyfourfan]

11:55 The white king keeps on chasing the black rook.

[@tdubmorris5757]

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

[@davethesid8960]

This video shattered my brain but every second was worth it. Great explanation of ordinals!

[@pauls5745]

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!

[@pravkdey]

Man the throne room and the towers and cannons makes me think of a story where these mad god -kings have these massive 3D printed armies that span galaxies and they're throwing them at each other in increasingly elaborate and ridiculous scenarios thinking they're brilliant and geniuses. Meanwhile the kings are still in throne room, the dimensions of which are the same as that of their ancient ancestors, perhaps even the ancestors of their ancestors. Anyway cool video haha

[@bessie8612]

Love that towards the end is just a bunch of Mathematicians imaging arbitrarily high ordinals and naming them after themselves

[@mastercrash0683]

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

[@chenivan6171]

very well made! haven't seen anything that makes me so invested in quite a while. props to u!

[@Tovosx2]

Bruh the editing and explanation is just too good 😭 Bro is severely underrated

[@gunhasirac]

This is truly remarkable. Thank you for putting all these together. This will be an incredibly good introduction to ordinals and how big omega_1 is. This has as much education value as entertainment value.

[@MeGaGiGaGon]

Amazing video! I love the delving into the infinites of ordinals, and can't wait to see what you produce next!

[@Dieto-eit]

I didn't even notice that 29 mins had flown by. I wish you and your project all the best!

[@droidanimado5803]

I've never seen so much dedication to a game of chess, good job (for leaving my brain in liquid form)

[@The4DRY4N]

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.

[@thelistener1268]

I heard about Aleph Nol and Omega in the Vsauce video about counting past infinity, but this finally made omega make sense to me.

[@ilikevideogames2807]

Demon: You get to leave hell if you beat white at infinite chess

[@Naviary]

Me: "Bishop to Busy Beaver functio of TREE(Rayo's number)!"

[@confusedindividual]

I love how we are potentially generations away from solving regular chess and we’re already thinking about stuff like this.

[@steeevealbright]

20:26, pitch perfect delivery lolol

[@asj3419]

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.

[@Naviary]

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!

[@DeathDragon99a]

Can’t wait for 3d infinite chess to get to mate in omega

[@1vader]

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.

[@5dnikita]

the best video!!! i am looking forward for w^w position

[@InTheBeginningTheUniverseWas]

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!

[@angelmendez-rivera351]

There is only a countably infinite amount of objects which can be described in any formal language with a finite alphabet of symbols.

[@isa._mus]

26:35 The only Way to Reach Omega 1 In mate is to make a Uncountable Infinity Making it Absolute

[@jgseg6828]

Probably the best video Ive seen to grasp an intuition about this amazing and weird branch of Mathematics. Kudos!

[@goodmorningennui]

I love how in Omega^2 the chess is no longer a game, it now resembles a computation with set values.

[@Patashu]

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

[@MichaelDarrow-tr1mn]

Mate in w^2 with finite pieces is known

[@Robert-jy9jm]

This is a masterpiece! It may very well be the best video I have ever watched!

[@ohboiyou]

Get the camera Mom, Naviary just uploaded

[@tomatoblate2170]

This was so well done. I have an infatuation with trans-infinite numbers, so seeing someone actually give examples of them on an infinite Chess board is just so cool

[@SolubleParrot9776]

1:06 for a second i thought this was bad apple😭

[@DoNotSin]

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

[@xaf15001]

Several universe died during 1 turn of Infinite Chess. The only epic chess battle you couldn't not miss.

[@DoNotSin]

@@xaf15001 you couldn't not miss?

[@weare2iq376]

@@DoNotSin Yeah, you had to miss it, you'd be dead long before turn 1 finished 🤣

[@thenumberpie314]

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?

[@Naviary]

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!

[@crystallinnen5600]

@@Naviary❤

[@sleepykitten2168]

This was an extremely well done video.

[@iamagi]

I held on as long as I could, but around 25:00 it was all Greek to me.

[@Sloppnheimer]

lol

[@TropicalPenguin24]

In all my years on the internet, this is easily my favourite video I have ever watched. Thank you.

[@Dronkhrrrrng]

can we play checkers instead

[@the1stwing]

Ik you're joking, but just a thought Infinite checkers would be nigh impossible since, unless one side manages to capture every opponent piece, as soon as the pieces pass each other, all they can do is move onto infinity since they'll never promote to kings, and even if they could, one player could stall for infinity

[@MatthewConnellan-xc3oj]

Yes.

[@maximized_1]

@@the1stwingit will be more impossible if the checker pieces actually repeat infinitely, just a fun little game on how many captures you can make with one move

[@wren_.]

what if the board was only infinite one way? The amount of squares it takes to make a king is the same, but the board extends infinitely in the horizontal direction

[@Error422win]

I’m just gonna play connect 4

[@matthewbolan8154]

Hey look that's me.

[@Naviary]

Thank you for your contributions!

[@tsevasa]

Minecraft youtubers are going crazy these days.

[@lamshywy8920]

I am your new subscriber who found your channel from this video

[@decract]

"I hate math" "I hate physic" "I hate chess"

[@Bananappleboy]

"Is non-exist"

[@chaathon2]

I got lost as soon as we got into the ω of ω, but even then it was still understandable, interesting, and the most insane video I've ever watched. I don't even know how I ended up in the Chess KZbin, but I'm enjoying my stay. Props for all the searching and explanation, that video was a blast to sit through!

[@TalkAboutaTrapstar]

This is clearly absolutely hilarious, and the mathematics and theory work are great. Excellent work!

[@bobczech7774]

0:01 8848 mountain everest moment

[@UserSilentMaelstrom]

Fr

[@Roovinggoove]

88

[@anirudhv2215]

Can you make a video explaining the Mate in Omega^Omega (theoretical) mate please? 🥺

[@Naviary]

😉👍

[@aav56]

​@@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.

[@Naviary]

@@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.

[@ihateyoutubehandles444]

​@@Naviaryis it possible for an Omega^^Omega checkmate?

[@Naviary]

@@ihateyoutubehandles444 That's just written as e_0 (epsilon zero), and yes!

[@cyan1280]

1:05 this transition was fire

[@chunqinghe3396]

Black king : *calling Bishop* Black bishop : Oh I am getting a call! Black king : Hello I want to live for a lot of time (Omega^5 seconds) So I need you to move to the 900 Quattourtivigintillionth Wing terminal “ Black bishop : “ You motherf-“

[@Rikitaskar1]

Bro this video is probably one of the best i've ever seen, it makes my math knowledge increase on every second