Ernst Zermelo



Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German
logic Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science investigating h ...
ian and
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem. Furthermore, his 1929 work on ranking chess players is the first description of a model for pairwise comparison that continues to have a profound impact on various applied fields utilizing this method.


Ernst Zermelo graduated from Berlin's Luisenstädtisches Gymnasium (now ) in 1889. He then studied
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
at the
University of Berlin Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin Berlin ( , ) is the capital and largest city of Germany ...
, the University of Halle, and the University of 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 the University of Göttingen, at that time the leading centre for mathematical research in the world, where he completed his habilitation thesis in 1899. 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 the University of Freiburg in 1926, which he resigned in 1935 because he disapproved of
Adolf Hitler Adolf Hitler (; 20 April 188930 April 1945) was an Austrian-born German politician who was dictator of Germany Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populo ...
's regime. At the end of
World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...
and at his request, Zermelo was reinstated to his honorary position in Freiburg.

Research in set theory

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 Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...
, was the continuum hypothesis introduced by Cantor in 1878, and in the course of its statement Hilbert mentioned also the need to prove the well-ordering theorem. Zermelo began to work on the problems of set theory under Hilbert's influence and in 1902 published his first work concerning the addition of transfinite cardinals. By that time he had also discovered the so-called Russell paradox. 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 powerset axiom and the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...
, was not accepted by all mathematicians, mostly because the axiom of choice was a paradigm of non-constructive mathematics. In 1908, Zermelo succeeded in producing an improved proof making use of Dedekind's notion of the "chain" of a set, which became more widely accepted; this was mainly because that same year he also offered an axiomatization of set theory. Zermelo began to axiomatize set theory in 1905; 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, Abraham Fraenkel and Thoralf Skolem independently improved Zermelo's axiom system. The resulting 8 axiom system, now called Zermelo–Fraenkel axioms (ZF), is now the most commonly used system for axiomatic set theory.

Zermelo's navigation problem

Proposed in 1931, the Zermelo's navigation problem is a classic optimal control problem. The problem deals with a boat navigating on a body of water, originating from a point O to a destination point D. The boat is capable of a certain maximum speed, and we want to derive the best possible control to reach D in the least possible time. Without considering external forces such as current and wind, the optimal control is for the boat to always head towards D. Its path then is a line segment from O to D, which is trivially optimal. With consideration of current and wind, if the combined force applied to the boat is non-zero, the control for no current and wind does not yield the optimal path.


* * * Jean van Heijenoort, 1967. ''From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931''. Harvard Univ. Press. **1904. "Proof that every set can be well-ordered," 139−41. **1908. "A new proof of the possibility of well-ordering," 183–98. **1908. "Investigations in the foundations of set theory I," 199–215. *1913. "On an Application of Set Theory to the Theory of the Game of Chess" in Rasmusen E., ed., 2001. ''Readings in Games and Information'', Wiley-Blackwell: 79–82. *1930. "On boundary numbers and domains of sets: new investigations in the foundations of set theory" in Ewald, William B., ed., 1996. ''From Kant to Hilbert: A Source Book in the Foundations of Mathematics'', 2 vols.
Oxford University Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...
: 1219–33. Works by others: *''Zermelo's Axiom of Choice, Its Origins, Development, & Influence,'' Gregory H. Moore, being Volume 8 of ''Studies in the History of Mathematics and Physical Sciences,'' Springer Verlag, New York, 1982.

See also

Axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...
Axiom of infinity In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the ...
* Axiom of limitation of size * Axiom of union * Boltzmann brain * Choice function * Cumulative hierarchy * Pairwise comparison * Von Neumann universe * 14990 Zermelo,
asteroid An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere ...


* * Grattan-Guinness, Ivor (2000) ''The Search for Mathematical Roots 1870–1940''. Princeton University Press. * * * Ebbinghaus, Heinz-Dieter (2007) ''Ernst Zermelo: An Approach to His Life and Work''. Springer.

External links

* *
Zermelo Navigation
{{DEFAULTSORT:Zermelo, Ernst 1871 births 1953 deaths 20th-century German philosophers 19th-century German mathematicians Mathematical logicians Writers from Berlin People from the Province of Brandenburg Set theorists University of Zurich faculty Humboldt University of Berlin alumni Martin Luther University of Halle-Wittenberg alumni University of Freiburg alumni University of Freiburg faculty University of Göttingen faculty German male writers 20th-century German mathematicians