The Warsaw School, Widening Applications, Models of Set Theory (1918–1940)

Abstract

When, shortly after 1918, a school of mathematicians emerged at Warsaw under Sierpinski’s tutelage, the Axiom finally gained the attention that it deserved, and thereby became the object of much careful research. Within a few years Polish mathematicians discovered interconnections between the Axiom and many other propositions in various branches of mathematics. Sierpinski’s survey of the Axiom’s uses was soon followed by Tarski’s research on definitions of finite set whose equivalence required the Axiom. Furthermore, Tarski discovered that each of several propositions in cardinal arithmetic implies the Axiom and hence is equivalent to it. On the other hand, Banach and Tarski extended Hausdorff’s paradox by demonstrating via the Axiom that any sphere S can be decomposed into a finite number of pieces and reassembled into two spheres with the same radius as S. Neither author regarded this result as any slight on the Axiom, but later mathematicians were to call it the Banach-Tarski paradox.