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Ernst Friedrich Ferdinand Zermelo (; ; 27 July 187121 May 1953) was a German
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
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 Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theo ...
. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the
well-ordering theorem In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set ''X'' is ''well-ordered'' by a strict total order if every non-empty subset of ''X'' has a least element under the order ...
. 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.


Life

Ernst Zermelo graduated from Berlin's Luisenstädtisches Gymnasium (now ) in 1889. He then studied
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
,
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
and
philosophy Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
at the
University of Berlin The Humboldt University of Berlin (, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany. The university was established by Frederick William III on the initiative of Wilhelm von Humbol ...
, the
University of Halle Martin Luther University Halle-Wittenberg (), also referred to as MLU, is a public research university in the cities of Halle and Wittenberg. It is the largest and oldest university in the German state of Saxony-Anhalt. MLU offers German and i ...
, and the
University of Freiburg The University of Freiburg (colloquially ), officially the Albert Ludwig University of Freiburg (), is a public university, public research university located in Freiburg im Breisgau, Baden-Württemberg, Germany. The university was founded in 1 ...
. He finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the
calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...
(''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 physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in ...
. In 1897, Zermelo went to the
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen (, commonly referred to as Georgia Augusta), is a Public university, public research university in the city of Göttingen, Lower Saxony, Germany. Founded in 1734 ...
, 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 the
University of Zurich The University of Zurich (UZH, ) is a public university, public research university in Zurich, Switzerland. It is the largest university in Switzerland, with its 28,000 enrolled students. It was founded in 1833 from the existing colleges of the ...
, which he resigned in 1916. He was appointed to an honorary chair at the
University of Freiburg The University of Freiburg (colloquially ), officially the Albert Ludwig University of Freiburg (), is a public university, public research university located in Freiburg im Breisgau, Baden-Württemberg, Germany. The university was founded in 1 ...
in 1926, which he resigned in 1935 because he disapproved of
Adolf Hitler Adolf Hitler (20 April 1889 – 30 April 1945) was an Austrian-born German politician who was the dictator of Nazi Germany from 1933 until Death of Adolf Hitler, his suicide in 1945. Adolf Hitler's rise to power, He rose to power as the lea ...
's regime. At the end of
World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...
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 The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...
,
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
challenged the mathematical community with his famous
Hilbert's problems Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pr ...
, 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 mathema ...
, was the
continuum hypothesis In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
introduced by
Cantor A cantor or chanter is a person who leads people in singing or sometimes in prayer. Cantor as a profession generally refers to those leading a Jewish congregation, although it also applies to the lead singer or choir director in Christian contexts. ...
in 1878, and in the course of its statement Hilbert also mentioned the need to prove the
well-ordering theorem In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set ''X'' is ''well-ordered'' by a strict total order if every non-empty subset of ''X'' has a least element under the order ...
. 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 In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set ''X'' is ''well-ordered'' by a strict total order if every non-empty subset of ''X'' has a least element under the order ...
(''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 In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set ''X'' is ''well-ordered'' by a strict total order if every non-empty subset of ''X'' has a least element under the order ...
, based on the powerset axiom and the
axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
, 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, and the original numbering. In 1922,
Abraham Fraenkel Abraham Fraenkel (; 17 February, 1891 – 15 October, 1965) was a German-born Israeli mathematician. He was an early Zionist and the first Dean of Mathematics at the Hebrew University of Jerusalem. He is known for his contributions to axiomatic ...
and
Thoralf Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skole ...
independently improved Zermelo's axiom system. The resulting system, now called Zermelo–Fraenkel axioms (ZF), is now the most commonly used system for
axiomatic 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 mathema ...
.


Zermelo's navigation problem

Proposed in 1931,
Zermelo's navigation problem In mathematical optimization, Zermelo's navigation problem, proposed in 1931 by Ernst Zermelo, is a classic optimal control problem that deals with a boat navigating on a body of water, originating from a point A to a destination point B. The boat ...
is a classic
optimal control Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations ...
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.


Publications

* * * 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 publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press 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, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
* Axiom of constructibility *
Axiom of extensionality The axiom of extensionality, also called the axiom of extent, is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory. The axiom defines what a Set (mathematics), set is. Informally, the axiom means that the ...
*
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 ...
* Axiom of limitation of size *
Axiom of pairing In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by as a special case of his axiom of elementary sets ...
*
Axiom of union An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
*
Axiom schema of specification In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (''Aussonderungsaxiom''), subset axiom, axiom of class construction, or axiom schema of restricted comprehension is ...
* Boltzmann brain *
Choice function Let ''X'' be a set of sets none of which are empty. Then a choice function (selector, selection) on ''X'' is a mathematical function ''f'' that is defined on ''X'' such that ''f'' is a mapping that assigns each element of ''X'' to one of its ele ...
* Cumulative hierarchy * Pairwise comparison *
Von Neumann universe In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory ( ...
* 14990 Zermelo,
asteroid An asteroid is a minor planet—an object larger than a meteoroid that is neither a planet nor an identified comet—that orbits within the Solar System#Inner Solar System, inner Solar System or is co-orbital with Jupiter (Trojan asteroids). As ...


Citations


References

* * * * *


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 Academic staff of the University of Zurich Humboldt University of Berlin alumni Martin Luther University of Halle-Wittenberg alumni University of Freiburg alumni Academic staff of the University of Freiburg Academic staff of the University of Göttingen German male writers 20th-century German mathematicians