Part 5/7:
The Axiom of Choice has triggered several counterintuitive results that elicit skepticism. The Vitali set, created by Giuseppe Vitali in 1905, is one such example. By categorizing real numbers based on their differences, Vitali demonstrated how one could create a set with bizarre properties—specifically, a set that is unmeasurable. This fundamentally undermined traditional concepts of size and dimension in mathematics.
Furthermore, the Banach-Tarski Paradox presented yet another mind-bending scenario wherein a solid ball could be subdivided and rearranged into two identical copies of the original ball. While the construction of these pieces is not practically feasible, it raises profound philosophical questions about the nature of infinity and measurement.