One example of a mathematical paradox is the Banach-Tarski paradox.
Understanding the Banach-Tarski Paradox
The Banach-Tarski paradox is a theorem in set theory that states the following:
- A solid ball in 3-dimensional space can be divided into a finite number of non-overlapping pieces.
- These pieces can then be moved and rotated (without deforming them) to form two identical copies of the original ball.
In simpler terms, you could theoretically take a solid ball (like a marble), cut it into a few pieces, and reassemble those pieces into two marbles identical to the first! This sounds impossible, hence the paradox.
Key Aspects of the Paradox
The Banach-Tarski paradox relies on concepts that seem to defy our everyday intuition.
- Non-Measurable Sets: The pieces the ball is divided into are not "measurable" in the traditional sense. This means we can't assign a volume to them.
- Axiom of Choice: The proof of the Banach-Tarski paradox relies on the axiom of choice, a controversial axiom in set theory.
- Theoretical, Not Practical: The paradox is a theoretical result. It doesn't mean you can actually take a physical ball and duplicate it. The pieces involved are infinitely complex and cannot be physically created.
The Paradox Explained Further
According to the reference, the Banach-Tarski paradox can be described as: "A ball can be cut into a finite number of pieces and re-assembling the pieces will get two balls, each of equal size to the first."
Why is it a Paradox?
It's a paradox because it contradicts our understanding of volume and how objects behave. We intuitively believe that if you break something into pieces and put it back together, you should have the same amount of "stuff" as you started with. The Banach-Tarski paradox shows that this isn't always true, especially when dealing with abstract mathematical concepts and non-measurable sets.
