> fond-math > théorie-des-ensembles > death-by-infinity-puzzles-and-the-axiom-of-choice-mathologer

Death by infinity puzzles and the Axiom of Choice

Mathologer - 2017-01-21

In this video the Mathologer sets out to commit the perfect murder using infinitely many assassins and, subsequently, to get them off the hook in court. The story is broken up into three very tricky puzzles. Challenge yourself to figure them out before the Mathologer reveals his own solutions. Featuring Batman, the controversial Axiom of Choice and a guest appearance by the Banach-Tarski paradox. 

The pictures that I used for the Banach-Tarski ball splitting action were grabbed off the brilliant VSauce's video on the Banach-Tarski paradox (https://youtu.be/s86-Z-CbaHA) I mainly did this for easy reference since most people here will have seen this video and in this way would be able to connect easily with what I am talking about here. 
I also mention that those sets that get pushed around in the Banach-Tarski paradox are constructed using the Axiom of Choice. Vsauce actually does not mention this although this is really a big deal as far as mathematics is concerned (understandable though since his video was already very long).  Here is a link to the spot in the Vsauce video where the Axiom of Choice is envoked (although you have to have a really close look to see how :)   https://youtu.be/s86-Z-CbaHA?t=14m2s

There is a very nice TEDed video about the finite version of the last of our puzzles: https://youtu.be/N5vJSNXPEwA  The solutions to both the infinite and the finite version are closely related. 

Oh and today's t-shirt is from here  http://shirt.woot.com/offers/infinite-doughnut?ref=cnt_ctlg_dgn_3

Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.



Mathologer - 2017-01-21

Very important: If there is anything about what I say that you are not sure about please ask. There are a lot of very knowledgable people roaming the comments section that you help out :)
There is a very nice TEDed video about the finite version of the last of our puzzles: https://youtu.be/N5vJSNXPEwA The solutions to both the infinite and the finite version are closely related.
I also mention that those sets that get pushed around in the Banach-Tarski paradox are constructed using the Axiom of Choice. Vsauce in his otherwise brilliant video on the Banach-Tarski paradox actually does not mention this although this is really a big deal as far as mathematics is concerned (understandable though since his video was already very long). Here is a link to the spot in the Vsauce video where the Axiom of Choice is envoked (although you have to have a really close look to see how :) https://youtu.be/s86-Z-CbaHA?t=14m2s

Jeremy Lunsford - 2018-06-29

Mathologer is it possible to,use this to gain advantage over a fair roulette wheel?

Nathaniel Prawdzik - 2018-07-30

Mathologer The strange thing is that Batman can't actually get to any of the assassins. That's because whichever assassin you think he can get to, has one prior which would kill him, meaning he can't get to the one you thought of. So the person hiring this should be clever enough to realize he can save a lot of money by not hiring any of the other assassins after the one that kills Batman. But we already said he can't get to any of the assassins, so the person hiring them should therefore not hire any assassins. But now Batman lives as a strange consequence of the fact that his death is guaranteed in this scenario.

Jeremy Lunsford - 2018-07-30

Nathaniel Prawdzik what if the assassin we think he can get to is the very first One? There's nobody before the very first one.

agilsaelan - 2018-09-01

I don't get how you get the equivalence classes and how you choose which equivalence class to apply. Can i get some explanation on these

Uday Sinha - 2018-09-11

Can't assassin just shout the number on the head of the assassin in front of them? That way, the assassin in the back has a 50% chance while all the others have 100%

605jik - 2017-01-21

Its the perfect murder because all those assassins would create an infinitely dense human meat ball which would create a black hole swallowing batman and any other targets you may ever have. Also police cant do anything about it. Pepper spraying the black hole will only make it stronger.

Mathologer - 2017-01-22

Ah yes, I'll add this to my list of properties that make this setup the perfect murder :)

Mighty Doge - 2017-02-14

It's perfect murder because this way the assassins would kill every possible target in universe and the universe itself :D

A transcendental Experience - 2019-11-20

It’s the perfect crime

שחר א. - 2020-02-18

Wait, what about the criminal committing it?!

