Death by infinity puzzles and the Axiom of Choice

by birtanpublished on August 28, 2020

welcome to the first mythology video of the year today is all about playing with infinity and the infamous axiom of choice to commit the perfect murder and a cheat death because I need the whole width of the screen for most of the

Video you won't see much of me except now and at the end of the video bit of an experiment okay so let's get started by imagining that you're an evil mastermind who wants to commit a perfect murder it's all going to happen on this

Lively Street just off the main street and you know that I'd known your victim Batman got to walk down the street we're going to use you are infinitely many in sellable assassins place the first assessment right here as soon as Backman

Touches this line this assassin definitely will kill him now just to make sure you put a second assassin right here and again as soon as Batman touches this line he'll die by the hand of the second assassin and then you keep

On going like this or put another assassin here same rules and infinitely many more approaching this zero line but never reaching it and now it's absolutely certain that once at 12 o'clock

Batman crosses the zero line you'll be dead so that's pretty good but how is this the perfect murder you can stop the video before the countdown is finished to try and come up with your own solutions

so what makes this the perfect murder is that although Batman is definitely dead we can also prove that none of the SS and kill them so for example assassin number one cannot possibly have killed

Him because to get to number one Batman has to pass number two alive and it's impossible because this guy is also infallible definitely kill him but then you never get to number two because you first days to get past number three

Which is impossible and number three is impossible and so forth so that shows none of these guys killed Batman but still he's definitely dead perfect murder they should all go free which brings us to part two of this video

Trial by hats okay well the police is not that impressed to figure out what has happened they don't really know how Batman died but I know that somehow all of these guys were involved so these

Guys are all found guilty of accessory to murder and I sentenced to death except they have a chance of walking free if they can answer a certain question correctly for the questioning the set up is like

This the infancy many assessments get lined up like this then randomly zero or one heads get put onto their heads and they can look around you can see all the other people's heads the only thing they can't see is what numbers on their own

Head and now at a certain time they all have to shout out zero or one all at the same time and whoever gets it right walk three which is have a look well this guy walk three and that guy walks free and all of these guys walk free and the

Other guys who get it wrong get killed looks like it's a 50/50 chance for every single one of them but the judge tells them well what you can do is you can get together the evening before and can try to come up with a strategy and others

Tell you there is a strategy that will ensure that if you follow it exactly only financially many of you will get killed one more thing these assessments are not only infallible but they're also terribly good at math so actually they

Can do anything that mathematics can do so for example they can memorize infinitely many numbers or it can make infinitely complex calculations things like this again you've got up to the end of the countdown to stop and ponder this

here's the strategy the infinitely many assessments get together and they list all possible infinitives or one sequences and then they declare 2:01 sequences close if

They only differ in finitely many digits so for example these two sequences here are close because they only differ in those places here so what that also means is that from a certain point on the tails of these two things are the

Same and now what they do is they bundle sequences together into boxes that if you take two sequences from different boxes they will not be close the other hand if you take two sequence from one box they will be close the reason why

This actually works why it can do this lies in the sex that clean clothes is something called an equivalence relation and so the box is obviously the equivalence classes don't worry about it but you can think about why it is

Actually true I can do such a splitting up of the sequences anyway we've done it now and so what we do now is we pick out one sequence from each box and maybe label the boxes it okay one from each box and now what the assessments all do

Is they memorize these sequences the memorized sequences now have a very special property and it's like this if you take an arbitrary zero one sequence there's exactly one among these memorized sequences that is close to the

Sequence all other ones are not close differing infinitely many places so these memorized sequences are now what we're going to use to get our strategy okay short our next morning everybody head to the head and now the essence

Just look around and basically to see everything except for one entry here one above their heads and they can figure out now by comparing what they see to the memorized sequences that one memorized sequences close to what they

See and in this case say it's this one here and now the strategy is for every single one of those essence to pretend that the memorized sequence is the real sequence so the first assessment would say zero

And he would walk free the second guy would say zero but now tough luck gets killed so guy would say one tough luck gets killed but then from this point on everything coincides and everybody else walks free

And in general if you adopt the strategy you can ensure that there's only finitely many assassins that get killed pretty neat right but that's not the end of it actually part three here another puzzle for you about cheating death well

