What is the Banach-Tarski paradox?

Alright, no problem! Imagine we're sitting together, having a coffee, and I'll tell you all about this super cool, yet mind-bending mathematical topic.

What is the Banach-Tarski Paradox? A Mathematical Magic Trick of "Creation from Nothing"

Hey, glad you're interested in this topic! The Banach-Tarski Paradox is a very famous and peculiar conclusion in mathematics that sounds completely counter-intuitive, almost like science fiction.

Imagine this scenario...

You have a solid, ordinary, billiard-ball-like one sphere.

Now, I tell you, I have a method to "decompose" it into a finite number of pieces (say, 5 or 6 pieces, not infinitely many!). Then, I reassemble these pieces, without stretching, compressing, or changing their shape, simply by moving and rotating them...

Finally, I can assemble two identical solid spheres, just like the original one!

Upon hearing this, your first reaction must be: "That's impossible! Isn't that creating something out of nothing? What about the law of conservation of energy?!"

Hold on, that's impossible!

Exactly, in our real world, this is absolutely impossible. You can't cut an apple into a few pieces and reassemble them into two apples. If you could, the world would be in chaos by now.

The "paradoxical" nature of this paradox lies here: it is completely correct logically, yet completely wrong physically and intuitively.

So, what exactly is the secret behind this mathematical magic trick? The key lies in two points:

The Secret to the Magic

1. The "pieces" that are "cut" are not ordinary pieces

When we usually cut an apple with a knife, each piece has a definite shape, volume, and surface. You can hold them in your hand.

However, the "cutting" in the Banach-Tarski Paradox is a purely mathematical concept of decomposition. The "pieces" it carves out are actually individual sets of points. These sets of points are extremely complex and bizarre; you simply cannot create them in reality.

You can imagine these "pieces" as:

• Infinitely Fine Dust: Each piece is like a cloud of dust diffused throughout the entire sphere, with its "dust" intermingling and intertwining with the "dust" of the other pieces.
• No Concept of "Volume": These sets of points are so scattered and peculiar that you cannot define their "volume". Mathematically, they are called "non-measurable sets". It's like asking, "What is the volume of a cloud of infinitely many randomly selected individual atoms?" The question itself cannot be answered.

Therefore, there is no "conservation of volume" here, because these fragments had no volume to begin with. We are merely rearranging points that have no volume, and these points happen to fill the space of two spheres in a new way.

2. Enter the "Axiom of Choice"

This is the "magic wand" of the entire trick.

The Axiom of Choice is a fundamental axiom in set theory. Simply put, it states:

Imagine you have infinitely many boxes, and each box contains at least one item. Then, I grant you the ability to simultaneously pick one item from every single box.

Sounds perfectly reasonable, right? "I can take one from each box," what's so difficult about that?

However, in the infinite world of mathematics, this "ability" becomes incredibly powerful. The entire proof of the Banach-Tarski Paradox deeply relies on this axiom to "select" those peculiar points we need from infinitely many sets of points, in order to construct our non-measurable "fragments".

Without the Axiom of Choice, this paradox would not hold. You simply couldn't construct those miraculous fragments, and thus couldn't perform this "one into two" trick.

So, what does this paradox tell us?

It's not saying that physical laws are wrong; rather, it tells us:

1. The world of pure mathematics and our physical world are two different things. A "sphere" in mathematics is merely a set of points, while a real-world sphere is made of atoms.
2. Our intuition about concepts like "size" and "volume" fails when dealing with infinity and certain abstract sets.
3. It demonstrates the incredibly powerful and counter-intuitive force hidden behind the seemingly harmless tool that is the "Axiom of Choice". This is also why some mathematicians were hesitant about accepting this axiom back in the day.

In a Nutshell

The Banach-Tarski Paradox states that: in pure mathematical theory, by utilizing the "Axiom of Choice" as a tool, we can decompose an idealized sphere into a finite number of extremely complex "point clouds" that have no definable volume, and then, merely through rotation and movement, reassemble these "point clouds" into two spheres identical to the original one. This is impossible to achieve in the real world.

Hope this explanation helps you understand the fascinating nature of this paradox! It's like a thought experiment that allows us a glimpse into the profound and peculiar world of mathematical infinity.