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  1. Zermelo's axiom of choice: Its origins, development, and influence (Studies in the history of mathematics and physical sciences 8) by Gregory H. Moore, 1982-11-17
  2. Ernst Zermelo: An Approach to His Life and Work by Heinz-Dieter Ebbinghaus, 2010-11-30
  3. Einführung in die Mengenlehre: Die Mengenlehre Georg Cantors und ihre Axiomatisierung durch Ernst Zermelo (Springer-Lehrbuch) (German Edition) by Oliver Deiser, 2009-10-29
  4. Ernst Zermelo - Collected Works/Gesammelte Werke: Volume I/Band I - Set Theory, Miscellanea/Mengenlehre, Varia (Schriften der Mathematisch-naturwissenschaftlichen ... Wissenschaften) (English and German Edition) by Ernst Zermelo, 2010-03-05
  5. Ernst Zermelo - Collected Works/Gesammelte Werke: Volume II/Band II - Calculus of Variations, Applied Mathematics, and Physics/Variationsrechnung, Angewandte ... und Physik (English and German Edition) by Ernst Zermelo, 2011-06-29
  6. Untersuchungen zur Variations-rechnung (German Edition) by Ernst Zermelo, 1894-01-01
  7. Gesammelte Abhandlungen mathematischen und philosophischen Inhalts: Mit erläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel Cantor-Dedekind (German Edition) by Georg Cantor, 1980-09-01
  8. Untersuchungen zur Variations-Rechnung, Inaugural-Dissertation... von Ernst Zermelo,... by Ernst Friedrich Ferdinand (1871-1953). ZERMELO, 1894-01-01
  9. Gesammelte Abhandlungen Mathematischen Und Philosophischen Inhalts by Georg, Herausgegeben Von Ernst Zermelo Cantor, 1966

41. Zermelo Set Theory - Linix Encyclopedia
zermelo set theory, as set out in an important paper in 1908 by ernst zermelo, zermelo, ernst. Untersuchungen ?er die Grundlagen der Mengenlehre I .
http://web.linix.ca/pedia/index.php/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 to its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering. Table of contents showTocToggle("show","hide") 1 The axioms of Zermelo set theory
2 Connection with standard set theory

3 The aim of Zermelo's paper

4 The axiom of separation
...
edit
The axioms of Zermelo set theory
AXIOM I. Axiom of extensionality Axiom der Bestimmtheit ) "If every element of a set M is also an element of N and vice versa ... then M N . Briefly, every set is determined by its elements". AXIOM II. Axiom of elementary sets ( Axiom der Elementarmengen a a a and only a as element. If a and b a b a and b but no object x distinct from them both." See Axiom of pairs AXIOM III. Axiom of separation Axiom der Aussonderung x ) is definite for all elements of a set M M possesses a subset M' containing as elements precisely those elements x of M x ) is true".

42. Zermelo Set Theory
zermelo set theory, as set out in an important paper in 1908 by ernst zermelo, zermelo, ernst. Untersuchungen ¼ber die Grundlagen der Mengenlehre I .
http://www.algebra.com/algebra/about/history/Zermelo-set-theory.wikipedia
Zermelo set theory
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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 to its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering.
Contents
The axioms of Zermelo set theory
AXIOM I. Axiom of extensionality Axiom der Bestimmtheit ) "If every element of a set M is also an element of N and vice versa ... then M N . Briefly, every set is determined by its elements". AXIOM II. Axiom of elementary sets (

