Ernst Zermelo graduated from Berlin's Luisenstädtisches Gymnasium (now Heinrich-Schliemann-Oberschule [de]) in 1889. He then studied mathematics, physics and philosophy at the University of Berlin, 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's regime.[2] At the end of World War II and at his request, Zermelo was reinstated to his honorary position in Freiburg.
Ernst Zermelo tombstone in Friedhof Günterstal, in Günterstal district of Freiburg im Breisgau
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.
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.
PublicationsEdit
Zermelo, Ernst (2013), Ebbinghaus, Heinz-Dieter; Fraser, Craig G.; Kanamori, Akihiro (eds.), Ernst Zermelo—collected works. Vol. I. Set theory, miscellanea, Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften, vol. 21, Berlin: Springer-Verlag, doi:10.1007/978-3-540-79384-7, ISBN978-3-540-79383-0, MR 2640544
Zermelo, Ernst (2013), Ebbinghaus, Heinz-Dieter; Kanamori, Akihiro (eds.), Ernst Zermelo—collected works. Vol. II. Calculus of variations, applied mathematics, and physics, Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften, vol. 23, Berlin: Springer-Verlag, doi:10.1007/978-3-540-70856-8, ISBN978-3-540-70855-1, MR 3137671
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: 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.
Dirk Van Dalen; Heinz-Dieter Ebbinghaus (June 2000). "Zermelo and the Skolem Paradox". The Bulletin of Symbolic Logic. 6 (2): 145–161. CiteSeerX10.1.1.137.3354. doi:10.2307/421203. hdl:1874/27769. JSTOR 421203. S2CID 8530810.
Grattan-Guinness, Ivor (2000) The Search for Mathematical Roots 1870–1940. Princeton University Press.
Kanamori, Akihiro (2004). "Zermelo and set theory". The Bulletin of Symbolic Logic. 10 (4): 487–553. doi:10.2178/bsl/1102083759. MR 2136635. S2CID 231795240.
Schwalbe, Ulrich; Walker, Paul (2001). (PDF). Games and Economic Behavior. 34 (1): 123–137. doi:10.1006/game.2000.0794. Archived from the original (PDF) on 1 April 2017.
^Zermelo, Ernst (1929). "Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung". Mathematische Zeitschrift. 29 (1): 436–460. doi:10.1007/BF01180541. S2CID 122877703.
^KAPLANSKY, IRVING (2020). SET THEORY AND METRIC SPACES. PROVIDENCE: AMER MATHEMATICAL SOCIETY. pp. 36–37. ISBN9781470463847.
External linksEdit
Wikimedia Commons has media related to Ernst Zermelo.
ernst, zermelo, ernst, friedrich, ferdinand, zermelo, ɜːr, german, tsɛɐ, ˈmeːlo, july, 1871, 1953, german, logician, mathematician, whose, work, major, implications, foundations, mathematics, known, role, developing, zermelo, fraenkel, axiomatic, theory, proof. Ernst Friedrich Ferdinand Zermelo z ɜːr ˈ m ɛ l oʊ German tsɛɐ ˈmeːlo 27 July 1871 21 May 1953 was a German logician and mathematician 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 1 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 ZermeloErnst Zermelo in the 1900sBorn 1871 07 27 27 July 1871Berlin German EmpireDied21 May 1953 1953 05 21 aged 81 Freiburg im Breisgau West GermanyNationalityGermanAlma materUniversity of BerlinKnown forZermelo set theory Zermelo Fraenkel set theory Zermelo s navigation problem Zermelo ordinal Zermelo s theorem Zermelo s theorem game theory SpouseGertrud Seekamp 1944 death AwardsAckermann Teubner Memorial Award 1916 Scientific careerFieldsMathematicsInstitutionsUniversity of ZurichDoctoral advisorLazarus Fuchs Hermann SchwarzDoctoral studentsStefan Straszewicz pl Contents 1 Life 2 Research in set theory 3 Zermelo s navigation problem 4 Publications 5 See also 6 References 7 External linksLife Edit nbsp Ernst Zermelo in Freiburg 1953 Ernst