An impossible game at the heart of math

  • Опубликовано: 3 месяца назад

    подписчиков: 18 тыс.

    Strategy stealing, the axiom of determinacy, and why it's incompatible with the axiom of choice. #SoME3
    Resources to learn more and other interesting notes:
    Play online:
    Wikipedia article:
    List of known best first moves:
    Also check out chapter 18 in the book "Winning Ways for Your Mathematical Plays"
    Zermelo's Theorem:
    The Axiom of Determinacy:
    Wikipedia article:
    Borel Determinacy Theorem:
    The argument in the video is essentially given as Proposition 28.1 in the book The Higher Infinite by Akihiro Kanamori, but it is stated in more complicated terms. You can also find some discussion of it at this link:
    The Axiom of Choice:
    Wikipedia article:
    Banach-Tarski Paradox:
    Vsauce Video on Banach-Tarski Paradox: • The Banach-Tarski Paradox

@spenjaminn3846 +501

fun fact: for the “infinitely many primes” game, alice doesn’t even need to know where the next prime number is. since it’s been proven that for any integer n there is at least one prime number between n and 2n, alice simply has to color up to 2n, where n is the current total number of squares colored

@jonathandawson3091 +300

A set is good if sum of reciprocal is infinite. Alice wins. Strategy: No matter what Bob chooses, choose the next consecutive series of numbers to get a total sum of 1 or more. This is always possible since sum(1/n) diverges.

@empmachine +43

Chomp is cool, I've never seen it before.

@martinshoosterman +111

I remember taking a logic class a few years ago where i first saw a theorem proven by constructing a game where 1 player wins if and only if the theorem is true. And then showing that the player has a winning strategy.

@ethannguyen2754 +64


@CtHtThomas +69

Shockingly good video - pop science levels of understandability and intrigue, but real arguments, definition sketches, and proof ideas!

@ShankarSivarajan +46

My favorite quote about the axiom of choice: "The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's Lemma."

@columbus8myhw +8

Such games (in addition to obviously requiring an infinite amount of time) also require "supercommunication", that is, the ability to communicate infinite amounts of information (in fact, in this case, uncountably infinite amounts of information, in order to describe which sets are good and which are bad). This sort of setup often leads to independence results where the axioms of set theory don't prove or disprove either outcome.

@TheManxLoiner +82

Fun fact / terminology: the choice of what sets are good or bad is known as an 'ultrafilter'. Definitely up there as coolest term in mathematics! But I had no idea about this strategy stealing property - thanks for the video. Excellent choice of topic - accessible to wide audience, unknown to most mathematicians (e.g. I did not know about it despite knowing about ultrafilters!) and mathematically super interesting.

@yf-n7710 +6

I didn't immediately realize you were taking the axiom of choice, but I did see it before you mentioned it, which I'm proud of. I've never taken any classes which go over ZF set theory; all my math classes which used sets just gave an overview of naive set theory and then said "technically this version leads to paradoxes but within the context of what we're learning here, it'll work".

@d_ho__ +19

I didn't (yet) get this video as an option in the SoME3 voting, but I will say that I'm pretty sure I would have voted for it. As a person with little math education beyond high school and a semester of college calculus, I found this very accessible and clear, and it was the first piece of math explanation that in any way explained an alternative to the axiom of choice (which I never quite understood why one would or wouldn't "take" it... with most of my prior research being prompted by random xkcd comics)

@myrus5722 +14

Absolutely absolutely amazing video! You get a ton of things about being a good math explainer correct, both on the technical “how to I instill this idea” side and the emotional “how do I make this content fun, engaging, and properly motivated” side. Please consider doing more videos on the fundamental axioms of math, it’s a very ripe topic for educational videos.

@5thearth +3

This problem feels basically related to the halting problem and/or incompleteness theorem--It's always possible to pose questions that can't be answered.

@doctorjerbear3177 +14

This game reminds me of the prisoner problem where prisoners are lined up, each has a white or black hat, each can see the hats in front of them, and they must guess the color of their own hat. If the prisoners get to plan a strategy beforehand, then for finitely many prisoners, all but the first prisoner can guess correctly. The first prisoner guesses "white" or "black" depending on whether he sees an odd or even number of black hats in front of him, and iteratively the rest of the prisoners in line can figure out their own hat color.

@johandaun874 +44


@a52productions +1

This is a really great video! I love the subtle connection between taking the top right square in chomp, whether to include the number one in the list of composites, and whether Alice chooses the number 1 in the complement of the determinacy game. These little parallels really help with understanding!

@nathanielhellerstein5871 +3

I'm a fan of Koenig's Lemma: any tree with infinitely many nodes and finite branching has an infinite branch. It's equivalent to what I call the Bush Theorem: any tree with finite branching and finite branches has finitely many nodes. I consider that intuitive. Koenig's Lemma implies the Compactness Theorem: for any set of statements, if every finite subset is consistent, then the set as a whole is consistent. This in turn implies nonstandard analysis, which has genuine infinitesimals. It also implies the Lowenheim-Skolem Theorem, which says that set theory has countable models. Since I've had it up to here with transfinite cardinals, this appeals to me.

@marvinmarvin38 +15

The "I knew you would've done that so I did the thing that would counter the thing that you would've done", but infinitely and thus first player being unable to make a move, or let's say the winning move .

@thetruetri5106 +1

We could just say red is a good set if it contains more numbers than the blue set. Then both blue and red would have a winning strategy by just choosing an amount of numbers bigger than the difference to the other set

@eammonful +12

Interesting video. Another channel just came out with a good video on infinite chess which I've seen come up in introductions to tge axiom of determinacy. Also you should tag this with


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