This is the longest episode of Grade -2 yet and I had a lot of fun making it. Thanks for watching! See the video description for links to my bonus channel, discord server, Patreon supporter names, and etc.
I once played a game of magic where multiple players started “going infinite”, and we agreed to notate everything algebraically. One person made “x” tokens, and the next guy made “2x” tokens, etc, where x was arbitrarily large, but unspecified.
I think there is a rule for this, where if two players can go reactively infinite to eachother, the active player decides the amount of loops they perform and then based on that, the non active player decides
I'm always arguing with myself about how purposeful the chaos in these videos is. Whether it's scripted, semi-intentional, or just a happy result of chaotic set dressing. That being said, the dominoes made me happy
A great example of something that seems infinite to me but is in fact merely arbitrarily large is my exponentially growing anxiety as I watch everything in your yard falling apart
Another type of infinity: the amount of havoc that Domotro can create inhis backyard during a single video while teaching us math. I have the feeling that with each new video, the amount of items thrown to the ground is greater than in the previous video. Eventually, there will be a video where there is no math and everything is chaos!
At the 5:00 mark, I was concerned he was going to drop a brick on his foot. Combo Class is the only math teaching position where steel-toed boots are recommended.
Yeah maybe I should start wearing boots. I haven’t bricked my foot yet but one time a metal clock fell on my foot and left a bruise for a few days haha
Thanks. Yeah hope more teachers use this content over time but so far it’s going good, since this channel is barely over 1 year old so I’m happy with its growth so far
This was actually a big problem on Arena's Alchemy format. There was a deck that used Vesperlark (a 2/1 that brings a creature with power 1 or less from the graveyard to the battlefield when it dies, non-optional) in combination with Davriel's Withering, an instant that gives a creature -2/-1 "perpetually", which was a concept they were trying for this digital-only format, where the card kept the new power and toughness even as it changed zones. This meant you could make your own Vesperlark a 0/0, and if it was the only creature in your graveyard, it would HAVE to get itself back, then it would instantly die again, be forced to get itself back, etc. If you had something that drained your opponent when your creature died, this meant assembling this combo wins you the game, but if it just wasn't looking good, even without the part that actually wins you could just as easily draw the game and try again. They had to change Davriel's Withering to say "an opponent controls" in order to reign the deck in.
@@Giantcrabz I think it would be considered unsportsmanlike if the deck wanted to draw rather than win specifically (certain cards/strategies have been banned in the past due to this), but if someone is simply using the resources they have available to stop themselves losing I believe most players would consider that valid
One of my favorite Magic stories involves an infinite loop. My opponent was using Sanguine Bond + Exquisite Blood- a loop where one card makes you gain life when an opponent loses life, and the other causes your opponent to lose life whenever you gain life. It's a loop without any way to manually break it, but it isn't typically truly infinite because as soon as a player drops to 0 life they lose the game. But when my opponent set it off against me, he didn't realize I had a Platinum Angel out- a card that says you can't lose the game, even when your life dips into the negatives. The game ended in a draw.
The way you explain these concepts are on par with my university professor. They make both intuitive sense while covering the natural progression of what we would want to know about their properties. Seriously, you are amazing and I wish I found you before discrete math 2.
Hey former MTG judge (Judges tell players how the rules work in tournaments), if you preform an "infinite loop", it matters for tournament logistics that these loops can be shortcut, so often after demonstrating that you could theoretically continue to do an action forever, you are then required to pick a number. what is more interesting and something you might like looking into is "non deterministic loops", like legacy 4 horsemen, which uses a combo to put the top cards of your library into your graveyard. you then eventually hit cards that from your graveyard win the game, but the hang up is that in order to play around people messing with your graveyard and to make the winning combo from your graveyard more efficient, you have to play a "shuffler", or a card that when put into your graveyard, shuffles your graveyard back into your deck. The issue is that in order to shortcut and "pick a number", you have to be able to definitively say that 100% doing an action that many times creates a specific "boardstate" (how the game pieces are in play and how resources have changed, like making 4 million mana, or doing 8 thousand damage), then you can take different actions other than the loop you demonstrated. This is theoretically not possible in a deck like 4 horsemen, because if you mill each card one at a time, you could theoretically mill your shuffle card infinite times in a row, and because it changes how the loop is preformed and what the loop does past that point, and because you cannot specifically give a number of activations required to win the game or produce the desired boardstate, it is not shortcutable and therefore is "soft banned" in tournaments because of rules on playing too slowly. If you ever want to talk about the crazy mathematics in magic the gathering, feel free to reach out, I can give you my discord or twitter or something, I was a magic judge for about 5 or 6 years and have a math degree, so I always love talking about this kind of stuff. (also, some of those cards look pretty old, you should check out some of the prices, the older cards can go for literal thousands of dollars these days depending on what they are and what set they were printed in. I would recommend something like TCGplayer, it's what most people use to price cards)
Hey late reply here, but yeah I read about the 4 horsemen loop and it sounded interesting! About the card value, I don't think I have any of value (some are as old as Arabian Nights but mostly just beat up commons/uncommons). There are a few cards I set aside that are worth like $5 each but I don't think I have many worth more than that. As for your offer to chat, yeah I am down! I have some other MTG topics I've been personally brainstorming that I would love a judge's opinion on. Send me an email, or you can message on me the Combo Class Discord (both linked in the video description)
My favourite combo not mentioned here is Delina, Wild Mage alongside Wyll, Blade of Frontiers or Pixie Guide It has the excellent properties of being Arbitrarily Large and Non-deterministic Even better, Delina needed her wording changed to add a "may" so that you wouldn't have to play it out until you eventually failed all of the dice rolls
There's a combo or two that can make arbitrarily large amount of _squirrel_ tokens in MtG... Scurry Oak and Rosie Cotton or Scurry Oak, Heliod (the same as the one with Walking Ballista) and any creature that gives you life when a creature enters the battlefield.
Lol, I like that we're talking about mtg here. I didn't know that there was a rule for resolving a game where the stack couldn't empty. I might toss in one of these combos into the next _chaos_ build I do. Like in commander, "Okay, is the game now way _too_ complicated to play? Are we tired?" Stages "unresolvable combo". "Congrats we're all winners!"
One example that I have encountered is that when we use like algorithms/infinite series to calculate irrational numbers, we can calculate them to an arbitarily large amount of precision. You can calculate as many finite number of digits as you need given enough time. But you can’t calulate it to infinite precision because we can never write down/store all of the digits since it is infinite, and because we do actually need to stop somewhere to see our result. I’ve had to be careful with this when I explain this to some people at times.
Another interesting fact about Magic: The Gathering is that it's "Turing complete", meaning it can be used to simulate a Turing machine (and therefore can technically simulate any computer program). Kyle Hill did a great video covering this on Because Science; search "I built a computer in Magic: The Gathering".
