Choice of Choice: Paradoxical Results in Zermelo-Fraenkel Set Theory

When people think of mathematics they think “right or wrong,” “empirically correct” or “empirically incorrect.” Formalized logically valid arguments are one important step to achieving this definitive answer; however, what about the underlying assumptions to the argument? In the early 20th century mathematicians set out to formalize these assumptions, which in mathematics are known as axioms. The most common of these axiomatic systems was the Zermelo-Fraenkel axioms. The standard axioms in this system were accepted by mathematicians as obvious, and deemed by some to be sufficiently powerful to prove all the intuitive theorems already known to mathematicians. However, this system wasn’t without controversy; Zermelo included the Axiom of Choice to prove his Well Ordering Theorem. This lead to unintended consequences. Imagine taking a solid, three-dimensional ball and breaking it apart into certain finite pieces. Instinctively, one would agree that no matter how these pieces are rotated, when you put them back together you should have the same ball. Surprisingly the Axiom of Choice tells us this isn’t the case, that there is a way to put these pieces back together and have two identical copies of the original ball. Delving further, one can start with something the size of a pea, and after specific rotations, end up with a ball the size of our sun. The Axiom of Choice also lets us conclude that there is a way to predict the future correctly at almost any point in time. However, as many an incorrect weatherman will tell you, this too goes against what we believe. So how does one reconcile his or her concept of what’s true and what the Axiom of Choice tells us to be true? Do we simply take away the Axiom of Choice? As you may expect, the answer isn’t quite so simple…