1ucasvb - 2017-01-21

This explanation is bogus because the assumption that Batman can be killed is incorrect.

anand singh - 2017-02-02


Danny Sano' - 2017-04-15

if this is a setup, even if it is an infinite setup, the outcome should be the same. if you add 2+2 in an infinite equation all of the results will still be 4. There is no variable for that so Batman will be killed anyways.

D3RRANG3D - 2018-03-01

1ucasvb well i would take Altaïr or Ezio over batman anyday

Lord Neo - 2018-05-20

Danny Sano' only if the sequence consists of positive terms i.e only if the series converges absolutely

Peter Hjørland - 2018-09-20

The reason it's the perfect crime is that the assassin who killed Batman is Chuck Norris.

Minecraftster148790 - 2017-01-21

Want to know an anagram of banach tarski? Banach tarski banach tarski

porteal - 2017-02-10

@Insanity Cubed I jumped the gun and commented before the video ended

Dennis W - 2017-03-18

The B in Benoit B. Mandelbrot stands for Benoit B. Mandelbrot

lelcetz - 2017-04-29

an anagram of banarch tarksi : iskart hcranab

Pi - 2017-06-26

Another anagram is tarski banana

The Scattered Man - 2018-06-15

lol also lol

benni p - 2017-01-22

Do you really think infinite Assassins could kill Batman?
You would need at least twice as many!

Mathologer - 2017-01-22


Pi - 2017-06-26

benni p no. Aleph aleph aleph aleph............ assassins

Legend hunter - 2019-11-21

@Mathologer reductio ad absurdum via ad infinidum lol

שחר א. - 2020-02-18

Pi probably Mem Tesha

שחר א. - 2020-02-18

Pi probably Mem Tesha

Tim Jadeglans - 2017-01-21

It's the perfect murder because of biggest army politics. You have the biggest army with infinite assassins therefore you make the laws

2nafish117 - 2017-01-23

why not divide a cake up in an infinite geometric series 1,1/2,1/4, etc

Mathologer - 2017-01-24

+Shashank Chandrashekar Why not indeed. I always do this with my cakes :)

Visual_Vexing - 2017-01-30

That might be something reminiscent to "Gone with the Blastwave".

T Perm - 2017-02-10

The position is epsilon.

Peter Hjørland - 2018-09-20

Well, the REAL reason it's the perfect murder is that the assassin was Chuck Norris in disguise. QED

Max Power - 2017-01-21

Today - 21-012-2017 - Mathologer referenced VSauce, and VSauce referenced 3Blue1Brown. For the love of order and purity in the universe, I sincerely hope 3B1B releases a video today referencing Mathologer.

Pablo López-Ortega - 2017-01-22

But then we'd have a circular reference and Excel would complain

Veexliat - 2017-09-21

He referenced minutephysics. I know its not mathematics, but hey i think we got the circular reference

Wurtt Mapper - 2018-01-13

You have to watch the bitcoin video.

Danilego - 2019-03-21

I miss the olden days when VSauce uploaded videos

9 54 61 - 2020-02-22

@Danilego now those days are back

SVVN - 2017-01-21

A problem may arise when each assassin realises that there is no possibility he's going to kill Batman, and can as such go home so as to not risk being caught later.

Pi - 2019-04-20

And Batman dies

kumoyuki - 2017-01-22

Re: Axiom of Choice - It is something that is intuitively obvious - and infinite sets are anything but intuitive. Personally, I am a constructivist when it comes to proofs, but then again I am also a computer scientist: if you can't construct a thing it is generally pretty useless (with the possible exception of Turing's Oracle)

Ian Kjos - 2017-01-22

kumoyuki As a computer scientist myself I share your bent toward construction. However, non-constructable things are useful in the construction of reductio ad absurdity. and these boxes are absurd.

Lens98052 - 2018-08-10

Agree. Because most of my proofs were constructivist in nature, my professor tutor in algebra exclaimed in exasperation to one of my proofs: "Why do you have to do everything the hard way?". I should have replied: "Because I want to understand why it is true, not just know that it is true".

