Ernst Zermelo | |
|---|---|
 Ernst Zermelo in the 1900s | |
| Born | (1871-07-27)27 July 1871 |
| Died | 21 May 1953(1953-05-21) (aged 81) |
| Nationality | German |
| Alma mater | University of Berlin |
| Known for | |
| Spouse | Gertrud Seekamp (1944 – death) |
| Awards | Ackermann–Teubner Memorial Award (1916) |
| Scientific career | |
| Fields | Mathematics |
| Institutions | University of Zürich |
| Doctoral advisor | |
| Doctoral students | Stefan Straszewicz [pl] |
Ernst Friedrich Ferdinand Zermelo (/zɜːrˈmɛloʊ/;German:[tsɛɐ̯ˈmeːlo]; 27 July 1871 – 21 May 1953) was a Germanlogician andmathematician, whose work has major implications for thefoundations of mathematics. He is known for his role in developingZermelo–Fraenkel axiomatic set theory and his proof of thewell-ordering theorem. Furthermore, his 1929[1] work on ranking chess players is the first description of a model forpairwise comparison that continues to have a profound impact on various applied fields utilizing this method.
Ernst Zermelo graduated from Berlin's Luisenstädtisches Gymnasium (nowHeinrich-Schliemann-Oberschule [de]) in 1889. He then studiedmathematics,physics andphilosophy at theUniversity of Berlin, theUniversity of Halle, and theUniversity of Freiburg. He finished his doctorate in 1894 at the University of Berlin, awarded for a dissertation on thecalculus of variations (Untersuchungen zur Variationsrechnung). Zermelo remained at the University of Berlin, where he was appointed assistant toPlanck, under whose guidance he began to studyhydrodynamics. In 1897, Zermelo went to theUniversity of Göttingen, at that time the leading centre for mathematical research in the world, where he completed hishabilitation thesis in 1899.
In 1910, Zermelo left Göttingen upon being appointed to the chair of mathematics at theUniversity of Zurich, which he resigned in 1916.He was appointed to an honorary chair at theUniversity of Freiburg in 1926, which he resigned in 1935 because he disapproved ofAdolf Hitler's regime.[2] At the end ofWorld War II and at his request, Zermelo was reinstated to his honorary position in Freiburg.
In 1900, in the Paris conference of theInternational Congress of Mathematicians,David Hilbert challenged the mathematical community with his famousHilbert's problems, a list of 23 unsolved fundamental questions which mathematicians should attack during the coming century. The first of these, a problem ofset theory, was thecontinuum hypothesis introduced byCantor in 1878, and in the course of its statement Hilbert also mentioned the need to prove thewell-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 oftransfinite cardinals. By that time he had also discovered the so-calledRussell paradox. In 1904, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved thewell-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 thewell-ordering theorem, based on the powerset axiom and theaxiom 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 anaxiomatization 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 onZermelo set theory for an outline of this paper, together with the original axioms, with the original numbering.
In 1922,Abraham Fraenkel andThoralf Skolem independently improved Zermelo's axiom system. The resulting system, now calledZermelo–Fraenkel axioms (ZF), is now the most commonly used system foraxiomatic set theory.
Proposed in 1931,Zermelo's navigation problem is a classicoptimal 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.
Works by others:
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