The German mathematician and educator Ernst Zermelo (1871–1953) was a pioneer of set theory. A branch of logic, set theory deals with collections of numbers or other objects; it would become a central field of mathematical study in the later 20th and 21st centuries.

Ernst Zermelo had a varied education that touched on physics, logic, and philosophy in addition to mathematics. Although his research dealt with questions of pure mathematics, his work had practical applications, from calculating the best path of a boat to refining the structure of automobile transmissions to the organization of chess tournaments. Zermelo did much of his most significant work at the beginning of his career, also teaching at the universities of Zurich and Freiburg. His work as an educator was cut short by ill health as well as by the rise of fascism in his native Germany: he was forced out of teaching over his refusal to perform the Nazi salute. Zermelo's Axiom of Choice—eventually elaborated into a group of seven axioms of set theory—became the foundation of many subsequent mathematical investigations.

Ernst Friedrich Ferdinand Zermelo was born in Berlin, Germany, on July 27, 1871. His father, Theodor Zermelo, was a popular high-school professor who influenced his son's choice of a career as an educator. Zermelo had five sisters, one older and four younger. His surname appears to have been unique to his family (after his wife's death, German telephone directories showed no one else with that name), and he gave various explanations of its origin, including that it was taken from the middle of the word *Walzermelodie* (“waltz melody”). Zermelo attended a Berlin *Gymnasium* (a college-preparatory high school with a classical curriculum) and graduated in 1889, the year his father died.

During the late 1800s, the German university system was oriented toward star lecturers rather than toward a unified curriculum and, like many other young Germans aspiring to academic careers, Zermelo attended classes at several different schools. He began his studies at Berlin University in 1890, then moved to the University of Halle-Wittenberg, where his teachers included mathematician Georg Cantor, who had made early investigations into set theory, and the philosopher Edmund Husserl. He then attended Freiburg University before returning to Berlin, where he completed his Ph.D. in 1894 with the dissertation *Untersuchungen zur Variationsrechnung* (“Investigations into the Calculus of Variations”). By the mid-1890s, Zermelo was publishing academic papers on major topics in the field of mathematics.

To become a university professor in Germany involved not only earning a Ph.D. but also undertaking the *Habilitation*, which included a series of national exams, producing and publishing papers, and giving a lecture. To accomplish this, Zermelo again moved among several universities, teaching in Hamburg, working at Berlin University as an assistant to famed physicist Max Planck, and taking courses at the University of Göttingen, which vied with Berlin for the distinction of having Germany's top mathematics program.

At Göttingen, Zermelo gained the mentorship of the influential German mathematician David Hilbert, who presented his paper on differential equations to the Göttingen Imperial Society of Sciences. In 1897, Zermelo passed his state exams, and two years later he completed his *Habil-itationsschrift*, a two-part thesis covering fluid dynamics and the geometry of spheres. After giving his required lecture, in which he took issue with the statistical mechanics ideas of Austrian physicist Ludwig Boltzmann, Zermelo was granted his teaching certificate and in 1899 was appointed as a lecturer at the University of Göttingen.

Zermelo's seven axioms remain basic to set theory, which has since become a major branch of mathematics. In his own time, however, they were considered controversial because of the unconventional types of proof he employed. Although he was turned down for teaching positions at Breslau University (now Wroclaw University in Poland) and the University of Würzburg, Zermelo's work continued to attract the attention of top mathematicians and philosophers. Among these was the British philosopher and mathematician Bertrand Russell, who invited the German professor to give two lectures in Cambridge, England, in 1912. By that time, Zermelo was teaching at the University of Zurich in Switzerland and had been diagnosed with tuberculosis. The disease was not fatal but required extended periods of rest, preventing him from honoring Russell's invitation.

Zermelo continued to write important mathematical papers during the 1910s, but his work and his teaching were negatively impacted by his worsening health and in 1916 he retired from his professorship in Zurich. He moved for a time to Göttingen, where in 1916 he gave a lecture on his theory of ordinal numbers. For much of the period at the end of and just after World War I, he was a patient at sanatoriums in the Swiss Alps, hoping to recover his health in the clean, crisp air. Although he was never completely successful in this regard, in 1921 his condition improved to the point that he moved to Freiburg, living in an apartment in the old part of the city where his needs were within walking distance. He now accepted a position as Full Honorary Professor at the Freiburg University Mathematical Institute and began teaching mathematics courses. He also sat in on a philosophy course given by Husserl and began a new German translation of Homer's *Odyssey*.