43. Zermelo
(Redirected from zermelo). ernst Friedrich Ferdinand zermelo (July 27, 1871 –May 21, 1953) was a German mathematician and philosopher.
http://www.algebra.com/algebra/about/history/Zermelo.wikipedia
Zermelo
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Ernst Zermelo
(Redirected from Zermelo Ernst Friedrich Ferdinand Zermelo July 27 May 21 ) was a German mathematician and philosopher Zermelo was born in Berlin Germany . His secondary school education was at the Luisenst¤dtisches Gymnasium in Berlin where he graduated in . He then studied mathematics physics and philosophy at the universities of Berlin Halle and Freiburg . He finished his doctorate in and was awarded by the University of Berlin for his dissertation on the calculus of variations Untersuchungen zur Variationsrechnung ). Zermelo remained at the University of Berlin where he was appointed assistant to Planck and under his guidance began to study hydrodynamics . In , Zermelo went to G¶ttingen , at that time the leading centre for mathematical research in the world, where he completed his habilitation thesis in In , 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

44. Math Lessons - Ernst Zermelo
Math Lessons ernst zermelo. ernst zermelo. ernst Friedrich Ferdinandzermelo (July 27, 1871 – May 21, 1953) was a German mathematician and philosopher
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Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo July 27 May 21 ) was a German mathematician and philosopher Zermelo was born in Berlin Germany . His secondary school education was at the Luisenstädtisches Gymnasium in Berlin where he graduated in . He then studied mathematics physics and philosophy at the universities of Berlin Halle and Freiburg . He finished his doctorate in and was awarded by the University of Berlin for his dissertation on the calculus of variations Untersuchungen zur Variationsrechnung ). Zermelo remained at the University of Berlin where he was appointed assistant to Planck and under his guidance began to study hydrodynamics . In , Zermelo went to Göttingen , at that time the leading centre for mathematical research in the world, where he completed his thesis in In , 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

45. Math Lessons - Ernst Zermelo
Math Lessons ernst zermelo. ernst zermelo. (Redirected from zermelo).ernst Friedrich Ferdinand zermelo (July 27, 1871 – May 21, 1953) was a German
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Ernst Zermelo
(Redirected from Zermelo Ernst Friedrich Ferdinand Zermelo July 27 May 21 ) was a German mathematician and philosopher Zermelo was born in Berlin Germany . His secondary school education was at the Luisenstädtisches Gymnasium in Berlin where he graduated in . He then studied mathematics physics and philosophy at the universities of Berlin Halle and Freiburg . He finished his doctorate in and was awarded by the University of Berlin for his dissertation on the calculus of variations Untersuchungen zur Variationsrechnung ). Zermelo remained at the University of Berlin where he was appointed assistant to Planck and under his guidance began to study hydrodynamics . In , Zermelo went to Göttingen , at that time the leading centre for mathematical research in the world, where he completed his thesis in In , 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

46. ZF Set Theory
ernst zermelo (18711953). Adolf Fraenkel (1891-1965). A theorem of Godel andCohen proves that ZF set theory can neither prove nor disprove CH,
http://www.wall.org/~aron/zf.html
Zermelo-Fraenkel Set Theory
Below:
Friedrich Frege (1828-1925)/
Bertrand Russell (1872-1970)
The very first set theories had problems. The first one with specific axioms (base propositions from which other propositions could be established) was designed by someone named Frege and looked like this:
  • Extensionality:
    Two sets are the same if and only if they have identical elements.
  • Abstraction:
    For any property, there is a set of all things having that property.(The property must be expressed as a finite formula)
  • Choice:
    If you have a set which is partitioned into a number of subsets, with each subset containing at least one element and no two subsets sharing any elements, then there is yet another subset with exactly one element from each piece (This is explained in more detail below).
Unfortunatly, this system was shown to be inconsistant because of Russell's paradox. One can use the Axiom of Abstraction to generate the set of all sets that do not contain themselves. But does this set contain itself or not? If it does, it does not, if it does not, then it does. Ooops. Bertrand Russell, who pointed out this flaw, advocated correcting this by his Type Theory. It's sets were classified into multiple levels. Level had only basic elements that are not sets (called urelements). The next level had sets of urelements. Level 2 sets could contain urelements or Level 1 sets. And so on. In this set theory, no loops can occur, there is no set of all sets that do not contain themselves, and the paradox is averted.

