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The Banach–Tarski Paradox
Lernstatistiken
GER-Niveau
Schwierigkeit
Untertitel (389 Segmente)
Hey, Vsauce. Michael here. There's a famous way
to seemingly create chocolate out of nothing.
Maybe you've seen it before. This chocolate bar is
4 squares by 8 squares, but if you cut it like this
and then like this and finally like this
you can rearrange the pieces like so
and wind up with the same 4 by 8
bar but with a leftover piece, apparently created
out of thin air. There's a popular animation of this illusion
as well. I call it an illusion because
it's just that. Fake. In reality,
the final bar is a bit smaller. It contains
this much less chocolate. Each square along the cut is shorter than it was in
the original,
but the cut makes it difficult to notice right away. The animation is
extra misleading, because it tries to cover up its deception.
The lost height of each square is surreptitiously
added in while the piece moves to make it hard to notice.
I mean, come on, obviously you cannot cut up a chocolate bar
and rearrange the pieces into more than you started with.
Or can you? One of the strangest
theorems in modern mathematics is the Banach-Tarski
paradox.
It proves that there is, in fact, a way to take an object
and separate it into 5
different pieces.
And then, with those five pieces, simply
rearrange them. No stretching required into
two exact copies of the original
item. Same density, same size,
same everything.
Seriously. To dive into the mind blow
that it is and the way it fundamentally questions math
and ourselves, we have to start by asking a few questions.
First, what is infinity?
A number? I mean, it's nowhere
on the number line, but we often say things like
there's an infinite "number" of blah-blah-blah.
And as far as we know, infinity could be real.
The universe may be infinite in size
and flat, extending out for ever and ever
without end, beyond even the part we can observe
or ever hope to observe.
That's exactly what infinity is. Not a number
per se, but rather a size. The size
of something that doesn't end. Infinity is not the biggest
number, instead, it is how many numbers
there are. But there are different sizes of infinity.
The smallest type of infinity is
countable infinity. The number of hours
in forever. It's also the number of whole numbers that there are,
natural number, the numbers we use when counting
things, like 1, 2, 3, 4, 5, 6
and so on. Sets like these are unending,
but they are countable. Countable means that you can count them
from one element to any other in a
finite amount of time, even if that finite amount of time is longer than you
will live
or the universe will exist for, it's still finite.
Uncountable infinity, on the other hand, is literally
bigger. Too big to even count.
The number of real numbers that there are,
not just whole numbers, but all numbers is
uncountably infinite. You literally cannot count
even from 0 to 1 in a finite amount of time by naming
every real number in between. I mean,
where do you even start? Zero,
okay. But what comes next? 0.000000...
Eventually, we would imagine a 1
going somewhere at the end, but there is no end.
We could always add another 0. Uncountability
makes this set so much larger than the set of all whole numbers
that even between 0 and 1, there are more numbers
than there are whole numbers on the entire endless number line.
Georg Cantor's famous diagonal argument helps
illustrate this. Imagine listing every number
between zero and one. Since they are uncountable and can't be listed in order,
let's imagine randomly generating them forever
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