Ernst Zermelo

Ernst Zermelo was a German mathematician and logician whose axiomatization of set theory remains the foundation of the standard framework used throughout modern mathematics. In 1904 Zermelo proved the well-ordering theorem, showing that every set can be well-ordered, a result that depended on a principle he made explicit as the axiom of choice, whose non-constructive character provoked significant controversy among mathematicians uneasy with proofs that guarantee an object's existence without a means of constructing it. In 1908 Zermelo published the first formal axiomatization of set theory, a system that Abraham Fraenkel and Thoralf Skolem later extended in 1922 by adding the axioms of replacement and regularity, producing the Zermelo-Fraenkel set theory that remains the most widely used foundational system in mathematics today. Zermelo held academic positions at Gottingen, Zurich and Freiburg, resigning his position at Freiburg in 1935 in opposition to the Nazi regime and being reinstated there after the Second World War. In 1929 Zermelo also produced an early mathematical model for ranking competitors through pairwise comparison, an idea later applied broadly beyond its original context of chess tournament ranking.