And due to being turing complete, we can construct game positions where it is *impossible to prove* whether the combo is infinite, or simply very large. Thanks, Godel.
Only found your channel recently and I love the Intellectual chaos. I have been using the term Finite for combos that are not truly infinite. There are true infinite combos but require more of a setup (or dumb luck) to make work. This can be important in some cases where you might gain and arbitrary large amount of life but lose the pieces to set it up again. Then a few turns later your opponent creates a sequence that causes an arbitrary large amount of life loss. I've had it happen where I was quintupling my life during each of my end steps in addition to other life gain effects during mine, and other player's turns. I've been watching more of your other videos but as a MtG player I had to comment on this one.
the way "arbitrarily large" was taught to me was to say that you pick a finite large number to do some math with while knowing that you can do the exact same math again with an even larger number. Another way i would put it is that it means to take a finite number to do someth9ing an infinite value can't or shouldn't do. and also if it doesn't matter what the exact value of the number is or even its magnitude. the important thing is that your arbitrarily large number could be larger by any amount and the math still holds true. it really only came up when messing with primes though and doing some set theory stuff
22:56 Squirrel!!! As for the other parts of this video. I did meet someone playing these arbitery large combo's. He always won in the first turn. And the whole room eventually stopped playing the game....with that guy.
Thanks! To note, if that person you mentioned was winning on the 1st turn more than 90% of the time, there’s a good chance they were cheating too (by fake shuffling or something). Even the greatest decks sometimes don’t draw the right cards for a turn-one win. But yeah those early combo wins caused a bunch of cards to be still banned or restricted in various tournament formats
Great video! Not sure if it's quite the same, but my 7th grade math teacher explained to us that rays or graphs which have a starting point but never end in the other direction aren't necessarily 'infinite' because they also have an infinite amount of unused space in the other direction, so he referred to those as 'infinitely long' I think he also explained that between any two numbers exist an arbitrary large number of points, but not actually infinite since a decimal can't go on for infinite digits and a fraction can't have infinity as a denominator. Math has always been my favorite subject but have stopped paying much attention since graduation admittedly, so thanks for creating this neat explanation!
Thanks. And a geometric interpretation would depend on what you’re looking for, like if you are asking about the amount of points there are an infinite amount of points in a line segment, but if you asked for the maximum length on a coordinate plane that a line segment could have, the answer would be arbitrarily large and not infinite
This distinction actually matters in game sometimes. The most common example is that Arbitrarily Large life pretty much always loses to Arbitrarily Large damage. If they were both infinity, then it would technically equal zero.
Another example is playing a true infinity combo, like the Sporemound combo. If you can somehow find a way to turn it into damage, like Impact Tremors for instance, then you can turn the draw into a win.
Arbitrarily many life can beat arbitrarily many damage as long as it can be activated at instant speed after the opposing player has chosen their arbitrarily large amount of damage. Spike feeder + pridemate/heliod is one of the most common "infinite" life combos and it falls into this category
*technically* if a player with infinite life takes infinite damage, the result is undefined under the common rules of math, and if you do define that it'll probably not be 0 but something like "bottom element"
@@dm9910unless the arbitrarily large damage is dealt on the turn of the player with the arbitrarily large life gain combo. In this case, the player with the life gain combo has to be the first to break the loop
Once upon a time, I perused the detailed and comprehensive Magic rules. Somewhere in one of the documents (I forget if it was the comprehensive rules or the tournament rules), it specifically states that you can't do "infinite" anything. It specifically gives an example of a loopable series of actions that generate a token and states that a player may, for example, say "I'll generate a million tokens." as shorthand for doing that loop a million times. At least, I think. It's been a while.
For a second, I thought at 4:30 you were going to say "So what does it mean to be 'arbitrarily large'? Well, this isn't a term that can describe your mother..."
Yeah back when I originally played, “colorless” mana symbols didn’t exist, and the term was used interchangeably with “generic” by players so I wasn’t sure if colorless or generic was the right term for that now, but I guess I said it right in the episode since it looks like krark clan ironworks’ official rules text now shows it as colorless symbols
I consider myself a level 0 judge (not an official one, but I do know the rules very comprehensively), and every time someone in a game in involved in says they make something with "infinite" power or gain "infinite" life, I always mention that you do have to actually pick a number of times to do it, even if it's "a trillion trillion", especially relevant with multiple of these loops belonging to different players, since if you gain an arbitrarily large amount of life but i can deal an arbitrarily large amount of damage, we need to be able to figure out who actually wins, and it typically comes down to who gets to name their number second
What would happen if they could activate both arbitrarily large combos on the stack, though? (Say the disciple of the vault combo with leyline of anticipation vs. Marrow-gnawer + thornbite staff + soul warden)
These kind of loops also happen in YuGiOh, in fact the official konami documentation mentions "infinite loops" and gives an example with a card called Pole Position. In this case if the loop does not eventually cause a victory condition, a judge determines what's the problem card and sends it to the graveyard so the game can continue.
Nice. It will show up again at some point, since I’m brainstorming a few future episodes (out of the many topics I’m brainstorming) that relate to Magic again. But that might not be until later this grade
@@vantheman8050 Yeah I will return to that at least in a livestream on the Domotro channel, and possibly in a main episode on this channel someday. I think I've proved a 21 card deck that would work for that puzzle, and have some ideas of how 22 or 23 might be possible but haven't gone through all the logic on those yet
The term "arbitrary large" is very useful in software development, though usually this is in practice impossible because we're limited by very large but very much finite amount of memory, address space, and time (and while time is for our purposes infinite, our patience is very limited so things that doesn't complete in less than a few milliseconds is unusable in frontend code; and things that doesn't complete in less than 30 seconds are useless in interactive server code; and number crunching that doesn't finish in less than a few months on a super computer is pretty much unworkable). The term "arbitrary" (but implicitly bounded by such time and space limits) is actually even more useful in programming.. It's trivial to "hardcode" something, i.e. something that spits out the same result every time with limited to no inputs; it's harder to make something that works well with sensible and carefully chosen inputs (such as what you could expect most competent and trusted employees to provide most of the time; but it's really hard to make something that works okay and at least don't break badly (or if it break anyway, doesn't cause a security problem) for any **arbitrary** input (i.e. provided by incompetent or malicious users of a website). And even the supposedly well intentioned and competent editors quite often end up accidentally providing inputs that breaks stuff or at least makes it look rather ugly. And one rather basic thing that often breaks layouts is simply prime numbers: any grid layout for a variable amount of stuff will somewhat break when provided a prime number of elements or simply a number of elements that doesn't divide into the current number of columns. For large datasets you can usually avoid this by simply always fetching a highly composite number of elements, because if the dataset is sufficiently large (such as all the videos on TH-cam) any user will "never" reach the final page. But for smaller size websites any frontend developer will have to spend a lot of time figuring out clever fallbacks to avoid having a single element all on it's own on the end of a grid whenever the number fetched is one more than a multiple of the number of columns.
