Welcome to the first Mathologer video
of the year. Today is all about playing
with infinity and the infamous Axiom of
Choice to commit the perfect murder and
to 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 are an evil mastermind
who wants to commit the perfect murder.
It's all going to happen on this lively
street just off the main street and you
know that at noon your victim, Batman, will
walk down this street.
We're going to use your infinitely many
infallible assassins. Place the first
assassin right here. As soon as Batman
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 hands of the
second assassin and then you keep on going
like this. So put another assassin here,
same rules and infinitely many more
approaching this 0 line but never reaching
it and now it is absolutely certain that
once, at twelve o'clock, Batman crosses the
zero line he will be dead.
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 solution.
So what makes this the perfect murder is
that although Batman is definitely dead
we can also prove that none of the assassins killed him.
So, for example, assassin number 1
cannot possibly have killed him because
to get to number 1 Batman has to pass
number 2 alive and that's impossible
because this guy is also infallible, will
definitely kill him. But then he'll never
get to the number 2 because he first
has to get past number 3 which is
impossible and number 3 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 2 of this video---
Trial by hats.
Well the police is not that impressed, they
figure out what's happened. They don't
really know how Batman died but they know
that somehow all of these guys were
involved so these guys are all found
guilty of accessory to murder and are
sentenced to death, except they have a
chance of walking free if they can
answer a certain question correctly.
For the questioning the setup is like this:
the infinitely many assassins get
lined up like this. Then randomly
0 or 1 hats get put onto their heads and they can
look around they can see all the other
peoples' hats,
the only thing that cannot see is what
numbers on their own hat. And now, at a
certain time, they all have to shout out
0 or 1 all at the same time and whoever
gets it right walks free, let's have a look. Well
this guy walks free 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
I'll just tell you there is a strategy that
will ensure that if you follow it
exactly only finitely many of you will
get killed.
One more thing: these assassins are not
only infallible but they're also
terribly good at maths so actually they
can do anything that mathematics can do.
For example they can memorize infinitely many
numbers or can make infinitely
complex calculations, things like this.
Again you've got until the end of the
countdown to stop and ponder this.
Here is the strategy. The infinitely many
assassins get together and they list
all possible infinite 0, 1 sequences and
then they declare two 0,1 sequences CLOSE
if the only differ in finitely many
digits. For example, these two sequences
here are CLOSE because they only differ
in those places here. 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 such that if
you take two sequences from different
boxes they will not be CLOSE. On the other
hand, if you take two sequences from one
box they will be CLOSE. The reason why
this actually works, why we can do this
lies in the fact that being CLOSE is
something called an equivalence relation
and so the boxes are basically the equivalence classes.
Don't worry about it but you can think
about what this is actually true, why you can do
such as 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 box with it, one from each box
and now what the assassins all do is they
memorise these sequences. The memorised
sequences now have a very special
property and it's like this: if you take
an arbitrary 0,1 sequence then there is exactly
one among these memorised sequences that
is CLOSE to the sequence, all other ones
are not CLOSE (differ in infinitely many
places). So these memorised sequences are
now what we're going to use to get our
strategy. Okay Showtime! Next morning
everybody gets their hat and now the
assassins just look around and basically they see
everything except for one entry here, the
one above their heads and they can figure
out now, by comparing what they see to
the memorised sequences, that one
memorised sequence that is 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 the assassins to pretend that the
memorised sequence is the real sequence.
So the first assassin would say zero and
he would walk free. The second guy
would say zero but now, tough luck, gets
killed. Third 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 this 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. There is part 3
here, another puzzle for you. It's about
cheating death. Well, actually, when you
have a really, really close look you
ask yourself as an assassin: Well what
does the strategy do for me? How much do
my chances of surviving this improve?
Well before, without any strategy, it was
obviously 50-50 and actually when you have a
really close look now, although we can
guarantee that there will only be finitely many
assassins that get killed, the chances of
every single one is still 50-50.That's
not great, so the assassins, you know: Why would
we bother with this strategy so they just
tell the judge: You have to do better than
that. Actually the judge gets it and he
says: OK, 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 a certain
direction. Then we get our hats and then
this guy, for example, can see all of
those hats and this guy here can see all
of those hats and this guy here 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 we ask you one at a time.
So we first ask this guy here and he
says 0 or 1. He can't say anything else he
can't give anything else away, just 0 or 1.
Everybody else can hear what he says and
then it's a second guy's turn. He says
whatever he wants to say, 0 or 1.
Everybody else hears it and then it's
this guy's turn, and so on. And now actually
with this setup, and I am just telling you guys,
if you come up with the right strategy
you can actually ensure that pretty much
everybody walks free. In fact, apart
from the first one who still got his
50-50 chance
everybody if you don't mess up can walk
free. So that's definitely worth pondering.
And now, again, you've got until my countdown ends to
come up with a strategy.
Here is the strategy. We're actually going to use
exactly the same memorised sequences as
before.
So again, the assassins can figure out which
one of these sequences that they
memorised is CLOSE to the one that's
above them because it just depends on
the tail and their position in here ... we all know
this sequence and now the strategy is
again based on this memorised 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 memorised 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 a 0 and an odd number
corresponds to a 1. So this is even so he
says 0 and he's also lucky, he 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 who just said
0 saw an even number of
differences. Now he only sees an odd
number of differences. So what that means
is there gotto to be a difference in the
spot that he is sitting in. So he knows that the
memorised sequence shows a 0, so there has
to be a difference, that means there has
to be a 1 on his hat and now I leave it to you
to you to figure out how the third guy
has to argue to figure out that there is a 0
on his head and maybe do the details in
the comments and just in general what's the
general strategy for the nth guy here. Maybe
what's also interesting, just in case one of
these guys messes up does this mean that
the other people are lost or not?
So that was fun, right.
Of course, since all this involves infinitely many
assassins and those assassins perform
superhuman infinite feats, 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 makes some hardcore
mathematicians uneasy. It's the bit where we've put
all the infinitely many 0,1 sequences in
those infinitely many boxes and then
choose one sequence from each box. Here
all the infinte sets and how we get them is
not the issue, it's the picking of the
sequence from each box that has raised
mathematical eyebrows. It all sounds very
innocent until you think about how you
would actually accomplish this even with
all the powerful maths tools at our
disposal. In this setup there simply
does not seem to exist a simple rule that
can help us choose one sequence from
each box. 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 like, for example, here 2, there
a 1, here a 2. That 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 a choice
in some fuzzy way anyway requires us to
accept the so-called Axiom of Choice
within the canon of axioms that maths
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 it 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 Banach-Tarski paradox.
It says that a solid ball can be split
into finitely many disjoint sets like
this which can subsequently be pushed
and rotated around in space such that they
recombine into two solids balls of
exactly the same size as the one we
started with. Pretty crazy right?
There's a great video by Vsauce about
the Banach-Tarski
paradox which is really a must-see
for everybody here. Anyway, so should we
accept the Axiom of Choice given that it
implies super-paradoxical results like
the Banach-Tarski 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
leave your thoughts in the comments:)