47. The Zermelo-Fraenkel Axioms: Discussion (from Set Theory) --  Encyclopædia Bri
ernst Friedrich Ferdinand zermelo University of St.Andrews Biographical sketchof this German mathematician known for his contributions to statistical
http://www.britannica.com/eb/article?tocId=24035

48. Zermelo Set Theory - Wikipedia, The Free Encyclopedia
zermelo set theory, as set out in an important paper in 1908 by ernst zermelo, zermelo, ernst. Untersuchungen über die Grundlagen der Mengenlehre I .
http://en.wikipedia.org/wiki/Zermelo_set_theory
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Zermelo set theory
From Wikipedia, the free encyclopedia.
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 to its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering.
Contents
edit
The axioms of Zermelo set theory
AXIOM I. Axiom of extensionality Axiom der Bestimmtheit ) "If every element of a set M is also an element of N and vice versa ... then M N . Briefly, every set is determined by its elements". AXIOM II. Axiom of elementary sets ( Axiom der Elementarmengen ) "There exists a (fictitious) set, the null set, ∅, that contains no element at all. If a a a and only a as element. If

49. Zermelo-Fraenkel Set Theory: Information From Answers.com
zermeloFraenkel set theory The zermelo-Fraenkel axioms of set theory (ZF) are the axiom system put forth by ernst zermelo in 1908 (zermelo set theory).
http://www.answers.com/topic/zermelo-fraenkel-set-theory
showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping Zermelo-Fraenkel set theory Wikipedia Zermelo-Fraenkel set theory The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice , all of ordinary mathematics is based in modern formulations. When the axiom of choice is included, the resulting system is ZFC The axioms are the result of work by Thoralf Skolem in , based on earlier work by Abraham Fraenkel in the same year, which was based on the axiom system put forth by Ernst Zermelo in Zermelo set theory The axiom system is written in first-order logic ; it has an infinite number of axioms because an axiom schema is used. An alternative, finite system is given by the von Neumann-Bernays-G¶del axioms (NBG), which add the concept of a class in addition to that of a set ; it is "equivalent" in the sense that any theorem about sets which can be proved in one system can be proven in the other.

50. 1953 Set: -CITE- 42 USC CHAPTER 6 - THE CHILDREN'S BUREAU 01/06/03 -EXPCITE .
zermelo ernst Friedrich Ferdinand zermelo (1871-1953) Biography of ernstzermelo (1871-1953) . In 1902 zermelo published his first work on set theory
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1 of 1953, set out as a note under section 3501 of this title. Federal Security Agency and office of Administrator abolished by section 8 of Reorg. Plan No. .
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Biography of Ernst Zermelo (1871-1953) . In 1902 Zermelo published his first work on set theory which was on the addition of transfinite cardinals. .
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51. Practical Foundations Of Mathematics
ernst zermelo had been enticed from applied mathematics into foundations by Hilbert.He was interested in cardinal arithmetic and in particular the
http://www.cs.man.ac.uk/~pt/Practical_Foundations/html/s22.html
Practical Foundations of Mathematics
Paul Taylor
Sets (Zermelo Type Theory)
These methods of construction were first set out as a basis of set theory by Ernst Zermelo in 1908. The subsequent work sought to formalise them in terms of a notion of membership in which any entity in the universe may serve either as an element or as a set, and where it is legitimate to ask of any two entities whether one bears this relation to the other. We shall make a distinction between elements and sets, though in such a formalism it is usual to refer to terms and types as we did in Section . We shall also modify what Zermelo did very slightly, taking the cartesian product XxY X Y cf Examples Our system conforms very closely to the way mathematical constructions have actually been formulated in the twentieth century. The claim that set theory provides the foundations of mathematics is only justified via an encoding of this system, and not directly. It is, or at least it should be, surprising that it took 60 years to arrive at an axiomatisation which is, after all, pretty much as Zermelo did it in the first place. V - 1pt. For a detailed account of the modern system and its history, see [