In fact, if I spend a 8 hours working on a feature; quite often I wrote the implementation for the best case sensible inputs in a few minutes and the rest of the time is spent thinking about and mitigating issues that might happen with problematic arbitrary inputs. Especially this is true of front end code that most of the time is rather unimportant for security, but is essential that it always looks good (or at least not horrible) no matter what nonsense is thrown at it. Only in the most bizarre cases would you prefer to just abort and print an error message rather than somehow handling bad input.
One question about "arbitrarily long" arithmetic progressions of primes. If we exclude the special case for when n is a multiple of a, can you have an infinite set of primes of the form a+bn for all n coprime to a? I suspect that it isn't but it's still fun to ask
Nope. That would mean that the density of primes cannot drop below the density of such numbers a+bn with n coprime to a. Yet, the density of primes goes as 1/ln(n)...
I have ended a game with an infinite draw before! i've also won a game with a deck that was designed to an arbitrary amount of time! if i played the right goblin cards, i was able to resort my entire deck, then shuffle it, then resort it an arbitrary number of times. because it wasn't infinite, it was not a draw, and i won because my opponent got so mad and he didn't want to wait for the time limit which would force me to concede. i was essentially gabling that he would concede before i was forced to, and i was right! i didn't play many tournaments, but i was obsessed with breaking the game and playing it "wrong".
Any other magic players anxiously scanning the pile of cards to make sure there wasn't anything expensive knowing how props get treated on this channel 😂😭
I noticed that you switched the chaotic collapses of your decoration, to planed domino effects. 4:46 This is less funny to me☹️ I liked the random, 11:29 although almost unavoidable collapses, way more. Those tickle my funny bone. Please put way more stuff in harms way and do it sloppy.
Off topic, BUT, a fraction creates a decimal number, and I was curious about NON-repeating finite decimals, HOW LONG CAN THEY BE? I used a grid of 1000x1000 numbers creating decimals from the fraction x/y, and the row X/512 is unique. It is the largest at 9 decimal places, and the pattern is interesting stepping like: 9, 8, 9, 7, 9, 8, 9, 6, etc, where these are the number of decimals with increasing numerator values. Excel is a pain with 15 significant digits, but there are no decimal values at lengths 10 and 11. Beyond this, Excel is rounding and ruins it. I haven't explored different bases, but that 512 stands out a lot. I did end up finding longer decimal lengths. 1024 and 2048 were interesting.
and don't forget "arbitrarily small" when considering the rational and the reals and the distinctions between them, along with the ordering problem of being given one rational fraction and trying the determine the _next_ one (and the corollary of always finding a rational [real] between two reals [rationals], so obviously '?' they alternate;-). Mathematicians get very frustrated at their inability to explain in common language how they managed to tie their shoelaces together (a minor distinction)!
Fascinating subject! What about a+bn when a = 0? You don't have to worry about the result being a multiple of 0. Is it possible there is infinite arithmetic progression of primes with a large enough b value, or has that also been proven not to exist? I'm sure Tao and Green probably covered this, but I'd be interested in seeing how this one case was proven (assuming I don't need anything higher level than basic calc to understand it)
If a=0, then when n=2 the result would be a multiple of when n=1, so those couldn’t both be prime. Overall yeah green/tao thoroughly proved the “can be arbitrarily long” part. To prove the “not infinitely long” part can be done pretty simply although I just included the general logic in this episode and not a thorough proof
If you like Magic, consider this combo (obviously, usually achieved having Hive Mind): - Wheel of Misfortune - Multiple players choosing it with Intervention Pact Result: each player says a number, the higher one wins that much life😂
I am not familiar with MTG but I remember there is a card that turns your life gain into your opponent's life loss. In this case, this combo becomes a large number naming contest.
I'm not sure "infinite" is entirely inaccurate for Magic combos that you can choose to end. Even if using it to win requires you to end it, barring external considerations (such as death by old age stopping you from continuing) it theoretically could be continued infinitely. I'm also thinking of the fighting game perspective, where any "infinite combo" will have some eventual end (opponent dies, player quits or messes up, the Sun explodes), but one might argue it is "infinite" in the sense that nothing within the combo sequence itself prevents it from being continued. Which is at least somewhat different from a combo that can be continued for a finite number of iterations, but something within the sequence will eventually force it to end.
If it ever terminates it ain't infinite... something magic players tend not to grasp. Arbitrarily Large Numbers, however are interesting. And some combos which could in theory be executed ad infinitum, do you no favor, as you need to be able to terminate them to use them. Comprehensive Rules 104.4b If a game that’s not using the limited range of influence option (including a two-player game) somehow enters a “loop” of mandatory actions, repeating a sequence of events with no way to stop, the game is a draw. Loops that contain an optional action don’t result in a draw. And 104.4f In a multiplayer game using the limited range of influence option, if the game somehow enters a “loop” of mandatory actions, repeating a sequence of events with no way to stop, the game is a draw for each player who controls an object that’s involved in that loop, as well as for each player within the range of influence of any of those players. Only those players leave the game; the game continues for all other players.
Someone put an ability on a mutavault that prevented me from winning the game. I couldnt remove the land so I just played Vito and Exquisite Blood and forced a draw
And the number of different ordering in a standards 52 card deck is 'astronomically' large. a fresh deal every second, after a billion years take a 1cm step along the equator, every time to reach the pacific take out a 5mL medicine spoon of water, every time it empties, and a sheet of A4 paper to a pile, when yo reach the moon start counting, when you reach a million visits, 'sigh' and realise your not even 90% of the way to dealing all the different arrangements. Even if you a deal every Planck interval (10e-43 sec) you can still only do the deals of a 40 card deck in the life of the universe!
Would it be possible to make a and n fractions (or even irrationals), then pick b in such a way as to generate a series of all integers, and if so can there be a formula of that type to always generate prime integers? If the former is possible then the case where a=n may no longer be a problem depending on which fraction (or irrational) they equal and what the choice of b was. My guess though is that the former is possible but the latter isn't.
I'ma start calling them arbatrairily large combos. Yugioh also has some infinate combos, although due to the rules you can't do them. If you were to make a play that would result in an endless loop then that play is illegal
if in magic, one player has a combo to make an arbitrarily large amount of life and the other player stacks a combo that can take an arbitrarily large amount of life away and they function both on the same turn and also at the same rate... is that now infinite?
No such situation can happen inside the rules of the game since the game uses a stack system for the actions taken. So no two things happen at the same time, they go in lifo order until that action resolves and ceases to exist.
@@friki1282 I am aware of that but my comment was more of an 'if this were to happen' kinda as an attempt to clarify bt arbitrarily large and infinite even further. But also I appreciate the comment, my best friend was once 15th highest rated magic player on the online game and he said the same thing to me, lmao!
