Zermelo was born in Berlin, Germany. His secondary school education was at the Luisenstädtisches Gymnasium in Berlin where he graduated in 1889. He then studied mathematics, physics and philosophy at the universities of Berlin, Halle and Freiburg. He finished his doctorate in 1894 and was awarded by the University of Berlin for his dissertation on the calculus of variations. Zermelo remained at the University of Berlin where he was appointed assistant to Planck and under his guidance began to study hydrodynamics. In 1897, Zermelo went to Göttingen, at that time the leading centre for mathematical research in the world, where he completed his thesis in 1899.
In 1900, in the Paris conference of the International Congress of Mathematicians, David Hilbert challenged the mathematical community with his famous Hilbert's problems, a list of 23 unsolved fundamental questions which mathematicians should attack during the coming century. The first of these, a problem of set theory, was the continuum hypothesis introduced by Cantor in 1878.
Zermelo began to work on the problems of set theory and in 1902 published his first work concerning the addition of transfinite cardinalss. In 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he the well-ordering theorem (every set can be well ordered). This result brought fame to Zermelo, who was appointed as professor in Göttingen, in December 1905. His proof of the well-ordering theorem, which was based on the axiom of choice, was not accepted by all mathematicians, partly because the lack of axiomatization of set theory at this time. In 1908, Zermelo succeeded in producing a much more widely-accepted proof.
In 1905, Zermelo began to axiomatize set theory; in 1908, he published his results despite his failure to prove the consistency of his axiomatic system. See the article on Zermelo set theory for an outline of this paper, together with the original axioms, with the original numbering.
It should be noted that, in 1922, Adolf Fraenkel and Thoralf Skolem independently improved Zermelo's axiom system. The resulting system, now called Zermelo-Fraenkel axioms (ZF), with ten axioms, is now the most commonly used system for axiomatic set theory.
In 1910, Zermelo left Göttingen when he was appointed to the chair of mathematics at the Zurich University. In 1916, he resigned his chair in Zurich. He was appointed to an honorary chair at Freiburg im Breisgau in 1926 but he renounced his chair in 1935 because of his disapproval of Hitler's regime. At the end of World War II Zermelo requested that he be reinstated to his honorary position in Freiburg and indeed he was reinstated to the post in 1946.
Zermelo died in Freiburg im Breisgau, Germany.