Mate-in-Omega, The Great Phenomenon of Infinite Chess
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- čas přidán 31. 05. 2024
- What does it mean for a chess position to be Mate-in-Omega? Let's find out!
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The positional examples of mate-in-omega are from Joel David Hamkins and Cory Evans' research! Read their paper here: math.colgate.edu/~integers/og2...
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Chapters:
0:00 Intro
1:02 The Checkmate Clock
2:06 The Infinite Board
3:00 Mate-in-Omega
4:23 Ordinal Numbers
5:26 Omega Example 2
6:56 Omega^2
8:06 Outro
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#SoME #SoME3 Summer of Math Exposition 3! - Hry
I've never realized how important the 50 moves rule is
🤣
Just imagine your immortal 400 elo opponent playing a variant of Chess with no threefold repetition, can't block checks and no 50 move rule. You only have one legal move to evade the check. Give him an infinite amount of time, and and that 400 elo opponent will keep checking you forever.
lol @@BlueProgamer212
But I can win after omega steps, why would I choose to invoke the 50 moves rule?
actually, the 50 moves rule wouldn't be applied to the second position with the infinite pawn wall, since the 50 move rule is reset on pawn push
the 1000 move rule is where it's at, it does not get affected by anything, if a game is 1000 moves long, the game ends in a draw
Imagine you're playing a casual game of infinite chess with your friend and just as you're about to win he pulls out a "mate-in-ω²" stalling tactic. I'd flip the damn infinite board over.
😂
Lol
Rotating it by a slight degree is already impossible
But where will you grab it from???
I wonder if a program can actually detect a mate-in-ω mate, let alone a mate-in-(ω^n) mate. Feels kinda similar to the halting problem
I had this very position in a tournament just last week. I announced mate in w^2 and my opponent said “I wondered if you’d see that,” and immediately resigned.
imagine typing w^2 instead of ω²
@@aprilvee9154 not everyone has that I think
@@spektator5418you just add another keyboard, it's not exactly rocket science
@@methatis3013 yeah and then every time you want to change language you need to go over omega squared layouts
@@shauas4224 idk how it works on other systems, but on my huawei I just swipe to change the keyboard. I have on top of my native one, German, Russian and Greek keyboards
Mathematicians: How an infinite hotel ran out of rooms
Naviary: How an infinitely far rook ran out of checks
Infinity never fails to amaze me with its complexity!
This is just inaccurate. Hilbert's Hotel never "ran out of rooms," and the rook was never infinitely away. This was explicitly clarified in the video. I am confused as to how people missed this part of the video.
@@angelmendez-rivera351 You are probably the type of guy who tries to look for every logical flaw in a joke just to ruin it
also I think if you give the hotel an infinite amount of buses each with an infinite amount of people it would be full, right ?
@@youssefchihab1613 The joke ruined itself by presenting an inaccurate statement as if it were fact. It defeats the entire point of the format.
The Hotel is still not "full" in the situation you describe
@@angelmendez-rivera351
I mean, the hotel usually starts full? That's kinda the whole point with showing how the union of two countable infinities is the same size as them.
Or for how the set of all rational numbers is the same size as the the whole numbers.
you've heard of forcing a draw, now get ready for forcing the heat death of the universe
😂
When you have mate in omega, but you opponent knows you have a flight the next day.
August 12, 2036 heat death of the universe
Bro's about to make chess longer than monopoly
@@cyberfunk4580no
3:24 The queen and the rook chasing down the king across an infinite board is hilarious
😂 Glad it got positive feedback!
put Benny Hill music over that
@@maximkovalkov1334 You thinking of a specific song from Benny Hill?
@@maximkovalkov1334 or the super mario 64 slide theme. XD
@@Naviary The iconic Benny Hill Theme. With clever and simple editing, it could be an instant youtube chess classic
the reason I love this video is because it approaches transfinite numbers from a perspective where it actually gives them a meaning/purpose, rather than just being a crazy fever dream of symbols
so incredibly based
Crazy fever dream of symbols without meaning/purpose, or pure maths, are kinda' fun though.
@@NotSomeJustinWithoutAMoustache depends, imo. I like abstract maths, but I watched the VSauce video about transfinite numbers and I was kinda like "Ok cool, but what does it really mean for a number to be beyond finite? Are we just defining it to be that and that's it?" I think it needs to have some sort of application, abstract or not, otherwise it's just making random stuff up.