The diagonal argument is clearly ridiculous. The matrix proposed by Cantor is based on a square matrix/list of permutations. It has to have equal length rows and columns. Any list of permutations will be iterative from left to right, but will grow by the factorial downwards. No such list of combinations or permutations exists in order to make the diagonal argument
I think my brain just broke. The largest prime would be arbitrarily large because just like with the highest number of digits in a prime there's always a bigger one but the bigger one always has a numeric value. Okay, with that detailed there's a problem. The largest prime starting at 2 (arbitrarily large): 2, 3, 5, 7, 11, 13 The number of primes(infinite): 1, 2, 3, 4, 5, 6 The number of digits in the largest prime(arbitrarily large): 1, 1, 1, 1, 2, 2 These patterns diverge with an arbitrarily large pattern at either side of the infinite pattern. I will convey that this pattern continues with the 104011th prime: Value: 1356409 Number of primes to this point: 104011 Number of digits: 7 Now, there's a bit of a problem with this logic obviously 104011 isn't the total number of prime numbers so isn't infinite, but 1356409 also isn't the largest prime. I don't exactly know how to think about all this.
You may be interested in learning about the concepts of cardinality and ordinality! To put it briefly, cardinality describes the size of numbers, while ordinality describes how they are ordered relative to each other. For numbers like integers, this distinction is trivial, as an integer's value and its place on the number line are one-to-one and describe essentially the exact same thing. But these concepts are much more useful when it comes to thinking about different types of infinities. In your example, "1356409" isn't just a value-it's a cardinal value! This is different than "104011," which is an ordinal value, because this number can be thought of representing the position in the sequence of prime numbers at which "1356409" appears. Cardinality describes a property of the number itself, whereas ordinality describes a property that only manifests in relation to other numbers. Because there's no defined size that describes infinity (at least in the context of this video), infinity also lacks a corresponding cardinality, so therefore any situation necessitating a cardinal representation (i.e., writing out the digits of a number) becomes impossible-this is why the number of digits is constrained to being arbitrarily large, not infinite. However, there is no such restriction on adding more items to the list of prime numbers, which is why "the number of primes (up to this point)" is allowed to be truly infinite. All in all, you did a really good job of intuiting the incongruity between cardinality and ordinality! I hope this explanation provides a useful way of thinking for you.
Is the difference between infinite and arbitrarily large actually well defined here? I mean... in magic you cannot define a variable as infinity. But... Can you in math either? There are very few examples where you can actually use infinity as a value in an equation.
Yeah it’s a well-defined concept within realms like natural numbers. It can be proven that within the system we use, there are an infinite amount of natural numbers but the amount of digits one can have is just up to arbitrarily large, due to describing quantities that can surpass any finite size but cannot be infinite. And then it can be proven that other types of number traits sometimes are infinite, arbitrarily large, or have a finite limit
But "going infinite" rolls off the tongue a bit better than "going arbitrarily large" But I guess there's a terrible commander strategy everyone will hate you for. Aim to actually go infinite, so everyone draws. Would make you a priority target, or get you banned from every table, but it's an amusing thought anyway.
was going to say, people not using 'arbitrarily large' could just be due to a branding issue. infinity is catchy. if AL was called fakefinity it'd probably catch on
If you mean money, it depends. This video cost probably $100-200 to make. Some videos cost less. I do a lot of parts of the process myself from scratch, but I do pay camera people for their time and sometimes need to buy props
I make less money than I spend on this channel currently but my other bonus channel (which has more subscribers due to shorts and is cheaper to produce) and my Patreon supporters add up to enough to cover the cost. I don’t mind sometimes spending money on making main episodes though, I see it as an investment and they are projects I’m proud of making
That’s what an arbitrarily large number means. “Arbitrarily large” isn’t the size of a single number, it’s a description of something that can be larger than any given finite number but not infinite, where you could pick googolplex or googolplex+1 or any finite number
There's no way it would fit in what is already a great video, but I hope that in your research you found the proof that MTG is Turing Complete - meaning that it is possible to set up a state in the game in which all actions are automatic and you can use it to execute any Turing machine (effectively any closed-system computer program). One of the ridiculous results of that is that it is possible to put the game into a deterministic state in which it will never terminate but you can't actually prove that's the case.
This is the longest episode of Grade -2 yet and I had a lot of fun making it. Thanks for watching! See the video description for links to my bonus channel, discord server, Patreon supporter names, and etc.
I'm glad the "etc." can be an arbitrarily large amount of Combo Class related content, but not infinite.
I like the word "unlimited" for situations where things can theoretically become arbitrarily large.
I'd say semifinite, @@johnfinkbiner3599.
Imagine doing a googol damage… bye bye Travis 👋🏻 you’ll never win!
Imagine visiting your family and suddenly
"Uncle, what was that noise?"
"Oh, that's just our neighbour teaching math"
I once played a game of magic where multiple players started “going infinite”, and we agreed to notate everything algebraically. One person made “x” tokens, and the next guy made “2x” tokens, etc, where x was arbitrarily large, but unspecified.
I believe you effectively rediscovered infinite ordinals my friend. That's really cool
Did someone make "x" ^2
Cards?
I think there is a rule for this, where if two players can go reactively infinite to eachother, the active player decides the amount of loops they perform and then based on that, the non active player decides
I'm always arguing with myself about how purposeful the chaos in these videos is. Whether it's scripted, semi-intentional, or just a happy result of chaotic set dressing.
That being said, the dominoes made me happy
A great example of something that seems infinite to me but is in fact merely arbitrarily large is my exponentially growing anxiety as I watch everything in your yard falling apart
Another type of infinity: the amount of havoc that Domotro can create inhis backyard during a single video while teaching us math. I have the feeling that with each new video, the amount of items thrown to the ground is greater than in the previous video. Eventually, there will be a video where there is no math and everything is chaos!
This video gives me mad scientist vibes. The dissheveled kind. Very entertaining!
How have I never seen this channel before. This is actually fantastic.
Thanks! Welcome
I used to have a huge blackberry bush in my back yard (about 3 acres). That intro takes me back.
Time to take up the infinite chess terminology and call it the omega mana/damage/etc combo
At the 5:00 mark, I was concerned he was going to drop a brick on his foot. Combo Class is the only math teaching position where steel-toed boots are recommended.
Yeah maybe I should start wearing boots. I haven’t bricked my foot yet but one time a metal clock fell on my foot and left a bruise for a few days haha
The amount of Rube Goldbergian preparation you put into your videos is just… 👌chef's kiss
How do these videos not get tens of thousands of views? They should be shown in every school in the world.
Tru
Thanks. Yeah hope more teachers use this content over time but so far it’s going good, since this channel is barely over 1 year old so I’m happy with its growth so far
I'd like to see a Magic game where somebody is on the brink of losing but manages to play an infinite combo to create a draw.
