Ernst Zermelo
Encyclopedia
Ernst Friedrich Ferdinand Zermelo (ʦɛrˈmeːlo; 1871–1953) was a German
mathematician
, whose work has major implications for the foundations of mathematics
and hence on philosophy
. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory
and his proof of the well-ordering theorem
.
, 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 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.
, 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, 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, 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, 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.
problem. The problems 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 to follow a straight line segment from O to D. With consideration of current and wind, the shortest path from O to D is in fact, not the optimal solution.
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, whose work has major implications for the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
and hence on philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
and his proof of the well-ordering theorem
Well-ordering theorem
In mathematics, the well-ordering 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 ordering. This is also known as Zermelo's theorem and is equivalent to the Axiom of Choice...
.
Life
He graduated from Berlin's Luisenstädtisches Gymnasium in 1889. He then studied mathematicsMathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
at the universities of Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
, Halle
Halle, Saxony-Anhalt
Halle is the largest city in the German state of Saxony-Anhalt. It is also called Halle an der Saale in order to distinguish it from the town of Halle in North Rhine-Westphalia...
and Freiburg
Freiburg
Freiburg im Breisgau is a city in Baden-Württemberg, Germany. In the extreme south-west of the country, it straddles the Dreisam river, at the foot of the Schlossberg. Historically, the city has acted as the hub of the Breisgau region on the western edge of the Black Forest in the Upper Rhine Plain...
. He finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on the calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
(Untersuchungen zur Variationsrechnung). Zermelo remained at the University of Berlin, where he was appointed assistant to Planck
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...
, under whose guidance he began to study hydrodynamics. In 1897, Zermelo went to Göttingen
Göttingen
Göttingen is a university town in Lower Saxony, Germany. It is the capital of the district of Göttingen. The Leine river runs through the town. In 2006 the population was 129,686.-General information:...
, 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 Freiburg im Breisgau in 1926, which he resigned in 1935 because he disapproved of Hitler's regime. At the end of World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
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 MathematiciansInternational Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
, David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
challenged the mathematical community with his famous Hilbert's problems
Hilbert's problems
Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th century mathematics...
, 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
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, was the continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
introduced by Cantor
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets,...
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
Well-ordering theorem
In mathematics, the well-ordering 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 ordering. This is also known as Zermelo's theorem and is equivalent to the Axiom of Choice...
(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
Well-ordering theorem
In mathematics, the well-ordering 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 ordering. This is also known as Zermelo's theorem and is equivalent to the Axiom of Choice...
, based on the powerset axiom and the axiom of choice, 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
Zermelo set theory
Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted...
for an outline of this paper, together with the original axioms, with the original numbering.
In 1922, Adolf Fraenkel and Thoralf Skolem
Thoralf Skolem
Thoralf Albert Skolem was a Norwegian mathematician known mainly for his work on mathematical logic and set theory.-Life:...
independently improved Zermelo's axiom system. The resulting 10 axiom system, now called Zermelo-Fraenkel axioms
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
(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 controlOptimal control
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...
problem. The problems 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 to follow a straight line segment from O to D. With consideration of current and wind, the shortest path from O to D is in fact, not the optimal solution.
See also
- Zermelo–Fraenkel set theoryZermelo–Fraenkel set theoryIn mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
- Zermelo set theoryZermelo set theoryZermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted...
- Well-ordering theoremWell-ordering theoremIn mathematics, the well-ordering 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 ordering. This is also known as Zermelo's theorem and is equivalent to the Axiom of Choice...
- Zermelo's theorem (game theory)Zermelo's theorem (game theory)In game theory, Zermelo’s theorem, named after Ernst Zermelo, says that in any finite two-person game of perfect information in which the players move alternatively and in which chance does not affect the decision making process, if the game cannot end in a draw, then one of the two players must...
- 14990 Zermelo14990 Zermelo14990 Zermelo is a main-belt asteroid discovered on October 31, 1997 by P. G. Comba at Prescott.- External links :*...
, asteroid