Ernst Zermelo

Ernst Zermelo
Ernst Zermelo
	Ernst Zermelo in the 1900s
Born	27 July 1871; Berlin,German Empire
Died	21 May 1953(aged 81); Freiburg im Breisgau,West Germany
Nationality	German
Alma mater	University of Berlin
Known for	Zermelo set theory; Zermelo–Fraenkel set theory; Zermelo's navigation problem; Zermelo ordinal; Zermelo's theorem; Zermelo's theorem (game theory);
Spouse	Gertrud Seekamp (1944 – death)
Awards	Ackermann–Teubner Memorial Award(1916)
	Scientific career
Fields	Mathematics
Institutions	University of Zürich
Doctoral advisor	Lazarus Fuchs; Hermann Schwarz;
Doctoral students	Stefan Straszewicz[pl]

Ernst Friedrich Ferdinand Zermelo(/zɜːrˈmɛloʊ/,German:[tsɛɐ̯ˈmeːlo];27 July 1871 – 21 May 1953) was a Germanlogicianandmathematician,whose work has major implications for thefoundations of mathematics.He is known for his role in developingZermelo–Fraenkel axiomatic set theoryand his proof of thewell-ordering theorem.Furthermore, his 1929^[1]work on ranking chess players is the first description of a model forpairwise comparisonthat continues to have a profound impact on various applied fields utilizing this method.

Life[edit]

Ernst Zermelo graduated from Berlin's Luisenstädtisches Gymnasium (nowHeinrich-Schliemann-Oberschule[de]) in 1889. He then studiedmathematics,physicsandphilosophyat 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 thesisin 1899.

In 1910, Zermelo left Göttingen upon being appointed to the chair of mathematics atZurich University,which he resigned in 1916. He was appointed to an honorary chair at theUniversity of Freiburgin 1926, which he resigned in 1935 because he disapproved ofAdolf Hitler's regime.^[2]At the end ofWorld War IIand at his request, Zermelo was reinstated to his honorary position in Freiburg.

Research in set theory[edit]

In 1900, in the Paris conference of theInternational Congress of Mathematicians,David Hilbertchallenged 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 hypothesisintroduced byCantorin 1878, and in the course of its statement Hilbert mentioned also 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 anaxiomatizationof 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 theoryfor an outline of this paper, together with the original axioms, with the original numbering.

In 1922,Abraham FraenkelandThoralf Skolemindependently improved Zermelo's axiom system. The resulting system, now calledZermelo–Fraenkel axioms(ZF), is now the most commonly used system foraxiomatic set theory.

Zermelo's navigation problem[edit]

Proposed in 1931, theZermelo's navigation problemis a classicoptimal controlproblem. 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[edit]

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,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 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 ofStudies in the History of Mathematics and Physical Sciences,Springer Verlag, New York, 1982.

References[edit]

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)."Zermelo and the Early History of Game Theory"(PDF).Games and Economic Behavior.34(1): 123–137.doi:10.1006/game.2000.0794.Archived fromthe 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

Citations[edit]

^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]

Works by or about Ernst ZermeloatInternet Archive
O'Connor, John J.;Robertson, Edmund F.,"Ernst Zermelo",MacTutor History of Mathematics Archive,University of St Andrews
Zermelo Navigation

[1] 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.

[2] KAPLANSKY, IRVING (2020).SET THEORY AND METRIC SPACES.PROVIDENCE: AMER MATHEMATICAL SOCIETY. pp. 36–37.ISBN 9781470463847.

[1]

[2]

v t e Set theory
Overview	Set (mathematics)
Axioms	Adjunction Choice countable dependent global Constructibility (V=L) Determinacy Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement(i.e. set difference) De Morgan's laws Disjoint union Identities Intersection Power set Symmetric difference Union
Concepts Methods	Almost Cardinality Cardinal number(large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram
Settypes	Amorphous Countable Empty Finite(hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite(Dedekind-infinite) Recursive Singleton Subset·Superset Transitive Uncountable Universal
Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck
Paradoxes Problems	Russell's paradox Suslin's problem Burali-Forti paradox
Set theorists	Paul Bernays Georg Cantor Paul Cohen Richard Dedekind Abraham Fraenkel Kurt Gödel Thomas Jech John von Neumann Willard Quine Bertrand Russell Thoralf Skolem Ernst Zermelo

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