This was actually a big problem on Arena's Alchemy format. There was a deck that used Vesperlark (a 2/1 that brings a creature with power 1 or less from the graveyard to the battlefield when it dies, non-optional) in combination with Davriel's Withering, an instant that gives a creature -2/-1 "perpetually", which was a concept they were trying for this digital-only format, where the card kept the new power and toughness even as it changed zones. This meant you could make your own Vesperlark a 0/0, and if it was the only creature in your graveyard, it would HAVE to get itself back, then it would instantly die again, be forced to get itself back, etc. If you had something that drained your opponent when your creature died, this meant assembling this combo wins you the game, but if it just wasn't looking good, even without the part that actually wins you could just as easily draw the game and try again. They had to change Davriel's Withering to say "an opponent controls" in order to reign the deck in.
In chess, this is possible :)
@@Giantcrabz no, going infinite to draw is a valid game action you are free to make if you so desire.
Try googling "LSV 3 oblivion rings"
@@Giantcrabz I think it would be considered unsportsmanlike if the deck wanted to draw rather than win specifically (certain cards/strategies have been banned in the past due to this), but if someone is simply using the resources they have available to stop themselves losing I believe most players would consider that valid
ah yes, classic marker towers, we've all made lightsabers out of them
glad to see your inner child refuses to die Demotro
Well this is proper earlier than I thought I'd be
This man is gonna blow up hes too good
I love picking blackberries... the concept of arbitrary largeness fits perfectly there! Learned a lot and thanks!
This man is the Linus for Mathematics. He drops EVERYTHING. I love the chaos.
One of my favorite Magic stories involves an infinite loop. My opponent was using Sanguine Bond + Exquisite Blood- a loop where one card makes you gain life when an opponent loses life, and the other causes your opponent to lose life whenever you gain life. It's a loop without any way to manually break it, but it isn't typically truly infinite because as soon as a player drops to 0 life they lose the game. But when my opponent set it off against me, he didn't realize I had a Platinum Angel out- a card that says you can't lose the game, even when your life dips into the negatives. The game ended in a draw.
The collapse of the set as you talked was amazing.
The way you explain these concepts are on par with my university professor. They make both intuitive sense while covering the natural progression of what we would want to know about their properties. Seriously, you are amazing and I wish I found you before discrete math 2.
Thanks :)
I love this! I learned alot and i love the part where everything falls over 😂
Hey former MTG judge (Judges tell players how the rules work in tournaments), if you preform an "infinite loop", it matters for tournament logistics that these loops can be shortcut, so often after demonstrating that you could theoretically continue to do an action forever, you are then required to pick a number. what is more interesting and something you might like looking into is "non deterministic loops", like legacy 4 horsemen, which uses a combo to put the top cards of your library into your graveyard. you then eventually hit cards that from your graveyard win the game, but the hang up is that in order to play around people messing with your graveyard and to make the winning combo from your graveyard more efficient, you have to play a "shuffler", or a card that when put into your graveyard, shuffles your graveyard back into your deck. The issue is that in order to shortcut and "pick a number", you have to be able to definitively say that 100% doing an action that many times creates a specific "boardstate" (how the game pieces are in play and how resources have changed, like making 4 million mana, or doing 8 thousand damage), then you can take different actions other than the loop you demonstrated. This is theoretically not possible in a deck like 4 horsemen, because if you mill each card one at a time, you could theoretically mill your shuffle card infinite times in a row, and because it changes how the loop is preformed and what the loop does past that point, and because you cannot specifically give a number of activations required to win the game or produce the desired boardstate, it is not shortcutable and therefore is "soft banned" in tournaments because of rules on playing too slowly. If you ever want to talk about the crazy mathematics in magic the gathering, feel free to reach out, I can give you my discord or twitter or something, I was a magic judge for about 5 or 6 years and have a math degree, so I always love talking about this kind of stuff. (also, some of those cards look pretty old, you should check out some of the prices, the older cards can go for literal thousands of dollars these days depending on what they are and what set they were printed in. I would recommend something like TCGplayer, it's what most people use to price cards)
Hey late reply here, but yeah I read about the 4 horsemen loop and it sounded interesting! About the card value, I don't think I have any of value (some are as old as Arabian Nights but mostly just beat up commons/uncommons). There are a few cards I set aside that are worth like $5 each but I don't think I have many worth more than that. As for your offer to chat, yeah I am down! I have some other MTG topics I've been personally brainstorming that I would love a judge's opinion on. Send me an email, or you can message on me the Combo Class Discord (both linked in the video description)
My favourite combo not mentioned here is Delina, Wild Mage alongside Wyll, Blade of Frontiers or Pixie Guide
It has the excellent properties of being Arbitrarily Large and Non-deterministic
Even better, Delina needed her wording changed to add a "may" so that you wouldn't have to play it out until you eventually failed all of the dice rolls
There's a combo or two that can make arbitrarily large amount of _squirrel_ tokens in MtG... Scurry Oak and Rosie Cotton or Scurry Oak, Heliod (the same as the one with Walking Ballista) and any creature that gives you life when a creature enters the battlefield.
Yeah I almost included a combo with “squirrel nest” in the episode but it didn’t fit as easily in the overall explanation
Great explanation for the difference between infinite and the infinitely smaller quantity of just arbitrarily large.
Lol, I like that we're talking about mtg here. I didn't know that there was a rule for resolving a game where the stack couldn't empty. I might toss in one of these combos into the next _chaos_ build I do. Like in commander, "Okay, is the game now way _too_ complicated to play? Are we tired?" Stages "unresolvable combo". "Congrats we're all winners!"
Come for excellent concepts, stay for the falling clocks
One example that I have encountered is that when we use like algorithms/infinite series to calculate irrational numbers, we can calculate them to an arbitarily large amount of precision. You can calculate as many finite number of digits as you need given enough time.
But you can’t calulate it to infinite precision because we can never write down/store all of the digits since it is infinite, and because we do actually need to stop somewhere to see our result.
I’ve had to be careful with this when I explain this to some people at times.
Way cool concept. Thanks Domotro! You are an asset to TH-cam.
Thanks :)
You've obiously never been Blackberry picking with me 🤣🤣🤣🤣🤣🤣🤣🤣
Another interesting fact about Magic: The Gathering is that it's "Turing complete", meaning it can be used to simulate a Turing machine (and therefore can technically simulate any computer program). Kyle Hill did a great video covering this on Because Science; search "I built a computer in Magic: The Gathering".
And due to being turing complete, we can construct game positions where it is *impossible to prove* whether the combo is infinite, or simply very large.
Thanks, Godel.