@@lizzyfleckenstein9837Making stuff up is abstract maths, that's pure mathematics. It's previously unthought of so you would have to invent names and figure stuff up about it, hell it's why math is even seen by some as an invention rather than a discovery, it quite literally is made up, even the useful ones. And it is fun to make the stuff up according to math rules. Another thing is sometimes a math concept is so abstract or exotic that it's use will not even be discovered far into the future, so I think the mindset with math "made-up stuff" isn't "this is useful" and "this is useless", but rather "this is useful now" and "this will be useful". Just because there's no known application in the math you're in doesn't mean there never will be, in fact it makes more sense to think no matter how abstract and utterly "made-up" your math stuffs are they will inevitably find a use and so you can just leave the discovery of the applications and practical uses to others, people in the future, and you (assuming you're the abstract mathematician) focus on what you're best at, figuring the rules of this thing you just made-up, so that the people in the future who will find a use for it will no longer need to tally the maths as some random guy with too much time years ago already computed and written down the rules of the abstract for them.
@@lizzyfleckenstein9837 Ordinals are very fundamental in set theory, they clearly have "abstract applications". For example, Cardinals are specific ordinals.
i was struggling to think of how to infinitely drag on my game when my rook got trapped in an infinitely long tower of pawns last week, this vid really helped me out!
I feel like logic gates could be possible, which would mean that in theory you could program chess inside of infinite chess.
Oh no.
You have started it.
The curse of Turing.
You know bad apple, doom, and AO3 in infinite chess are coming.
I'm having trouble making an infinite chess program Turing complete, as you necessarily start with a finite number of pieces and each time a logic gate would have to capture a piece. I think you can program some sort of an Adder but the program seems incapable of infinite recursion.
That's not to say that you can't program anything at all, though.
@@nanamacapagal8342Doom on Infinite Chess would be hilarious to play
@@iCarus_Aunless you introduce a custom piece…
@@iCarus_A I wouldn't rule out Turing completeness yet. After all, a Minsky machine has only a finite number of moving parts, but it's still Turing complete.
Hypothesis: maybe the logic gates should be encoded not in captures that *actually* happen, but in captures that *might have* happened, but are in fact avoided, assuming perfect play. That way you can keep the number of pieces constant.
I feel like having a forced mate in ω has to be a win condition similar to someone resigning against you and checkmate. You'd have to make that a separate rule though, since in normal chess if you see forced mate you just play it and win, or the opponent sees it and resigns. Since you can make omega arbitrarily large, you'd have to be able to declare that you see mate in omega, have a judge agree, and give you the win. Very cool and very entertaining to overthink
In the 1800s it was a practice to declare mate-in-X. This is clearly the way.
I think you'd rather the rule be "mate-in-w where w is not less than ω" as it's easy to see how a "mate-in-ω+1" (or any "mate-in-W+ω where w is a positive discrete ordinal") can be missed and you'd rather not have to specify by exactly how much...
Why not just play the mate out? Even though ω is infinite, mate in ω actually implies that for any strategy the opponent takes it will take a finite number of moves to mate them.
@@adayah2933 Because being finite doesn't guarantee that it won't be excessively large. If I move the rook 5 million squares away and we decide to play the mate out, both players as well as the audience have to sit through 5 million turns of everyone already knowing in advance who is going to win.
@@taelim6599 Right, but how would you convince the judge that the position is a mate-in-ω? Sometimes proving it is harder than just playing it out.
I feel like it would be funny if, in a situation like this, you could declare "mate in ω" and whoever has more time left on their clock wins. It wouldn't be balanced, but it would be funny
This seems to be quite reasonable.
Yeah, but if you’re playing with increment, then you’d just get like 5 * ω extra time… 🤔
I think it would be more reasonable to give the person with mate in ω the win. But I have a feeling there are symmetric cases. In that case it goes to the person who declares it.
Mate in Omega^Omega, in other words, mate in omega omega-ed
Remember, there aren't actually infinitely many moves played, the actual number played is finite, just unbounded.
This is very cool. I like how this really illustrates that ordinals have no descending chains - no matter how high the checkmate clock is, the game will always end in finitely many moves!