Elliot Grey - 2017-01-22

the Mathassin's Creed:
{thing} ≠ true; permit {ξ}

Mathologer - 2017-01-22


ChrisLuigiTails - 2017-05-22


Pi - 2017-06-26

Or 0/0

NoriMori - 2018-01-08

Omg. Hahaha!

Deep Patel - 2020-03-09

I didn't get the joke. Can you help me understand it?

Fester Blats - 2017-01-21

I didn't understand any of this.

Mathologer - 2017-01-24

If you are interested, just ask for clarification. There are lots of very knowledgable people roaming this comments section who'll be happy to explain things :)

Henry Dickinson - 2019-03-17

Maybe this question is stupid but, why can we assume that the assassins' hat sequence is 'close' to any of the sequences on the boxes at all, if they were taken out at random?

Logan Coen - 2017-01-24

Good news: this is math. We can take anything we wish as truth, so long as it does not contradict anything else we take to be true. Arguing about whether or not it makes physical sense is completely inconsequential until you construct a physical system where it might be applicable, which seems unlikely to happen with the types of things that make someone question the axiom of choice

Mathologer - 2017-01-24

Good news indeed :)

OmnissiahZelos - 2017-01-24

Good news everyone!

Jamie G - 2017-01-21

I'm glad I do engineering, we just let you guys do the tricky stuff XD

swordandgunandsteel - 2017-01-21

Jamie G >:(

Justin Ward - 2017-01-21

Jamie G Write some proofs you lazy bum.

AntsanParcher - 2017-01-21

Jamie G: What are you complaining about? They leave all the fun to us!

letao12 - 2017-01-22

Haha so true. We wait for you to figure out how it's supposed to work, then we make one that works like that :)

Dustin Rodriguez - 2017-02-16

But you do the things they have to cheat to do, like create a device that can move through an infinite number of positions in a finite amount of time despite Zeno's Paradox.

Jace Wright - 2017-11-14

So... An infinite army of mathematically genius assassins with infinite memory and processing power and can each assassinate batman without a problem get caught and have to do what the judge says or get executed by a judge who plays weird games with exceptions.

Seems legit.

KubeSquared - 2018-03-13

The problem is that acording to the Law of Conservation of Ninjutsu, there is always a finite amount of fighting prowess that is evenly spread out across a group of Ninjas (or assassins if you will). So your army of infinitely many assassins are each infinitesimally effective. So Batman will just defeat them all.

Mathologer - 2018-03-14


Naw Dawg - 2019-06-21

Even though the Assassins are infinitesimally effective it still takes some infinitesimal amount of energy to beat them, and Batman only has so much energy.

Sho Am - 2019-07-25

According to the same law, since Batman also uses a finite amount of ninjutsu, and there are an infinite amount of infinitesimal energies.. Banarch-Tarski strikes again and the universe is split into a multiverse where Batman is killed and is alive!

Ah Zhou - 2017-01-21

does it matter if a finite or infinite/2 number of assassins die, since there's always infinitely more to replace them?

Pablo López-Ortega - 2017-01-22

Well, having to deal with infinity corpses is a lot harder than a finite number of corpses XD

Raphael Schmidpeter - 2017-02-12

Well it does definitely matter to the assasins who were killed additionally when you kill infinitely many of them instead of just finitely many (also notice that infinite/2 is not an amount)

Mahissimo - 2017-01-22

I'd say the Axiom of Choice is true as long as you don't have an Electoral College :(

orochimarujes - 2017-01-31

@Marcel Hermes
Many countries don't elect their leaders by popular majority. England, Germany, Japan, and Canada (and many others) elect their leader based on what members of the biggest party in their various houses of legislation select. It is common for the leaders of those countries to get less than 40% of the popular support. The majority of our presidents have gotten over 50% of popular support, and only two presidents have gotten less than 40% (one of them was Lincoln).