Only found your channel recently and I love the Intellectual chaos. I have been using the term Finite for combos that are not truly infinite. There are true infinite combos but require more of a setup (or dumb luck) to make work. This can be important in some cases where you might gain and arbitrary large amount of life but lose the pieces to set it up again. Then a few turns later your opponent creates a sequence that causes an arbitrary large amount of life loss. I've had it happen where I was quintupling my life during each of my end steps in addition to other life gain effects during mine, and other player's turns. I've been watching more of your other videos but as a MtG player I had to comment on this one.
the way "arbitrarily large" was taught to me was to say that you pick a finite large number to do some math with while knowing that you can do the exact same math again with an even larger number. Another way i would put it is that it means to take a finite number to do someth9ing an infinite value can't or shouldn't do. and also if it doesn't matter what the exact value of the number is or even its magnitude. the important thing is that your arbitrarily large number could be larger by any amount and the math still holds true. it really only came up when messing with primes though and doing some set theory stuff
22:56 Squirrel!!!
As for the other parts of this video. I did meet someone playing these arbitery large combo's.
He always won in the first turn.
And the whole room eventually stopped playing the game....with that guy.
Looking forward to the next video.
Thanks! To note, if that person you mentioned was winning on the 1st turn more than 90% of the time, there’s a good chance they were cheating too (by fake shuffling or something). Even the greatest decks sometimes don’t draw the right cards for a turn-one win. But yeah those early combo wins caused a bunch of cards to be still banned or restricted in various tournament formats
Great video! Not sure if it's quite the same, but my 7th grade math teacher explained to us that rays or graphs which have a starting point but never end in the other direction aren't necessarily 'infinite' because they also have an infinite amount of unused space in the other direction, so he referred to those as 'infinitely long'
I think he also explained that between any two numbers exist an arbitrary large number of points, but not actually infinite since a decimal can't go on for infinite digits and a fraction can't have infinity as a denominator. Math has always been my favorite subject but have stopped paying much attention since graduation admittedly, so thanks for creating this neat explanation!
Thanks. And a geometric interpretation would depend on what you’re looking for, like if you are asking about the amount of points there are an infinite amount of points in a line segment, but if you asked for the maximum length on a coordinate plane that a line segment could have, the answer would be arbitrarily large and not infinite
This distinction actually matters in game sometimes. The most common example is that Arbitrarily Large life pretty much always loses to Arbitrarily Large damage. If they were both infinity, then it would technically equal zero.
Another example is playing a true infinity combo, like the Sporemound combo. If you can somehow find a way to turn it into damage, like Impact Tremors for instance, then you can turn the draw into a win.
Arbitrarily many life can beat arbitrarily many damage as long as it can be activated at instant speed after the opposing player has chosen their arbitrarily large amount of damage. Spike feeder + pridemate/heliod is one of the most common "infinite" life combos and it falls into this category
*technically* if a player with infinite life takes infinite damage, the result is undefined under the common rules of math, and if you do define that it'll probably not be 0 but something like "bottom element"
@@dm9910unless the arbitrarily large damage is dealt on the turn of the player with the arbitrarily large life gain combo. In this case, the player with the life gain combo has to be the first to break the loop
Demotro is pushing it with this one, the wightboard pushing the bricks, and then the bricks pushing the gloab is too comical for me.
Once upon a time, I perused the detailed and comprehensive Magic rules. Somewhere in one of the documents (I forget if it was the comprehensive rules or the tournament rules), it specifically states that you can't do "infinite" anything. It specifically gives an example of a loopable series of actions that generate a token and states that a player may, for example, say "I'll generate a million tokens." as shorthand for doing that loop a million times.
At least, I think. It's been a while.
wow thank you for talking about how people misuse "infinite" in mtg, that's such a niche thing that bothers me more than it should lol
Great video as always! i dont play magic the gathering but, i know too well about those arbitrarily large damage combos 😅
Watching this channel on 1.25 speed is the best way
For a second, I thought at 4:30 you were going to say "So what does it mean to be 'arbitrarily large'? Well, this isn't a term that can describe your mother..."
This man make me learn math by putting mtg in it
Haha one by one I will trick everyone into appreciating numbers
About the mana: "Generic" isn't a type of mana, but rather a type of mana *cost.*
Yeah back when I originally played, “colorless” mana symbols didn’t exist, and the term was used interchangeably with “generic” by players so I wasn’t sure if colorless or generic was the right term for that now, but I guess I said it right in the episode since it looks like krark clan ironworks’ official rules text now shows it as colorless symbols
I consider myself a level 0 judge (not an official one, but I do know the rules very comprehensively), and every time someone in a game in involved in says they make something with "infinite" power or gain "infinite" life, I always mention that you do have to actually pick a number of times to do it, even if it's "a trillion trillion", especially relevant with multiple of these loops belonging to different players, since if you gain an arbitrarily large amount of life but i can deal an arbitrarily large amount of damage, we need to be able to figure out who actually wins, and it typically comes down to who gets to name their number second
What would happen if they could activate both arbitrarily large combos on the stack, though? (Say the disciple of the vault combo with leyline of anticipation vs. Marrow-gnawer + thornbite staff + soul warden)
@@Pzcat0after a loop involving multiple players is established the active player (who's turn it is) must make a decision that ends the loop
@@Il_DilettanteThe real answer is that the player without priority should’ve played Stifle
So the big question I'm sure everyone's asking is "Is there a class of sequences between 'arbitrarily large' and infinitely large?'
These kind of loops also happen in YuGiOh, in fact the official konami documentation mentions "infinite loops" and gives an example with a card called Pole Position. In this case if the loop does not eventually cause a victory condition, a judge determines what's the problem card and sends it to the graveyard so the game can continue.
you made me curious about Magic so I did some quick research and found out that someone proved that this game is Turning-complete. wow. xD
loving the MTG stuff
Nice. It will show up again at some point, since I’m brainstorming a few future episodes (out of the many topics I’m brainstorming) that relate to Magic again. But that might not be until later this grade
@@ComboClass do you plan on further exploring the largest possible deck to always win on turn 1 with? I’ve been thinking about this puzzle a lot
@@vantheman8050 Yeah I will return to that at least in a livestream on the Domotro channel, and possibly in a main episode on this channel someday. I think I've proved a 21 card deck that would work for that puzzle, and have some ideas of how 22 or 23 might be possible but haven't gone through all the logic on those yet
The term "arbitrary large" is very useful in software development, though usually this is in practice impossible because we're limited by very large but very much finite amount of memory, address space, and time (and while time is for our purposes infinite, our patience is very limited so things that doesn't complete in less than a few milliseconds is unusable in frontend code; and things that doesn't complete in less than 30 seconds are useless in interactive server code; and number crunching that doesn't finish in less than a few months on a super computer is pretty much unworkable).
The term "arbitrary" (but implicitly bounded by such time and space limits) is actually even more useful in programming.. It's trivial to "hardcode" something, i.e. something that spits out the same result every time with limited to no inputs; it's harder to make something that works well with sensible and carefully chosen inputs (such as what you could expect most competent and trusted employees to provide most of the time; but it's really hard to make something that works okay and at least don't break badly (or if it break anyway, doesn't cause a security problem) for any **arbitrary** input (i.e. provided by incompetent or malicious users of a website).