If anybody found the concept of ordinals interesting, I highly highly recommend watching "How to Count Past Infinity" by Vsauce. I believe a few of the visuals regarding ordinals in this video was from his video.
You can see the watermark placed in the bottom left. I recognized it was Vsauce and looked for it.
That video started my path down infinities
Speaking of large numbers and infinities a CZcamsr called Orbital Nebula has a series on large finite and infinite numbers
No shit, ofc he was inspired by Michael.
@@Xnoob545oh hey xnoobspeakable
I love the creativity of how he asked to subscribe
😁
It was unexpected, but convenient. Only he missed the astonishing tactic "I'm alredy subscriber".
"I Made Chess, but It's Infinite" -- well you had my curiosity...
"Mate-In-Omega, The Great Phenomenon of Infinite Chess" -- ...but now you have my attention
when my brain clicked as you finished explaining mate in omega^2 i was like "oh noooooooooo" like out loud because i realised that it really does have the potential to be infinite and thats just insane
this is some actual quality content, you will defenetly get famous one day
On average people arent smart enough to stay interested in these complex themes for long. But there is a small enthusiastic audience nontheless :)
fame matter not, good matter
agree
imagine you are playing chess against a computer and "checkmate in ω•ω moves"" appears on the screen
Yes!
This is one of the best chess CZcamsrs I’ve seen keep up the good work!
Edit: Wow ty for the likes
Thank you! 😃
@@Naviaryno problem :D
@@Naviary About the Mate in Omega square problem: Wouldnt it be better for white to take the rook between the pawns with th right pawn so that the white king could hide between the pawns and not get checked every move? By doing this it wouldnt be Mate in Omega Square but Mate in Omega + X except if the black King tries to block the hole, then it would also take forever since the black King is above the white King and therefore would be faster at the hole. Or it ends in a stalemate because of move repetition as the black king tries to not let the white king upwards.
I refer to the problem where black has two rooks. One free and the other between the pawns.
@trempton4106 Someone else also noticed this. Yes blacks free rook can cut off white king from heading to the hole, while black's king calmly makes its way to the hole to prevent the white pawns from pushing further. White can no longer make progress and it will be draw by indefinite play. Black likes this as a draw is an improvement from losing, so whites best option still is to capture with a pawn from the left wall.
I think actually the rook needs to be the one to block the pawns from pushing, and the black king prevent the white king from coming near the hole.
I learned the hard way not to move a piece to far otherwise it becomes unclickable. Hard to snipe your opponent when you can’t click your own bishop.
just zoom into it
I've tried moving super far out, its possible, but its invisible and also you don't click where you would expect you would need to.
@@Izzythemaker127 moving 10000 squares away has basically the same effects as moving 272 squares
@@MichaelDarrow-tr1mn Have you tried moving it literal billions of squares out. If you waste moves by moving your piece further and further back it lets you zoom out more and more so that its possible to move it so far you cant click on it again.
@@rtxagent6303 but why would you do that
i can just imagine 2 omnipotent gods playing with infinite time and they never win or even draw and just move pieces for millenia
So basically, the Ellimist vs. Crayak.
That Omega^2 example blew my mind. Great video and amazing concept!
People have come up with an omega^4 position.
@@galoomba5559Where?
Finally, an “application” where ordinal numbers actually make sense!
I formally learned about cardinals in school, so I understand those well enough… But I only heard about ordinals later on, and they never really made sense to me lol
Damn bro, you learned about cardinals in school? I can't believe schools waste time teaching stuff like that when they could be teaching actually useful stuff. (This is sarcasm)
Props to you explaining omega with an infinite chess board. Can't wait to see the other part.
I've had people try to explain cardinal vs ordinal numbers to me in the past, but this video really made it *click* for me! Thank you!!
seeing the phrase "mate in omega" on my recommended sent me through the 5 stages of grief instantaneously
As a mathematician who has actually worked with ordinals before, this was surely something quite interesting! I'm sharing this video with my colleagues and I hope they can get interested in playing chess as well!
For people who dont understand Omega :
The first position is mate in Omega but once black move their rook (for exemples 100 case) this is no more mate in Omega but mate in 100.
Same for the second position. Until black move, this is mate in Omega square. But if black move the rook 100 case, this now mate in 100•Omega. (100 times Omega)
Hope it help
I'll have to go into detail of how exactly the checkmate clock descends with every move in future mate-in-omega episodes!