America is not a country carved up into states, it is a set of states united as a country, all of which joined the union individually and not without keeping a large amount of their autonomy. That is important. Although there are national government positions, there are no national popular elections. All are state level. The people vote directly for Senators, Representative, and Governors, but not for President, Speaker of the House, or Supreme Court judge (all powerful national positions). The constitution does not say the people vote for president, it says the states get a number of votes for president equal to the amount of senators and representatives allotted to that state, and that the states (meaning their legislators, their governors, their voters, ect) are free to determine how those votes are distributed. It just so happens that almost every state has chosen a first-past-the-post winner-take-all system.

Mahissimo - 2017-01-31

Well, I get your point, and historically, this "every state for themself" made sense. But among other, mostly weird things, the new president talks about unifieing the country (after splitting it...), so there is one man for everybody. People go and vote directly their president, why not count their votes directly? Well, it is, what it is now, but what is the deeper sense in today's world?

For your second part, I just can speak for my German country. We have indeed a different system, as I can see it in some form or another all around the globe. You are right, that our leader's party gets on average maybe 30 to 35% of the votes, but it's still a majority... In the US, you have your two big parties, of course the winner of the popular vote gets over 50% (that's still a math video, so yes, this has to be right). In Germany, we have lots of parties, but at the end, only 4 or 5 of them find their way to the parliament, where they vote the chancelor, who is typically the leader of the party with the most votes. The big difference is, that the parliament itself is seated with delegates based on their relative percentage, not with a winner takes it all mentality. So, if 35% vote the CDU, and therefor Angela Merkel, there will be about 35% delegates representing the CDU. (It's a really interesting way to do this the fairest way possible by using maths, even written in our election laws.) At the end of the day, this means, that every single vote, be it in Bavaria, Hesse or Berlin, has the same value. And we still have a federal system with lots of specific powers for the states. In the US in our days, republicans from California are as screwed as democrats in Texas, because their votes are simply thrown away. Or not even only that, in fact, they have been voted for their opponents maybe their whole life. I don't get it, sorry...

Houshalter - 2018-04-24

Mahissimo the smallest states entered the union on the condition they wouldn't just be outvoted by the big populous states. The electoral college is a compromise. If you get rid of it, most of the states would be better off leaving the union.

rainy7 - 2018-07-09

Without the Electoral College, candidates wouldn't even go to the low populated states. With nearly 6 times the population, a candidate would only need votes from California; one would not need a single vote from Montana to win. I don't want California and New York being the only states to elect a president.

Bijan Sheibani - 2018-07-14

rainy7 Well, with the electoral college, they don't go to most states.

NoriMori - 2017-01-21

Saw the Assassin's Creed-style assassins in the thumbnail and was sold. XD

Terry Endicott - 2017-01-21

Famous quote: “The Axiom of Choice is obviously true, the Well–ordering theorem is obviously false; and who can tell about Zorn’s Lemma?" Jerry Bona
I like Zorn's Lemma - Algebra is so much more beautiful with it than without.

Mathologer - 2017-01-22


oooBASTIooo - 2017-02-16

To me the axiom of choice is pretty much not questionable, if you want to work with infinite objects.
If you think about it, an equivalent definition of AC is that a product of empty sets is nonempty (since each element of the product constitutes a choice function).

so, imo, you either have to do without infinite objects (or at least without uncountably infinite objects) or you have to accept AC...

Apple Pie - 2017-02-22

Terry, you could always say things like "Let R be a ring with a maximal ideal" or "Let K be a field with an algebraic closure" :)

Per Trygve Myhrer - 2017-01-21

The problem with this is that batman is so popular they would have to bring him back

Pablo López-Ortega - 2017-01-22

You can't kill the Batman, you can only temporarily exile him from the DC Universe.

ChristianBach7 - 2017-02-05

Let's just not kill Batman in the first place... mkay..?

chamington - 2017-01-22

3:42 Love the Bill Wurtz feel

Psychosmurf5471 - 2017-03-20

"it's simple. We Achilles the Batman."

oldcowbb - 2017-01-21

it's high noon

Pi - 2017-06-26

oldcowbb what does HIGH noon even mean

Jimmy De'Souza - 2017-01-21

The Axiom is fine as long as you have paradox absorbing crumple zones installed I guess.