52. Lexikon Ernst Zermelo
ernst zermelo aus der freien EnzyklopädieWikipedia und steht unter der GNU Lizenz. Die Liste der Autoren ist unter
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Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo 27. Juli in Berlin 21. Mai in Freiburg im Breisgau ) war ein deutscher Mathematiker Zermelo besuchte das Luisenst¤dtische Gymnasium in Berlin bis zum Abitur . Er studierte Mathematik Physik und Philosophie an den Universit¤ten von Berlin, Halle (Saale) und Freiburg . Er promovierte und wurde f¼r seine Doktorarbeit von der Universit¤t Berlin ausgezeichnet. Dort studierte er unter Max Planck Hydrodynamik ging Zermelo nach G¶ttingen , damals das Weltzentrum der Mathematik. bekam Zermelo den Lehrstuhl f¼r Mathematik an der Universit¤t Z¼rich , den er wieder aufgab. 1919 bewies er, dass endliche Spiele wie Schach (es gibt gewisse Spielabbruchbedingungen, so dass kein Schachspiel unendlich lange dauern kann) eine eindeutige L¶sung besitzen. Das heiŸt entweder WeiŸ gewinnt oder Schwarz oder das Spiel endet remis. Dieses Ergebnis war eines der ersten in der

53. Ed Mares’s Web Page
She is named after the mathematician, ernst zermelo. zermelo was one of thefounders of Adobe ImageReady. And this is ernst zermelo the set theorist.
http://www.vuw.ac.nz/staff/edwin_mares/
Ed Mares’s Web Page
I am originally from Toronto, Canada. I have retained my Canadian accent (or so my Kiwi friends tell me when they can’t understand what I’m saying) and the (probably futile) hope that the Maple Leafs will again win the Stanley Cup some day. After leaving Toronto, I moved around a bit, living in the States, Australia, and on both ends of Canada before settling in New Zealand. Now I live in lovely Wellington and teach at Victoria University. Victoria is a wonderful place to do logic. We have a Centre for Logic, Language and Computation , which has members from our maths, philosophy, computer science, and linguistics programmes. There is a lot of interesting work going on here on modal logic, non-classical logics, model theory, algebraic logic, recursion theory, complexity theory, and set theory. My current research is on relevant logic, Bertrand Russell, and on the logic of belief revision. My interest in Russell is to reconstruct his view ca. 1913 to make a coherent epistemology and philosophy of mathematics. I started down this road by reading Bernie Linsky’s book, Russell’s Metaphysical Logic . This project is just beginning but it promises to be long and difficult but fun. My work on belief revision started with the question of how we should amend traditional epistemologies given the view that it is sometimes legitimate to believe in a contradiction. I realised early on that even if we think that to believe the negation of a proposition does not preclude believing the proposition itself, we may still hold that there are some propositions that we find abhorrent, that we want to preclude believing. These abhorrent propositions are those that we reject. The theory of belief revision has us revise our beliefs and rejections in response to new information to retain a coherent (although perhaps inconsistent) whole, in which the set of beliefs do not entail (disjunctions of) any of the propositions that are rejected. This project has also led me to think and write about paraconsistency, negation, and rejection.

54. Russell's Paradox [Internet Encyclopedia Of Philosophy]
zermelo, ernst. Investigations in the Foundations of Set Theory I. In FromFrege to Gödel, ed. by Jean van Heijenoort. Cambridge, MA Harvard University
http://www.iep.utm.edu/p/par-russ.htm
Russell's Paradox Russell's paradox represents either of two interrelated logical antinomies. The most commonly discussed form is a contradiction arising in the logic of sets or classes. Some classes (or sets) seem to be members of themselves, while some do not. The class of all classes is itself a class, and so it seems to be in itself. The null or empty class, however, must not be a member of itself. However, suppose that we can form a class of all classes (or sets) that, like the null class, are not included in themselves. The paradox arises from asking the question of whether this class is in itself. It is if and only if it is not. The other form is a contradiction involving properties. Some properties seem to apply to themselves, while others do not. The property of being a property is itself a property, while the propery of being a cat is not itself a cat. Consider the property that something has just in case it is a property (like that of being a cat ) that does not apply to itself. Does this property apply to itself? Once again, from either assumption, the opposite follows. The paradox was named after Bertrand Russell, who discovered it in 1901.
Table of Contents (Clicking on the links below will take you to those parts of this article)