While most of Zermelo's ideas fell into the realm of pure mathematics, some of them had practical applications. In a 1919 paper, he presented a method for evaluating the results of a chess tournament, a matter of more than theoretical interest because he was an avid chess player. In another paper, he demonstrated that in chess and games like it, a game could not last forever; one side or the other would always have a strategy for victory or would be able to force a draw. The so-called Zermelo's Navigation Problem, which he proposed in 1931, involved methods of choosing the best route for a boat under a series of variable conditions. In 1911 he wrote a paper describing a primitive kind of automatic transmission; he refined this idea in 1932, while living in a house high in the Black Forest outside Hamburg, again for his health. He traveled the area in a small, three-wheeled car, and his idea for an improved means of transferring the energy from a power source to the drive wheels was similar to that adopted by General Motors in 1939 as its Hydra-Matic transmission system.

Zermelo's career at the University of Freiburg was interrupted in 1935 as Germany's Nazi government began to assert its control over the country's universities. Jewish faculty had already been dismissed, and the university's rector, philosopher Martin Heidegger, had ordered that the Nazi salute, along with the slogan “Heil, Hitler,” be given at the opening of university ceremonies. One group of students accused Zermelo of refusing to return a Nazi salute when given one on campus, and a research assistant named Schlotter reported that, during the performance of a Nazi anthem at one event, Zermelo “felt himself visibly uncomfortable. First, the little hand did not want to rise at all. He looked around even more timidly until during the third verse he saw no other possibility but to hold his little hand waist-high, again with visible reluctance. He gave the impression he was swallowing poison, desperate movements of his lips could be observed more and more rarely. He seemed to be relieved when the ceremony had ended.” In March of 1935, as a result of these accusations, Zermelo was forced to resign his position and was subsequently banned from the faculty.

Zermelo lived largely in seclusion during the late 1930s and for the duration of World War II, spending much of his time leasing a home in the Black Forest. For the entertainment of trusted friends, he wrote satirical poems criticizing the Nazi regime, and sometimes he corresponded with other mathematicians. He enjoyed taking long walks in the forest as well as to nearby mountain towns, and in one of those towns he met a younger woman named Gertrud Seekamp. One afternoon in 1944, out of the blue, he proposed marriage to Gertrud, and after recovering from the shock, she agreed. Sometimes he addressed her by the formal pronoun “Sie” instead of the informal “du”; he explained that she and his sisters were the only people who he had ever been able to address as “du.”

Shortly after the end of World War II, Zermelo requested reinstatement to the University of Freiburg faculty, and it was granted in 1946. Unfortunately, in addition to his other ailments, he now suffered from glaucoma, and his diminished eyesight prevented him from returning to teaching. By the early 1950s, he was completely blind. Although he received a small pension from the University of Zurich, finances were tight, and when he attempted to return to the Alps for his health, he was denied entry to Switzerland because his pension was not considered large enough to support both him and his wife there. Zermelo died quietly in Freiburg on May 21, 1953.

*Complete Dictionary of Scientific Biography*, Scribner's, 2008.

Ebbinghaus, Heinz-Dieter, *Ernst Zermelo: An Approach to His Life and Work*, Springer Verlag, 2007.

*MacTutor History of Mathematics*,
http://www-history.mcs.st-andrews.ac.uk/Biographies/Zermelo.html
(November 24, 2017), “Ernst Friedrich Ferdinand Zermelo.”

*Stetson University*, http://www2.stetson.edu/~efriedma/periodictable/html/Zr.html (November 24, 2017), “Ernst Friedrich Ferdinand Zermelo.”

*Stochasticon*,
http://stochastikon.no-ip.org:8080/encyclopedia/en/zermeloErnst.pdf
(November 24, 2017), “Ernst Zermelo.”

*University of Freiburg Department of Mathematics*,
http://home.mathematik.uni-freiburg.de/mildenberger/freiburg2014/zermelo.html
(November 24, 2017), “Ernst Zermelo and Freiburg.”□

(MLA 8^{th} Edition)