Zermelo graduated from Berlin s Luisenstadtisches Gymnasium now Heinrich Schliemann Oberschule de in 1889 He then studied mathematics physics and philosophy at the University of Berlin 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 Gottingen at that time the leading centre for mathematical research in the world where he completed his habilitation thesis in 1899 In 1910 Zermelo left Gottingen 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 s regime 2 At the end of World War II and at his request Zermelo was reinstated to his honorary position in Freiburg nbsp Ernst Zermelo tombstone in Friedhof Gunterstal in Gunterstal district of Freiburg im BreisgauResearch in set theory EditIn 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 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 Gottingen 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 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 EditProposed 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 Publications EditZermelo Ernst 2013 Ebbinghaus Heinz Dieter Fraser Craig G Kanamori Akihiro eds Ernst Zermelo collected works Vol I Set theory miscellanea Schriften der Mathematisch Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften vol 21 Berlin Springer Verlag doi 10 1007 978 3 540 79384 7 ISBN 978 3 540 79383 0 MR 2640544 Zermelo Ernst 2013 Ebbinghaus Heinz Dieter Kanamori Akihiro eds Ernst Zermelo collected works Vol II Calculus of variations applied mathematics and physics Schriften der Mathematisch Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften vol 23 Berlin Springer Verlag doi 10 1007 978 3 540 70856 8 ISBN 978 3 540 70855 1 MR 3137671 Jean van Heijenoort 1967 From Frege to Godel 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 1219 33 Works by others Zermelo s Axiom of Choice Its Origins Development amp Influence Gregory H Moore being Volume 8 of Studies in the History of Mathematics and Physical Sciences Springer Verlag New York 1982 See also EditAxiom of choice Axiom of infinity Axiom of limitation of size Axiom of union Boltzmann brain Choice function Cumulative hierarchy Pairwise comparison Von Neumann universe 14990 Zermelo asteroidReferences EditDirk Van Dalen Heinz Dieter Ebbinghaus June 2000 Zermelo and the Skolem Paradox The Bulletin of Symbolic Logic 6 2 145 161 CiteSeerX 10 1 1 137 3354 doi 10 2307 421203 hdl 1874 27769 JSTOR 421203 S2CID 8530810 Grattan Guinness Ivor 2000 The Search for Mathematical Roots 1870 1940 Princeton University Press Kanamori Akihiro 2004 Zermelo and set theory The Bulletin of Symbolic Logic 10 4 487 553 doi 10 2178 bsl 1102083759 MR 2136635 S2CID 231795240 Schwalbe Ulrich Walker Paul 2001 Zermelo and the Early History of Game Theory PDF Games and Economic Behavior 34 1 123 137 doi 10 1006 game 2000 0794 Archived from the original PDF on 1 April 2017 Ebbinghaus Heinz Dieter 2007 Ernst Zermelo An Approach to His Life and Work Springer ISBN 3 642 08050 2 Zermelo Ernst 1929 Die Berechnung der Turnier Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung Mathematische Zeitschrift 29 1 436 460 doi 10 1007 BF01180541 S2CID 122877703 KAPLANSKY IRVING 2020 SET THEORY AND METRIC SPACES PROVIDENCE AMER MATHEMATICAL SOCIETY pp 36 37 ISBN 9781470463847 External links Edit nbsp Wikimedia Commons has media related to Ernst Zermelo Works by or about Ernst Zermelo at Internet Archive O Connor John J Robertson Edmund F Ernst Zermelo MacTutor History of Mathematics Archive University of St Andrews Zermelo Navigation Retrieved from https en wikipedia org w index php title Ernst Zermelo amp oldid 1172759908, wikipedia, wiki, book, books, library,
