Death by infinity puzzles and the Axiom of Choice

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.

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

Enjoy!

Burkard

[Mathologer]

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 :)

[Minecraftster]

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

[bennip]

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

[MaxPower]

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.

[ucasvb]

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

[Psychosmurf]

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

[sven]

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.

[timonix]

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

[terryendicott]

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.

[KubeSquared]

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.

[skyr]

I'm pretty sure he would see a pile of assassin's larger than the universe.

[johannesh]

If one of the assassins makes a wrong choice, those after him will still know their number because they will hear him getting executed for being wrong

[muskyoxes]

The axiom of choice was accepted, not because it seemed reasonable or necessary, but only because they'd have to throw out a bunch of theorems if they rejected it

[Sons]

The Banach-Tarski paradox sounded SUPER COUNTER-INTUITIVE when I first heard about it, but after calmly comparing with the fact that it is possible to rearrange all points in [0, 1] to make [0, 2] (which is doubling the length and is SUPER TRIVIAL), I now think the only "weirdness" in this paradox is the "possible just by dividing the cube to finite sets and rearranging" part and not the "possible to double the volume" part.
The latter part, I think actually is the origin of the (false) counter-intuitiveness for many non-experts, while it's actually not really the main point. After realizing this, I'm more "pro-axiom of choice" now.

[KasranFox]

I think the axiom of choice, like almost anything else in mathematics, is a tool. It behaves in certain ways, it implies certain things about what it governs, and when you're doing math it pays to be conscious of when you are using it and to ponder what would change if you didn't.

[mheermance]

It's not the perfect murder because an infinite number of assassins has an infinite amount of mass. So the assassins, the planet they're standing on, and Batman collapses into a black hole. From the point of view of an outside observer we know Batman can't escape the event horizon, so he's dead. But neither can the assassins, so they all got the death penalty as well!

[mr.nobody]

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.

[timothymclean]

The question is, of course, if Batman with infinite prep time could defeat an infinite number of infalible assassins.

[jonathanroach]

You'll just end up with an infinite amount of knocked up and tied up assasins... because BATMAN!!!

[yanntal]

3:04 What about the following strategy that avoids the axion of choice?
Assassin in position k looks at 4^k assassins in front of him and 4^k behind him and searches for the smallest least common string of length k bits (so 2^k options) that he could add to by picking 0 or 1 appropriately?
Since most numbers are normal, and since we can look at the problem as getting a random number in the range [0, 1] in binary mostly right. The probability that infinitely many assassins are dead should be 0 given that the hats have no patterns to them.