Banach-Tarski Paradox

Pure mathematics is abstract. There are many concepts which outside of a theoretical space, seem to be impossible. An example could be different dimensions. One could argue that even if mathematics holds true in higher dimensions because there is no tangible way hold a 17-dimensional cube, it is irrelevant, or maybe even impossible. This same kind of logic fits in with the Banach-Tarski paradox. First proposed by Stefan Banach and Alfred Tarski in 1924, this paradox states that “given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball“ (Wikipedia.) Simplifying, this paradox states that a single solid ball can be rearranged into two balls of the same size as the original.

That doesn’t seem to make any sense. If I had a cube of volume 27 units cubed, no matter how hard I tried to rearrange the cube, the total volume would never change from 27 units cubed. One reason that this Paradox has any foundation is that the “pieces“ of the share aren’t solid. They are instead scatterings of infinite points making up a “solid“ piece. This is where the theory first becomes paradoxical because it is playing with infinity.

This paradox relies on the “axiom of choice.“ This axiom states that for every possible set, it is possible to choose exactly one element. This is most useful with the infinite set. Imagine you had a bag with marbles where only one is red. This axiom states that if you reached into the bag, it is possible to choose that red marble. Logically, the odds of choosing that red marble are almost 0%.

In the Banach-Tarski paradox, this axiom of choice allows for the creation of non-measurable sets or sets where the volume is virtually negligible. Visually, each “piece“ of the ball would be so jagged so that the volume is almost nothing. Just like how the probability of choosing a red marble in a bag of infinite marbles is almost nothing, the same applies to these non-measurable sets. Without the axiom of choice, the Banach-Tarski paradox would have no real foundation.

This is where this theorem becomes a paradox. Because of the axiom of choice and the forming of non-measurable sets, theoretically, there would exist a way to divide a ball to create two balls of the same size. Logically though, there is no real application of this theory. Just like many faulty proofs that equal 1 to 0 by dividing by infinity somewhere in their solving, Banach-Tarski utilizes this same paradoxical principle of infinity. Mathematically, this principle may work, but because it is reliant on jagged pieces with negligible volume, there is no way to represent the Banach-Tarski theory in the real world.