@@Naviary make mate-in-ω^Blasphemorgulus video
Honestly, it's weird that we consider avoiding mate as long as possible to be "perfect play" in a forced mate situation. If we assume the opponent is perfect too, then the outcome is the same regardless, forced mate. If the opponent is a potentially flawed player then the "perfect play" move would be the most confusing, to make it look like it isn't forced mate. Often times, this will be the one that prolongs the game most, but that's just a tendency like how playing the move that gives the opponent the least move choices possible is usually a good move.
you raise an interesting point. here is some clarification: mate in X assumes perfect play from both players, and it calculates how long the best moves from the losing player would allow them to survive. if they make an improper move, the mate in X drastically drops. if the losing player can take a position where they cannot be checked on the next move, then there is no mate in X.
bro your channel and work is just amazing, you gonna blow up
Thanks! :)
I barely understand normal chess now i am trying to understand mate in ω^2
Great content . Short crisp and to the point.
Mate in Omega situations really are very interesting to even think about let alone the infinite concept
This omega squared cycle of checks reminds me of that one episode of Doctor Who where Peter Capaldi's Doctor got stuck inside a world that would repeatedly reset itself but not quite, allowing himself to escape after a few billion years!
Awesome work! Actually love what your infinite chess is able to bring to the table. The mate in omega concept is very cool 🤔
Wow this is somthing special very facinating concept I have heard about this but it never made sense to me but you have fixed that! Looking foward for more keep it up!
This is such a good video!!! I genuinely can’t believe that you’ve only made two so far- it seems like you’ve been doing this for a while!
Infinity and omega are such cool concepts and I find the things you can do with them fascinating. Combine that with chess, a game I love, and you have the perfect mixture.
This is one of the most entertaining videos I've seen ever. I love physics and math, and seeing it combined with chess and presented in such a clean way is so good. You've made a sub out of me for sure
This was fantastic, this reminded me of the fast growing hierarchy as soon as you explained how omega worked. And I was looking forward to omega ^ omega or even epsilon.
How about an inaccessible cardinal
congrats on making your second video, last video was amazing and I had some fun playing infinites chess with my friends, continue uploading this amazing content
Keep up the quality content man! There's a good crowd wanting such thought-provoking videos for sure
This is so cool! I’d love to see custom pieces in infinite chess, maybe with even a custom piece editor like the CEO piece maker. Could have so much cool mate-in-ω potential with pieces like the Thief from Faerie Chess.
bro this is some really good content, i really hope this vid pops off like before
It's interesting, magic the gathering has a set of rules for dealing with similar omega based scenarios.
Wherein one player demonstrates an infinite loop, but then must declare a number of times to repeat until they choose a different action. Then the other player may choose a different, smaller number, to allow the cycle to repeat before they deviate their action and break the cycle. Once that happens, the game shortcuts and jumps ahead to the end of the loop where they make a different action.
It seems like a similar ruleset could be adapted to infinite chess.
It is not uncommon for games in some formats to wind up with a player having a billion life totals because they created an infinite loop
Incredible content. Just wow! super hyped for next video!!!
Me watching your first video's miniature : "Yeah, whatever, just another chess variation, pretty sure I've seen something like this somewhere."
Me watching this video's miniature : "Ok, now that looks dope".
You are an incredible youtuber man, keep up the great work!
Thank you!
You explained such a hard topic to understand so well, big kudos ❤🎉
"How to count past Infinity" mentioned!!! great video
Subbed for the ordinal vs cardinal explanation alone quick and easy and makes me want to go check out the formal definition. Keep up the good work!
Ordinal numbers really are fascinating. I have a few theories for mate in omega^3 or mate in omega tetrated to 2. But mill keep it in mind and see if I were right or wrong
Mate in w^w? Describe the position because the best known is w^4
How would w^^2 exist
@@istoleyourorangejuice A complex and convoluted pawn structure, probably an "omega" amount of bishops, rooks and queens, and a few knight forks as well.
this is a really good video. please please please make a follow up explaining omega^3 or higher mates
This was one of the best videos i saw in a while.
I spent some time reading up about ordinals a while ago, and I loved the theory behind them. That they can be used like this makes them even cooler. Great video.
You should cover non-computable strategies.