Actually when you have a really really close look you ask yourself as an assassin well what does this strategy do for me how much do my chances of surviving this improve well before without any strategy with obviously

50/50 and actually when you have a really really close look now although we can guarantee this is only finitely many assassins that could kill the chance of every single one is still 50/50 that's not great so assassins

You know why would we bother with the strategy so they just tell the judge they have to do better than that actually the judge gets it and he says okay we're going to change things a little bit why don't we do this instead

Of lining you up like this I'm going to line you up like that everybody's facing now certain direction then we get your heads and then this guy for example can see all of those hats and this guy here you can see all of those hats and this

Guy here you can see all of those hats and so on that seems more restrictive than before but you know wait for it wait for it remember before everybody had to say their number simultaneously this time what you have to do is we kind

Of ask you one at a time so the first ask this guy here and he says zero one he can't say anything else he can't keep anything else away the zero or one everybody else can hear what he says and then it's the second guys turn he says

Whatever he wants to say they're all one everybody else he is it and then it's this term and so on and now actually with this setup I'm just telling you guys if you come up with the right strategy you can actually ensure that

Pretty much everybody walk through it in fact apart from the first one who still got to 50/50 chance everybody if you don't mess up can walk free so it's definitely worth pondering and now again got until my count on ends to come up

With the strategy so here's the strategy we're actually going to use exactly the same memorized sequences as before so again assassins can figure out which one of

These sequences that they memorize is close to the one that's above them because it does depend on the tail on and their position in here we all know this sequence and now the strategy is again based on this memorized sequence

So what we do is we just have a look at the first guy here this guy now compares what he sees to the memorized sequence and just counts how many differences he sees so he sees one difference here and one difference there

That's an even number of differences so an even number of differences we say corresponds to zero and odd number corresponds to one so he sees even so he says zero and she's also lucky walks free in this way so how does this now

Help the other people to figure out exactly what's on their heads well let's have a look at the second guy he knows that the one would have said they're Oh saw an even number of differences now he only sees an odd number of differences

So what that means is there's got to be a difference in the spot that he's sitting in so he knows the dimer slit sequence shows a zero so there has to be a difference that means there has to be a one on his head and I leave it to you

To figure out how the third guy has to argue to figure out that there's a zero on his head and maybe do the details in the comment and just in general what's the general strategy for n ska here remember what's also interesting just in

Case one of these guys messes up does this mean that other people are lost so that was fun right and of course since all it involves infinitely many attachments and those a

Second performance superhuman infinite feat don't expect to see anything like this in the news anytime soon on the other hand in the world of pure mathematics all this makes sense having said that there's one aspect to

All this that even make some hardcore mathematicians aleady it's the bit where we put all the infinitely many zero one sequences in those infinitely many boxes and then choose one sequence from each box see all the infinite sets and how we

Get them are not the issue it's the picking of the sequence from each box that has raised mathematical eyebrows it also has very innocent until you think about how you would actually accomplish this even with all the powerful math

Tools at our disposal in this setup that simply does not seem to exist a simple rule that can help us choose one sequence from each box so what do I mean by this well for example if the boxes where each filled with positive integers

We could simply choose the smallest number in each box right so for example here two there one here two they would pin down things nicely in this case though there does not seem to be any nice rule that could help and for us to

Be able to make it choice in some fuzzy way anyway requires us to accept the so-called axiom of choice within the kanan of axioms that math is based on informally the axiom really just says that given any collection of non-empty

Sets we can choose one element from each set now what some mathematicians find problematic is that is exactly the axiom of choice that makes some of the most mind-boggling paradoxical and counterintuitive theorems of mathematics

Possible the most famous example is the Bartowski paradox it says that a solid ball can be split into finitely many disjoint sets like this which can subsequently be pushed and rotate around in space such that they can be combined

Into two solid balls of exactly the same size as the one we started with pretty crazy right there's a great video by Vito's about the Bartowski paradox which is really a must be for everybody here anyway so should we accept the

Axiom of choice given that implied super paradoxical results like the Bartowski paradox well it's our choice and at least most mathematicians I know including myself subscribe to this axiom and so what do

You think about all this maybe illegal thoughts in the comments

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