Ernst Friedrich Ferdinand Zermelo (
July 27 1871, Berlin, German Empire –
May 21 1953, Freiburg im Breisgau, West Germany) was a
German mathematician, whose work has major implications for the
foundations of mathematics and hence on
philosophy.
He graduated from Berlin's
Luisenstädtisches Gymnasium in
1889. He then studied
mathematics, physics and
philosophy at the universities of
Berlin, Halle and
Freiburg. He finished his doctorate in
1894 at the
University of Berlin, awarded for a dissertation on the
calculus of variations (
Untersuchungen zur Variationsrechnung). Zermelo remained at the University of Berlin, where he was appointed assistant to
Planck, under whose guidance he 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
habilitation 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 cardinals. In 1904, he succeeded in taking the first step suggested by Hilbert towards the
continuum hypothesis when he proved the
well-ordering theorem (
every set can be well ordered). This result brought fame to Zermelo, who was appointed Professor in Göttingen, in
1905. His proof of the
well-ordering theorem, based on the
axiom of choice, was not accepted by all mathematicians, partly because
set theory was not
axiomatized 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.
In
1922, Adolf Fraenkel and
Thoralf Skolem independently improved Zermelo's axiom system. The resulting 10 axiom system, now called
Zermelo-Fraenkel axioms (ZF), is now the most commonly used system for
axiomatic set theory.
In
1910, Zermelo left Göttingen upon being appointed to the chair of mathematics at
Zurich University, which he resigned in
1916.
He was appointed to an honorary chair at
Freiburg im Breisgau in
1926, which he resigned in
1935 because he disapproved of Hitler's regime. At the end of
World War II and at his request, Zermelo was reinstated to his honorary position in Freiburg.
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