Abstract
The ordered field ℜ of real numbers is of course up to isomorphism the unique Dedekind continuous ordered field. Equally important, though apparently less well known, is the fact that ℜ is also up to isomorphism the unique Archimedean complete, Archimedean ordered field. The idea of an Archimedean complete ordered field was introduced by Hans Hahn in his celebrated investigation Über die nichtarchimedischen Grössensysteme which was presented to the Royal Academy of Sciences in Vienna in 1907. It is a special case of his more general conception of an Archimedean complete, ordered abelian group, a conception that was motivated by, and substantially generalizes, the idea of ℜ as an Archimedean ordered field which admits no proper extension to an Archimedean ordered field; that is, the idea of an Archimedean ordered field which satisfies Hilbert’s Axiom of (arithmetic) Completeness (Hilbert 1900a, p. 183; 1903a, p. 16).
In 1907 H. Hahn published his truly monumental analysis of totally (= linearly) ordered groups and fields. Residue Class Fields of Rings of Continuous Functions
(Norman Ailing 1976, p. 56)
The most exhaustive investigation of infinitesimals and infinite numbers is that due to Hahn (1907), who generalized and extended the researches of Levi-Civita by showing that a system of non-Archimedean numbers can be associated with any aggregate which is simply ordered in accordance with Cantor’s definition, and that such a system is the most general non-Archimedean system possible. 100 YEARS OF MATHEMATICS
(George Temple 1981, p. 22)
This work [(Hahn 1907]) is the origin of the theory of ordered algebraic structures. ZAHLEN UND KONTINUUM Eine Einführung in die Infinitesimal-mathematik
(Detlef Laugwitz 1986, p. 222)
This paper was partially written during the author’s tenure as a Research Fellow at the Center for the Philosophy and History of Science, Boston University. The author gratefully acknowledges the support of the Center as well as the support provided by the National Science Foundation (Scholars Award #SBR-9223839). During the same period of time, the author wrote the sections of (Ehrlich 1994b) entitled The Ordered Field of Reals: The Late 19th-Century Geometrical Motivation, but inadvertently failed to gratefully acknowedges the support of the Center and the National Science Foundation at that time. We take this belated opportunity to do so.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
Ailing, N.: 1961, ‘A Characterization of Abelian rlti groups in Terms of Their Natural Valuation’, Proceedings of the National Academy of Sciences 47, 711–713.
Ailing, N.: 1962a, ‘On Exponentially Closed Fields’, Proceedings of the American Mathematical Society 13, 706–711.
Ailing, N. 1962b. ‘On the Existence of Real Closed Fields That are tin-sets of Power Transactions of the American Mathematical Society 103 341–352.
Ailing, N.: 1976, ‘Residue Class Fields of Rings of Continuous Functions’, In Symposia Mathematica, Volume XVII, Academic Press, London, pp. 55–67.
Ailing, N.: 1985, ‘Conway’s Field of Surreal Numbers’, Transactions of the American Mathematical Society 287, 365–386.
Ailing, N.: 1987, Foundations of Analysis over Surreal Number Fields, North-Holland, Amsterdam.
Artin, E.: 1957, Geometric Algebra, Interscience Publishers Inc., New York.
Artin, E.: 1965, The Collected Papers. Edited by S. Lang and J. Tate, Addison, Wesley Publishing Co., Reed.
Artin, E. and O. Schreier: 1926, ‘Algebraische Konstruktion reeller Körper’, Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Univeristät, Leipzig 5, 85–99. Reprinted in (Artin 1965, pp. 258–272 ).
Artin, E. and O. Schreier: 1927, ‘Eine Kennzeichnung der reell abgeschlossenen Körper’, Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Univeristät, Leipzig 5, 225–231. Reprinted in (Artin 1965, pp. 289–295 ).
Ax, J. and S. Kochen: 1965a, ‘Diophantine Problems Over Local Fields: I.’, American Journal of Mathematics 87, 605–630.
Ax, J. and S. Kochen: 1965b, ‘Diophantine Problems Over Local Fields: II.’, American Journal of Mathematics 87, 631–648.
Ax, J. and S. Kochen: 1966, ‘Diophantine Problems Over Local Fields: III.’, Annals Of Mathematics (Series 2) 83, 437–456.
Baer, R.: 1927, ‘über nicht-archimedisch geordnete Körper’, Sitzungsberichte der Heidelberger Akademie der Wissenschaften 8, 3–13.
Baer, R.: 1927/1928a, ‘über ein Vollständigkeitsaxiom in der Mengenlehre’, Mathematische Zeitschrift 27, 536–539.
Baer, R.: 1927/1928b, ‘Bemerkungen zu der Erwiderung von Herrn P. Finsler’, Mathematische Zeitschrift 27, 543.
Baer, R.: 1936, ‘The Subgroup of the Elements of Finite Order of an Abelian Group’, Annals of Mathematics 37, 766–781.
