Name: 巴納赫-塔爾斯基悖論 (The Banach–Tarski Paradox)
Uploaded: 2021-01-14T06:39:27.000Z
Duration: 24 min 14 s
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Hey, Vsauce. Michael here. There's a famous way

情態副詞

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

bar but with a leftover piece, apparently created

out of thin air. There's a popular animation of this illusion

the final bar is a bit smaller. It contains

this much less chocolate. Each square along the cut is shorter than it was in

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.

theorems in modern mathematics is the Banach-Tarski

It proves that there is, in fact, a way to take an object

rearrange them. No stretching required into

that it is and the way it fundamentally questions math

and ourselves, we have to start by asking a few questions.

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.

and flat, extending out for ever and ever

without end, beyond even the part we can observe

That's exactly what infinity is. Not a number

of something that doesn't end. Infinity is not the biggest

there are. But there are different sizes of infinity.

in forever. It's also the number of whole numbers that there are,

natural number,  the numbers we use when counting

but they are countable. Countable means that you can count them

finite amount of time, even if that finite amount of time is longer than you

or the universe will exist for, it's still finite.

Uncountable infinity, on the other hand, is literally

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

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

with no repeats. Each number regenerate can be paired

with a whole number. If there's a one to one correspondence between the two,

that is if we can match one whole number to each real number

on our list, that would mean that countable

and uncountable sets are the same size. But we can't do that,

If we go diagonally down our endless list

of real numbers and take the first decimal of the first number

and the second of the second number, the third of the third and so on

if it happens to be a nine, we can generate a new

real number that is obviously between 0 and 1,

but since we've defined it to be different

from every number on our endless list and at least one place

In other words, we've used up every single whole number,

the entire infinity of them and yet we can still

come up with more real numbers. Here's something else that is true

but counter-intuitive. There are the same number

and odd numbers. At first, that sounds ridiculous. Clearly, there are only half

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巴納赫-塔爾斯基悖論 (The Banach–Tarski Paradox)

entire

illusion

matter

sense