The third great crisis in mathematics

Topology and measure theory, along with other branches of mathemtics, are founded upon the theory of sets. The beginnings of the theory were steeped in paradoxes. One concerns the very concept of a set itself, such as proving that ``0 is ordinary if and only if it is not.'' The concepts have been debated for more than a century and from them have evolved the axiom of choice, axiom of selection, infinite set theory, well-ordering theorem, and the Banach-Tarski paradox, among others.