That's a whole world I need to study more! 🤔
Super well-made video and the topic is really interesting. I usually don't like calls to subscribe but yours is so well integrated that it would have gotten me if i didn't already click the infamous button
Thank you!
what a great video! I like how it captures some fascinating intersections between chess and math
I love how you used Ordinals to explain an aspect of Infinite Chess, at the same time as using Infinite Chess to explain ordinals. Like, I already knew about ordinals but this made them even more intuitive than before. Definitely gonna try out the game myself. love this
When I stood up and said "Gentleman I have an announcement to make" at the local championship they kicked me out :(
Keep up the good work!
Thanks!
@@Naviary :)
Thank you! I was trying to understand this a while ago but I gave up, this explained it perfectly!
This video is brilliant!
Mate in Omega squared blew my mind!!
For that example, I think moving the rook on the right down two might be the best move, as it prevents instant checkmate and gives the king more than one line to move in
Fascinating concept! More infinite chess please!
-U too-
Love the Infinite Series shout outs, really helped me too.
This gave a really nice intuition for ordinals, well done!
Best infinite chess content! I’m sure you will get famous some day!
Amazing work!
As a video editor I want to provide one bit of constructive criticism, and I hope that I'm not offending with this especially since I know criticism was unasked for. But like I said, this is awesome content and I think this one tidbit of advice is worth it if it can improve your content even more.
The song in the intro carries an association with low quality content. Many popular songs from NCS have this association because absolute beginners have been grabbing these songs to use in their projects for years. I know that you know how important it is to establish credibility in the intro since you clearly put quite a bit of effort into it, but in my opinion the song choice is counterproductive to that. Many editors say that sound is just as, if not more important than visuals, and while my eyes were seeing high-effort content in the intro, my ears were hearing an amateur production. The reason I felt the need to make this comment is I really enjoyed this video, and I feel like this one seemingly small choice might be slightly undermining all of the great work you put into this video. I would recommend finding an intro song that is similar to the detective music that you use throughout the video, but with more grandeur to serve the intro. You could definitely find something like that on CZcams with enough searching, but if you're having difficulties with discovering songs on YT I would recommend looking into a service such as Epidemic Sound that you can pay a small subscription to and get access to a ton of songs (that you will have the rights to) with easy filtering options. Anyways, I wrote much more than I expected haha, once again I apologize for the unsolicited criticism and I hope to see more content from you!
I hadn't thought of that. I did put effort into making a custom version of it for the intro, but I didn't think people associate it with amateur content. Thanks for the advice! I'll re-think using NCS tracks.
So... NCS is the audio version of Comic Sans?
@@MOORE4U2essentially
ARE YOU CALLING MIKAN BASIC
This is an amazing explanation!
One of the most original way to introduce the ordinals I've seen so far, great vidéo
0:08 "there is something magnetic about it"
MAGNus carlsen
That bishop 1 quintillion tiles away:
Imagine sitting at the chess table, ready to fight your opponent and you didnt even made a move and he already called it mate-in-ω²"
Really interesting and well made video. It definetly taught me something new
I bet someone would discover mate-in-a-fraction very soon with this
Don’t think so. Chess works in whole numbers only, so a fractional love simply can’t happen
5:21
-Never let them know ur next move-
-*Q moves up 4 squares-
then rooks start to check
@@ayberkgirgin3864 how?
Unless u mean white rook then yes the next step is checkmate it with queen
@@hy7864 black rooks
@@ayberkgirgin3864 how?
Imagine checkmating your opponent on this infinite board and then just "Haha, you didnt see my bishop from 80928021938 squares away"
THANK YOU! I've never understood ordinals until I watched your "mate-in-Omega-squared".
Here before this blows up like the last one
If we take the 50 moves rule into consideration, that pawn tower indeed allows White to checkmate in any arbitrary number of moves set by Black.
Great video! Keep it up
Amazing work! I've never been able to understand ordinal numbers, but something clicked while watching your video. Your examples were really clear and ordinal numbers finally make sense. Thank you so much!
Not really a great explanation for ordinals since they doesn't represent how much there is of something , so an checkmate in omega doesn't make sense, same as a checkmate in third
@@olixx1213 Would you say that “mate on move omega+n,” where n is the number of already made moves, is more accurate?