55. Zermelo Set Theory Definition Of Zermelo Set Theory In Computing Dictionary - By
Meaning of zermelo set theory computer term. What does zermelo set theory mean? zermelo, ernst Friedrich Ferdinand zermeloFraenkel (set theory)
http://computing-dictionary.thefreedictionary.com/Zermelo set theory
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Zermelo set theory
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Cite / link Email Feedback (mathematics) Zermelo set theory - A set theory with the following set of axiom s: Extensionality: two sets are equal if and only if they have the same elements. Union: If U is a set, so is the union of all its elements. Pair-set: If a and b are sets, so is a, b. Foundation: Every set contains a set disjoint from itself. Comprehension (or Restriction): If P is a formula with one free variable and X a set then x: x is in X and P is a set. Infinity: There exists an infinite set Power-set: If X is a set, so is its power set. Zermelo set theory avoids Russell's paradox by excluding sets of elements with arbitrary properties - the Comprehension axiom only allows a property to be used to select elements of an existing set.

56. Zermelo Fränkel Set Theory Definition Of Zermelo Fränkel Set Theory In Computi
Meaning of zermelo Fränkel set theory computer term. What does zermelo Fränkelset theory mean? zermelo set theory zermelo, ernst Friedrich Ferdinand
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Cite / link Email Feedback (mathematics) Zermelo Fränkel set theory - A set theory with the axiom s of Zermelo set theory (Extensionality, Union, Pair-set, Foundation, Restriction, Infinity, Power-set) plus the Replacement axiom schema: If F(x,y) is a formula such that for any x, there is a unique y making F true, and X is a set, then F x : x in X is a set. In other words, if you do something to each element of a set, the result is a set. An important but controversial axiom which is NOT part of ZF theory is the Axiom of Choice Mentioned in No references found Computing browser Full browser zeen Zen zenix ZENO ... Zermelo, Ernst Friedrich Ferdinand

57. Earliest Known Uses Of Some Of The Words Of Mathematics (Z)
Attributions to ernst zermelo Untersuchungen über die Grundlagen der Mengenlehre.I Math. Annalen (1908) occur in A. Fraenkel, Zu den Grundlagen der
http://members.aol.com/jeff570/z.html
Earliest Known Uses of Some of the Words of Mathematics (Z)
Last revision: June 28, 2004 z-AXIS. Axis of z appears in "On the Attractions of Homogeneous Ellipsoids" by James Ivory, Philosophical Transactions of the Royal Society of London, Vol. 99. (1809), pp. 345-372. [JSTOR]. Axis z is found in 1866 in the 8th edition of Elements of analytical mechanics by William Holms Chambers Bartlett: "Take the axis z M from the axis z by r " [University of Michigan Digital Library]. The terms z and the z DISTRIBUTION were introduced by R. A. Fisher in "On a Distribution Yielding the Error Functions of Several well Known Statistics" Proceedings of the International Mathematics Congress, Toronto (1924). Fisher's development of the analysis of variance in this paper and in his book Statistical Methods for Research Workers (1925) was based on the z distribution. Fisher’s z is related to the modern F by z F [James A. Landau]. See also F DISTRIBUTION and VARIANCE. ZENO’S  PARADOXES are due to Zeno of Elea (c. 490 BC - c. 425 BC) but none of his works survive and the paradoxes are known through the writings of Aristotle and Simplicius , a 6 th century AD commentator on Aristotle. The four paradoxes, relating to motion, the dichotomy, the Achilles, the arrow, and the stadium, are discussed in