WrenAkula - 2017-01-22

I was just looking for videos about the axiom of choice earlier today; what a coincidence! :-D

Mr. Nobody - 2017-01-21

The second assassin whose number differs from the series knows that the last guy's number is different (too), and since the first one said there was an even number of differences, the second assassin with the wrong number knows he has to have his number wrong too, cause he doesn't see any difference with all the other assassins and the series.
I know I didn't explain it very well, sorry, but it's just how it came out of my mouth... well, my fingers.

Jimmy De'Souza - 2017-01-22

There are infinitely many sequences, as there are infinitely many assassins, and each can be a 1 or a 0. Additionally there are infinitely many possible heads, and the head matters as you are using that to determine what changes have occured.

I guess that is the thing that is throwing me off, unless you assume they can magically pick the right sequence (which would also mean they would know the number on thier head innately so the problem becomes moot (from a non mathematical standpoint anyway :D)), none of them will ever have a better than 50% chance.

Ah, I am probably just too stupid :D.

Steve's Mathy Stuff - 2017-01-22

There are infinitely many possible sequences, but there are only two that matter, the actual sequence on the hats and what I'm calling the reference sequence. The actual sequence is in one of the boxes. The assassins have memorized one sequence from each box. The reference sequence is the one that is in the same box as the actual sequence. Because it is from the same box, It differs from the actual sequence at only finitely many points.

Nicholas Iverson - 2017-01-24

It seems there can only be 1 set of 0's and 1's, that corresponds to the actual set of 0's and 1's and is also close such that when you see an odd number of differences your number will be 1, and if you see an even number of differences or none your number is 0, and that would be a sequence that ends in infinitely many 0's or there's a rule we didn't mention that states if your time to say your number is called you see no differences and the previous assassin saw no differences you opt to have yourself killed to signify the remainder of the numbers are the close portion. Otherwise you'd have a situation where you send false information?

Steve's Mathy Stuff - 2017-01-24

If a pair of sequences, one known and one unknown, has an even number of differences past a certain point only if the number at that point is 0 in the unknown sequence, then, yes, both sequences have to end in all 0's. That's why a strategy that involves guessing 0 when the differences are even cannot work and was not, I suspect, seriously considered by anyone.

Zakalite - 2017-01-25

Steve's mathy stuff I remember you replied to me on that numberphile sandpiles video

Bobba Chao - 2017-02-01

When I saw this in my recommandation and I looked at the picture and the title I was like, is this some gametheory video? Then I see mathloger... So here I am

Icy001 - 2017-01-21

3:51 You probably shouldn't say "list"! The set of infinite sequences of 0's and 1's is not listable!

Mathologer - 2017-01-22

Well, it is, you just well-order it :)

oooBASTIooo - 2017-02-16

Well, that doesn't suffice. listable essentially is the same as recursive enumerable (i.e. you are well below anything using the axiom of choice). And you cannot find a (finite) recursive algorithm that outputs exactly the elements of any given such equivalence class, because this would yield a well-ordering of the class which probably could yield a proof of countable choice in ZF.

The Scattered Man - 2018-06-15

Yeah... ummm... Set Theory was a long time ago for me, but I seem to recall finding out that the Reals could be well-ordered but were still UnCountable... and I never really DID understand that part. (At least not intuitively.) I believe it, I just don't understand it. :) [How can everything have an "immediate successor", but you STILL can't put them into 1-1 correspondence with the natural numbers?...it's odd. Or even. Or just strange. :) ]

ventura rodríguez vallejo - 2018-07-24

The Scattered ANY real number has infinite successors (not just one single "immediate successor"). Actually, NO REAL number has one immediate successor or immediate predecessor, only a proximal to both a minimum and a maximum in a given interval, arbitrarily "close" to the number. This is the way the set is considered as well-ordered.
The same goes for Rational numbers (a sub-set of Reals). This means that both sets of numbers have a DENSITY strictly higher than 0, as opposed to the set of Integers, wich density is strictly equal to 0.
Even though, Rational numbers are countable, while Reals are not.
The surprising proof of this was established by Cantor in his transcendental work on sets and infinite collections. The very beginning of Set Theory.