And even the supposedly well intentioned and competent editors quite often end up accidentally providing inputs that breaks stuff or at least makes it look rather ugly. And one rather basic thing that often breaks layouts is simply prime numbers: any grid layout for a variable amount of stuff will somewhat break when provided a prime number of elements or simply a number of elements that doesn't divide into the current number of columns. For large datasets you can usually avoid this by simply always fetching a highly composite number of elements, because if the dataset is sufficiently large (such as all the videos on TH-cam) any user will "never" reach the final page. But for smaller size websites any frontend developer will have to spend a lot of time figuring out clever fallbacks to avoid having a single element all on it's own on the end of a grid whenever the number fetched is one more than a multiple of the number of columns.
In fact, if I spend a 8 hours working on a feature; quite often I wrote the implementation for the best case sensible inputs in a few minutes and the rest of the time is spent thinking about and mitigating issues that might happen with problematic arbitrary inputs. Especially this is true of front end code that most of the time is rather unimportant for security, but is essential that it always looks good (or at least not horrible) no matter what nonsense is thrown at it. Only in the most bizarre cases would you prefer to just abort and print an error message rather than somehow handling bad input.
One question about "arbitrarily long" arithmetic progressions of primes. If we exclude the special case for when n is a multiple of a, can you have an infinite set of primes of the form a+bn for all n coprime to a? I suspect that it isn't but it's still fun to ask
Nope.
That would mean that the density of primes cannot drop below the density of such numbers a+bn with n coprime to a. Yet, the density of primes goes as 1/ln(n)...
I have ended a game with an infinite draw before! i've also won a game with a deck that was designed to an arbitrary amount of time! if i played the right goblin cards, i was able to resort my entire deck, then shuffle it, then resort it an arbitrary number of times. because it wasn't infinite, it was not a draw, and i won because my opponent got so mad and he didn't want to wait for the time limit which would force me to concede. i was essentially gabling that he would concede before i was forced to, and i was right! i didn't play many tournaments, but i was obsessed with breaking the game and playing it "wrong".
Any other magic players anxiously scanning the pile of cards to make sure there wasn't anything expensive knowing how props get treated on this channel 😂😭
Haha. Don’t worry, I’m pretty sure none of my cards are very valuable and they survived the scene safely anyway
Do a video about aleph null vs continuum
loved the episode, as an avid magic player, also good job explaining magic in 30 seconds for someone who doesnt know the game
In my country, we call it "maths".
love your channel bud. t, math teacher
I thought this guy was the same guy from explosions and fire, i was wrong
this channel is like bill nye on some level, and CSI miami on another, and finally midnight gospel on a little bit.
There are 2 things that still bother me when people say them; "infinite" when they meant "arbitrarily large", and "try and" when the meant "try to".
I noticed that you switched the chaotic collapses of your decoration, to planed domino effects. 4:46 This is less funny to me☹️
I liked the random, 11:29 although almost unavoidable collapses, way more. Those tickle my funny bone.
Please put way more stuff in harms way and do it sloppy.
Guess it's time for some Halo Arbitrarily Large.
To arbitrary largeness... and beyond!
Off topic, BUT, a fraction creates a decimal number, and I was curious about NON-repeating finite decimals, HOW LONG CAN THEY BE? I used a grid of 1000x1000 numbers creating decimals from the fraction x/y, and the row X/512 is unique. It is the largest at 9 decimal places, and the pattern is interesting stepping like: 9, 8, 9, 7, 9, 8, 9, 6, etc, where these are the number of decimals with increasing numerator values. Excel is a pain with 15 significant digits, but there are no decimal values at lengths 10 and 11. Beyond this, Excel is rounding and ruins it. I haven't explored different bases, but that 512 stands out a lot. I did end up finding longer decimal lengths. 1024 and 2048 were interesting.
Clearly there should be an Ancestral Walk: Mox Time Lotus card.
I like your large Orange Calculator.
and don't forget "arbitrarily small" when considering the rational and the reals and the distinctions between them, along with the ordering problem of being given one rational fraction and trying the determine the _next_ one (and the corollary of always finding a rational [real] between two reals [rationals], so obviously '?' they alternate;-).
Mathematicians get very frustrated at their inability to explain in common language how they managed to tie their shoelaces together (a minor distinction)!
Fascinating subject! What about a+bn when a = 0? You don't have to worry about the result being a multiple of 0. Is it possible there is infinite arithmetic progression of primes with a large enough b value, or has that also been proven not to exist? I'm sure Tao and Green probably covered this, but I'd be interested in seeing how this one case was proven (assuming I don't need anything higher level than basic calc to understand it)
those are just multiples of b
If a=0, then when n=2 the result would be a multiple of when n=1, so those couldn’t both be prime. Overall yeah green/tao thoroughly proved the “can be arbitrarily long” part. To prove the “not infinitely long” part can be done pretty simply although I just included the general logic in this episode and not a thorough proof
@@ComboClassOh sheesh. It's so obvious I don't know how I missed it.
It happens to the best of us :)
this channel should do a video on chaos theory 😅
If you like Magic, consider this combo (obviously, usually achieved having Hive Mind):
- Wheel of Misfortune
- Multiple players choosing it with Intervention Pact
Result: each player says a number, the higher one wins that much life😂
I am not familiar with MTG but I remember there is a card that turns your life gain into your opponent's life loss. In this case, this combo becomes a large number naming contest.
I'm not sure "infinite" is entirely inaccurate for Magic combos that you can choose to end. Even if using it to win requires you to end it, barring external considerations (such as death by old age stopping you from continuing) it theoretically could be continued infinitely. I'm also thinking of the fighting game perspective, where any "infinite combo" will have some eventual end (opponent dies, player quits or messes up, the Sun explodes), but one might argue it is "infinite" in the sense that nothing within the combo sequence itself prevents it from being continued. Which is at least somewhat different from a combo that can be continued for a finite number of iterations, but something within the sequence will eventually force it to end.
Nobody tell him about yugiohs "darkly big rabbi"...
What's the largest value known where n goes up to a-1?
If it ever terminates it ain't infinite... something magic players tend not to grasp. Arbitrarily Large Numbers, however are interesting. And some combos which could in theory be executed ad infinitum, do you no favor, as you need to be able to terminate them to use them.
Comprehensive Rules 104.4b If a game that’s not using the limited range of influence option (including a two-player game) somehow enters a “loop” of mandatory actions, repeating a sequence of events with no way to stop, the game is a draw. Loops that contain an optional action don’t result in a draw.
And
104.4f In a multiplayer game using the limited range of influence option, if the game somehow enters a “loop” of mandatory actions, repeating a sequence of events with no way to stop, the game is a draw for each player who controls an object that’s involved in that loop, as well as for each player within the range of influence of any of those players. Only those players leave the game; the game continues for all other players.