58. Steps Towards A Logic Of Natural Objects
It is perhaps of interest to note that ernst zermelo, in his fundamental paper (1908)on zermelo, ernst (1908). Untersuchungen über die Grundlagen der
http://faculty.baruch.cuny.edu/lkirby/naturalobjects.html
Back to Laurence Kirby's home page Steps towards a logic of natural objects Laurence Kirby Baruch College, City University of New York
This article appeared in Epistemologia vol XXV (2002), pp.225-244. 1. Introduction
The natural objects that I propose to consider are broadly those physical objects which are studied and referred to by science and by a common sense view, informed by science, of the world. Natural objects are, philosophically speaking, individuals ; they are involved as units in dynamic, causal processes. I shall draw a distinction between natural objects and the abstract objects of mathematics, in particular set theory. Natural objects encompass atoms and molecules; cells and organisms, including you and me; the objects of everyday life such as chairs and automobiles; nations, continents, ecosystems, mountain ranges, geological faults; planets, stars and galaxies. Each natural object, when regarded internally, is a dynamic system with various interacting parts and components (some of which may be natural objects in their own right); when regarded externally, a natural object acts as a unit with respect to a larger system or systems (which may again be natural objects) of which the given object forms a part or component. It is sometimes argued that objects such as atoms or galaxies are theoretical constructs, as much so as mathematical objects (or even, according to some, more so). It is true that any reference to an object rests on epistemological assumptions. The approach here will be not to belittle these important epistemological questions but to leave them aside, and accept as a working assumption the practical viewpoint of people who are dealing with the world: that natural objects exist, act, and are acted upon, independently of the observer although any description of them or of their actions is dependent on the describer.

59. Ernst Zermelo - Bedeutung Definition Erklärung Glossar Zu Ernst Zermelo
Translate this page ernst Friedrich Ferdinand zermelo (* 27. Juli 1871 in Berlin, ? 21. Mai 1953 inFreiburg im Breisgau) war ein deutscher Mathematiker.zermelo besuchte das
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Ernst Zermelo
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Buchtipp zum Thema: Ernst Friedrich Ferdinand Zermelo 27. Juli in Berlin 21. Mai in Freiburg im Breisgau ) war ein deutscher Mathematiker Zermelo besuchte das Luisenstädtische Gymnasium in Berlin bis zum Abitur Er studierte Mathematik Physik und Philosophie an den Universitäten von Berlin, Halle (Saale) und Freiburg. Er promovierte und wurde für seine Doktorarbeit von der Universität Berlin ausgezeichnet. Dort studierte er unter Max Planck Hydrodynamik ging Zermelo nach Göttingen, damals das Weltzentrum der Mathematik. bekam Zermelo den Lehrstuhl für Mathematik an der Universität Zürich, den er wieder aufgab.

60. AIP International Catalog Of Sources
zermelo, ernst, 1871. Eidgenössische Technische Hochschule Zürich. Center forHistory of Physics (American Institute of Physics)
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Item Information Holdings More by this author Debye, Peter J. W. (Peter Josef William), 1884- Subjects Bohr, Niels Henrik David, 1885-1962. Born, Max, 1882-1970. Broglie, Louis de, 1892- Casimir, H. B. G. (Hendrik Brugt Gerhard), 1909- Ehrenfest, Paul, 1880-1933. Einstein, Albert, 1879-1955. Friedrich, Walter, 1883-1968. Heisenberg, Werner, 1901-1976. Hilbert, David, 1862-1943. Langevin, Paul, 1872-1946. Laue, Max von, 1879-1960. London, Fritz, 1900-1954. Nernst, Walther, 1864-1941. Planck, Max, 1858-1947. Sommerfeld, Arnold, 1868-1951. Zermelo, Ernst, 1871- Center for History of Physics (American Institute of Physics) Massachusetts Institute of Technology. Atomic theory. Nuclear physics. Quantum theory. Science History. Browse Catalog by author: Debye, Peter J. W. (Peter Josef William), 1884-

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