 Set theory is one of the greatest achievements of modern mathematics. Basically all mathematical concepts, methods, and results admit of representation within axiomatic set theory. Thus set theory has served quite a unique role by systematizing modern mathematics, and approaching in a unified form all basic questions about admissible mathematical arguments—including the thorny question of existence principles. This entry covers in outline the convoluted process by which set theory came into being, covering roughly the years 1850 to 1930.