@@noobatredstone3001 why would it be
Omega is an ordinal
@@olixx1213 If you say “move 3,” 3 is ordinal, right? Meaning that saying “move omega” should work
@@noobatredstone3001 3 is a cardinal
An ordinal would be third move
Coming from somebody who spent two hours going 1e+69 squares out in infinite chess, I’m excited to see what happens with this game
🤣
Wow, 1.3888...e65 moves per second, you're really fast at clicking.
@@linuslucke3838 what I would do is I’d move the piece out really far then on my next turn I’d use the zoom out button to move it exponentially further
@@linuslucke3838 not all chess pieces move one square per move
This mate in ω^2 made me laugh so hard. So stupid yet so fascinating.
Vihart introduced me to the countably-infinite ordinals all those years ago, but today you've finally given me a concrete way to understand when and how to apply the concept.
this is amazing! kudos and thank you!
Okay. This got me thinking, in theory it would be possible in a layout to have an mate-in-ω for both sides? Imagine a crazy stall tactic of both teams going at it, and it would theatrically be determined who moved further, but if they moved the same distance, would it be a tie? Or still mate-in-ω? Or even a possibility of a setup of mate-in-ω with a tie being the only way to break it?
maybe one can construct a symmetric variant of the mate-in-ω-position from the video, where both would lose to a mate-in-ω if they do the first move. but whoever does the first move is bound to lose, because the other player will not have to move their rook at all. they will just play the checks to get the mate-in-ω. so in that case i think whoever moves first loses to mate-in-ω. it would be interesting however to find a tie-in-ω-layout.
Such a cool comment! Thank you! Now I'm going to my thinking booth…
No, it is impossible for one side to have a mate-in-α and for the other to have a mate-in-β for any ordinals α, β.
Proof (by contradiction): suppose white has a mate-in-α and black has a mate-in-β. Without loss of generality it is white's turn and (α, β) is minimal.* On the one hand, there is a move after which white has a mate-in-γ for some γ
Is there a paper I can read associated with this? Seems like a really fun way to work with set theory and spaces
Here you can find a paper about transfinite game values in infinite chess! arxiv.org/abs/1302.4377
this is genuinely such a high quality and entertaining video how do you not have over a mil subs
Thanks, really fascinating ! I would love a video on why new pieces are necessary to the game and what they bring in terms of strategy.
So, if an engine could be made to play infinite chess (should be possible, there are no fancy rules), would it be able to find mate in ω or would it freeze as a kind of "halting problem" issue? If you capped its search depth to prevent freezing, how would it evaluate a mate in ω position?
It would need some kind of complex algorithm to detect the patterns in omega positions. Brute force calculating moves may not work.
Luckily i watched vsause video about infinity numbers.
really good, well made!
that's soooooooooooooooo interesting, can't wait for the next video! instantly subcribed
Is there a bound to how high the ordinal checkmate clock can go? I'm fairly certain a game must have at most a countable checkmate clock so W_1 is out of the question, but can it go lower than that? If so, are there alternate pieces that could make it go higher?
Correct, I believe it can go as high as possible but not including omega_1! Vedras in the discord server knows more!
@@Naviary What about something like the Feferman-Schütte ordinal? What would that mean in terms of announcements?
@@emmabellhelium I'm not knowledgable with how that ordinal is made.
I wouldn’t be too surprised if, just as there is a countable ordinal that is the supremum of all computable ordinals, and where this is the main description of it,
I wouldn’t be too surprised if there wasn’t really much of a nice way to specify the supremum of all “ordinals alpha such that a there exists position in infinite chess with finitely many pieces where white has mate in alpha” other than, basically the same definition?
Though... maybe if restricting to boards with finitely many pieces, might reduce how big the ordinals expressible this way enough, that it would have a more normal description?
But, if allowing any computable board state...
Hm.
Mate in tetrated omega?
And he's back, with another video!
Damn, this is a very cool fact I did not know I needed to know. Amazing video!
Thank you. I was searching for this kind of video as I read many years ago about infinite chess, I think it was in Scientific American?
Magnus Carlsen still will win the endgame! 🙂 Even it will take long, especially at the end.....
This is a really good way to explain infinite ordinals. Really gives an intuitive understanding.
Earned a sub for excellent content!
Neat, I love how ive seen this same exact video concept like 3-4 years ago but just as a hypothetical, chess truely has been around forever.