22:57 ❤
8:00 why does this feel like a fourier transform? maybe something can be done with this
Someone put an ability on a mutavault that prevented me from winning the game. I couldnt remove the land so I just played Vito and Exquisite Blood and forced a draw
And the number of different ordering in a standards 52 card deck is 'astronomically' large.
a fresh deal every second, after a billion years take a 1cm step along the equator, every time to reach the pacific take out a 5mL medicine spoon of water, every time it empties, and a sheet of A4 paper to a pile, when yo reach the moon start counting, when you reach a million visits, 'sigh' and realise your not even 90% of the way to dealing all the different arrangements.
Even if you a deal every Planck interval (10e-43 sec) you can still only do the deals of a 40 card deck in the life of the universe!
Would it be possible to make a and n fractions (or even irrationals), then pick b in such a way as to generate a series of all integers, and if so can there be a formula of that type to always generate prime integers? If the former is possible then the case where a=n may no longer be a problem depending on which fraction (or irrational) they equal and what the choice of b was. My guess though is that the former is possible but the latter isn't.
You mean rounding to an integer.
Nope, the density of primes becomes arbitrarily low, and your proposed recipe doesn't.
Post MaLabCoat has a a point here
Has anyone arranged TH-cam math educators on a continuum with Combo Class on one end and 3Blue1Brown on the other end?
“Chaotic good” vs “lawful good” alignments haha (I think 3blue1brown is great by the way)
420420, nice
I have something arbitrarily large
I'ma start calling them arbatrairily large combos. Yugioh also has some infinate combos, although due to the rules you can't do them. If you were to make a play that would result in an endless loop then that play is illegal
Tolarian Community College: Mathematics Department.
if in magic, one player has a combo to make an arbitrarily large amount of life and the other player stacks a combo that can take an arbitrarily large amount of life away and they function both on the same turn and also at the same rate... is that now infinite?
No such situation can happen inside the rules of the game since the game uses a stack system for the actions taken. So no two things happen at the same time, they go in lifo order until that action resolves and ceases to exist.
@@friki1282 I am aware of that but my comment was more of an 'if this were to happen' kinda as an attempt to clarify bt arbitrarily large and infinite even further. But also I appreciate the comment, my best friend was once 15th highest rated magic player on the online game and he said the same thing to me, lmao!
I also appear to be early
The diagonal argument is clearly ridiculous. The matrix proposed by Cantor is based on a square matrix/list of permutations. It has to have equal length rows and columns. Any list of permutations will be iterative from left to right, but will grow by the factorial downwards. No such list of combinations or permutations exists in order to make the diagonal argument
There's way too many R's in arbrirtrary and I never know where to insert them all when writing...
I think my brain just broke. The largest prime would be arbitrarily large because just like with the highest number of digits in a prime there's always a bigger one but the bigger one always has a numeric value. Okay, with that detailed there's a problem.
The largest prime starting at 2 (arbitrarily large): 2, 3, 5, 7, 11, 13
The number of primes(infinite): 1, 2, 3, 4, 5, 6
The number of digits in the largest prime(arbitrarily large): 1, 1, 1, 1, 2, 2
These patterns diverge with an arbitrarily large pattern at either side of the infinite pattern.
I will convey that this pattern continues with the 104011th prime:
Value: 1356409
Number of primes to this point: 104011
Number of digits: 7
Now, there's a bit of a problem with this logic obviously 104011 isn't the total number of prime numbers so isn't infinite, but 1356409 also isn't the largest prime. I don't exactly know how to think about all this.
You may be interested in learning about the concepts of cardinality and ordinality! To put it briefly, cardinality describes the size of numbers, while ordinality describes how they are ordered relative to each other. For numbers like integers, this distinction is trivial, as an integer's value and its place on the number line are one-to-one and describe essentially the exact same thing. But these concepts are much more useful when it comes to thinking about different types of infinities.
In your example, "1356409" isn't just a value-it's a cardinal value! This is different than "104011," which is an ordinal value, because this number can be thought of representing the position in the sequence of prime numbers at which "1356409" appears. Cardinality describes a property of the number itself, whereas ordinality describes a property that only manifests in relation to other numbers.
Because there's no defined size that describes infinity (at least in the context of this video), infinity also lacks a corresponding cardinality, so therefore any situation necessitating a cardinal representation (i.e., writing out the digits of a number) becomes impossible-this is why the number of digits is constrained to being arbitrarily large, not infinite. However, there is no such restriction on adding more items to the list of prime numbers, which is why "the number of primes (up to this point)" is allowed to be truly infinite.
All in all, you did a really good job of intuiting the incongruity between cardinality and ordinality! I hope this explanation provides a useful way of thinking for you.
I think it would be more accurate to say that the largest prime does not exist because the primes do continue to infinity
Is the difference between infinite and arbitrarily large actually well defined here? I mean... in magic you cannot define a variable as infinity. But... Can you in math either? There are very few examples where you can actually use infinity as a value in an equation.
Yeah it’s a well-defined concept within realms like natural numbers. It can be proven that within the system we use, there are an infinite amount of natural numbers but the amount of digits one can have is just up to arbitrarily large, due to describing quantities that can surpass any finite size but cannot be infinite. And then it can be proven that other types of number traits sometimes are infinite, arbitrarily large, or have a finite limit
11:09 *HOW*
But "going infinite" rolls off the tongue a bit better than "going arbitrarily large"
But I guess there's a terrible commander strategy everyone will hate you for.
Aim to actually go infinite, so everyone draws.
Would make you a priority target, or get you banned from every table, but it's an amusing thought anyway.
Fakefinity
was going to say, people not using 'arbitrarily large' could just be due to a branding issue. infinity is catchy. if AL was called fakefinity it'd probably catch on
did you just feed a nut to your pet squirrel
How much do you spend per video?
If you mean money, it depends. This video cost probably $100-200 to make. Some videos cost less. I do a lot of parts of the process myself from scratch, but I do pay camera people for their time and sometimes need to buy props
@@ComboClass Well I hope TH-cam pays you well then
I make less money than I spend on this channel currently but my other bonus channel (which has more subscribers due to shorts and is cheaper to produce) and my Patreon supporters add up to enough to cover the cost. I don’t mind sometimes spending money on making main episodes though, I see it as an investment and they are projects I’m proud of making
I just name a googleplex if I want the games biggest amount, because arbitrary “numbers,” are not allowed as stated in the rules.
That’s what an arbitrarily large number means. “Arbitrarily large” isn’t the size of a single number, it’s a description of something that can be larger than any given finite number but not infinite, where you could pick googolplex or googolplex+1 or any finite number
Tree(3) has entered the chat
There's no way it would fit in what is already a great video, but I hope that in your research you found the proof that MTG is Turing Complete - meaning that it is possible to set up a state in the game in which all actions are automatic and you can use it to execute any Turing machine (effectively any closed-system computer program). One of the ridiculous results of that is that it is possible to put the game into a deterministic state in which it will never terminate but you can't actually prove that's the case.
Yeah I did read the paper about that and have considered making an episode about that someday in the future. Super cool