שחר א. - 2020-02-18

1:33 But those facts can be true even without doing it!!

EternalBooda - 2017-01-21

0:23 Imagine?

ozdergecko - 2017-01-21

The perfect murder is...
what murder?

NUKE - 2017-02-07

I would love to see an introduction to the proof of Fermat's Last Theorem by Andrew Wiles from Mathologer!

Stephen Furr - 2017-01-21

+1 for the Michael Stevens's beard graphic at 12:00! Hilarious!

Mathologer - 2017-01-22

Could not resist :)

Kasuga Ryuichi - 2017-01-21

Love your videos~

Roger Barraud - 2019-12-09

.: The judge is The Riddler.

Kevin Octa - 2017-01-24

I love this channel. Possibly my favorite maths channel. I hope there will be plenty of uploads in 2017 :D THANKS MATHOLOGER!

Mathologer - 2017-01-25

Glad you like what I do and thank you for saying so :)

TuEsStercus LoquorLatinam - 2018-03-25

I'd like to think I am a mathassin. I have slayed many a test with my logic and mathematics, especially after not studying.

Nick Demos - 2017-01-26

Thank you Burkard. It's good to finally have an awesome Australian mathematician! Happy Australia day btw :)

Mathologer - 2017-01-27


64standardtrickyness - 2018-12-30

wait how can a system that kills finitely many assasins kill each of the infinitely many assasins with probability 1/2?

Penny Lane - 2017-01-21

What did Batman ever do to you?

David Wührer - 2017-01-22

He's Batman.

Andy Brice - 2017-03-14

He took it upon himself to try and save Gotham. Gotham cannot be saved, it must be destroyed.

Jainil Shah - 2017-01-22

What software do you use to make animations? Love your videos by the way.

Mathologer - 2017-01-22

In this video mostly just Magic moves transitions that are built into Apple's Keynote and a few animations in Adobe Premier. The individual pictures that get manipulated I mostly do in Adobe Illustrator and Photoshop :)

Fiat Lux - 2018-05-22

Practical application, of course I have to make some adjustments but this is actually kind of helpful.

Ocie Mitchell - 2017-01-21

Love the tshirt. I think Homer would approve.

Mathologer - 2017-01-22

I got mine from here http://shirt.woot.com/offers/infinite-doughnut?ref=cnt_ctlg_dgn_3 :)

unfa - 2017-02-14

As a sound guy I have to say that the "pause and ponder" music is a bit too loud in relation to the rest of the show.

Awesome videos! Thank you :)

Matheus Gondim - 2017-01-27

Hello there!
First of all, I'd like to thank you for your videos; they've been always really enjoyable to watch.

I'd like to ask you about your thoughts on intuitionistim. I take it that you don't accept all its premises since you declared your subscription to the axiom of choice; still, I'd really think it would be nice to hear (or read) what are your main opinions about this take on logic and math foundation.

Thanks in advance,

Best regards,

Mathologer - 2017-01-29

I'd say live and let live. Happy to entertain intuitionism as an alternative way of looking at things. I personally don't have a problem with the fact that there are multiple ways of building mathematics and that things are constantly evolving in this respect :)

Prithiraj Mallik - 2018-09-29

When did axiom of choice come to play

Overkillius - 2017-07-23

I don't get this on a such a level that I can't think of what questions to ask.

CriticalMass 21 - 2017-01-21

but infinite number of asassins will fill the entire universe , even if the asassins is smaller than a quark....

Benjamin Bedert - 2017-01-21

That is only true if they are all equally large. They would for example only fill a finite part of space if the volume of assasin N is 1/2^N ( for that matter any geometric progression with ratio r for which 0<r<1 would work ).

king_ Tesseract - 2018-10-19

@Benjamin Bedert which works in theoretical maths, but out here in the real world an assassin can't be smaller than a planck length in any direction. Or else by observing themselves they'll create a black hole.

Matron - 2020-02-01

Too many paradox’s
hurts my mind.
Stay healthy