Baer, R.: 1943, Review of (Levi, F. 1942). Mathematical Reviews 4, 192.
Baer, R.: 1955, Review of (Clifford, A. 1954). Zentralblatt für Mathematik und ihre Grenzgebiete 56, 255.
Banaschewski, B.: 1956, ‘Totalgeordnete Moduln’, Archiv der Mathematik 7, 430–440.
Bergman, G.: 1978, ‘Conjugates and nth roots in Hahn-Laurent group rings’, Bulletin of the Malaysian Mathematical Society (2) 1, 29–41.
Bergman, G.: 1979, ‘Historical Addendum to: “Conjugates and nth Roots in Hahn-Laurent Group Rings”‘ Bulletin of the Malaysian Mathematical Society (2) 2 41–42.
Bettazzi, R.: 1890, Teoria Delle Grandezze. Pisa. Reprinted in Annali Della Università Toscane, Parte Seconda, Scienze Cosmologiche 19 (1893), 1–180.
Bindoni, A.: 1902, ‘Sui numeri infiniti ed infinitesimi attuali’, Atti della Reale Accademia dei Lincei, Classe di scienze fisiche, matematiche e naturali, Rendiconti, Roma 11, 205–209.
Birkhoff, G.: 1942, ‘Lattice Ordered Groups’, Annals of Mathematics 43, 298–331.
Birkhoff, G.: 1948, Lattice Theory (2nd Revised Edition). American Mathematical Society Colloquium Publications Volume XXV, American Mathematical Society, New York, NY.
Birkhoff, G.: 1967, Lattice Theory (3rd Edition). American Mathematical Society Colloquium Publications Volume X XV, American Mathematical Society, New York, NY.
Blumberg, H. 1920, ‘Hausdorff’s Grundzüge der Mengenlehre’, Bulletin of the American Mathematical Society 27, 116–129.
Bourbaki, N.: 1968. Elements of Mathematics, Theory of Sets. Hermann, Publishers In Arts And Sciences, Paris. Translation of Éléments De Mathématique, Théorie Des Ensembles,Hermann, Paris (1954).
Burali-Forti, C.: 1893, ‘Sulla Teoria della grandezze’, Rivista di Matematica 3, 76–101. Reprinted in Formulaire de Mathématiques, Volume 1, Bocca, Torino (1895), ed. by G. Peano.
Carnap, R. and F. Bachmann: 1936, ‘Ober Extremalaxiome’, Erkenntnis 6, 166–188.
Carnap, R. and F. Bachmann: 1981, ‘On Extremal Axioms’, History and Philosophy of Logic 2, 67–85.
English translation by H. G. Bohnert of (Carnap and Bachmann 1936 ).
Cavaillès, J.: 1938/1962, Remarques sur la formation de la théorie abstraite des ensem-bles,Paris (1938); Reprinted in Philosophie Mathématique
Hermann, Paris (1962). Clifford, A. H.: 1954, ‘Note on Hahn’s Theorem on Ordered Abelian Groups’, Proceedings of the American Mathematical Society 5, 860–863.
Clifford, A. H.: 1958, ‘Totally Ordered Commutative Semigroups’, Bulletin of the American Mathematical Society 64, 305–316.
Cohen, L. W. and C. Goffman: 1949, ‘The Topology of Ordered Abelian Groups’, Transactions of the American Mathematical Society 67, 310–319.
Cohen, L. W. and C. Goffman: 1950, ‘On Completeness in the Sense of Archimedes’, American Journal of Mathematics 72, 747–751.
Conrad, P.: 1953, ‘Embedding Theorems for Abelian Groups with Valuations’, American Journal of Mathematics 75, 1–29.
Conrad, P.: 1954, ‘On Ordered Division Rings’, Proceedings of the American Mathematical Society 5, 323–328.
Conrad, P.: 1955, ‘Extensions of Ordered Groups“ Proceedings of the American Mathematical Society 6, 516–528.
Conrad, P.: 1958, ‘A Note on Valued Linear Spaces’, Proceedings of the American Mathematical Society 9, 646–647.
Conrad, P.: 1964. Review of (Gemignani, G. 1962). Mathematical Reviews 27, 32.
Conrad, P. and J. Dauns: 1969, ‘An Embedding Theorem for Lattice-Ordered Fields’, Pacific Journal of Mathematics 30, 385–397.
Conrad, P., J. Harvey and H. Holland: 1963, ‘The Hahn Embedding Theorem for Lattice-Ordered Groups’, Transactions of the American Mathematical Society 108, 143–169.
Conway, J. H.: 1976. On Numbers and Games, Academic Press.
Dehn, M.: 1900, ‘Die Legendreschen Sätze über die Winkelsumme im Dreieck’, Mathematische Annalen 53, 404–439.
Dehn, M.: 1905, ‘Ober den inhalt sphärischer Dreiecke’, Mathematische Annalen 60, 166–174.
Dehn, M.: 1908, Review of (Hahn, H. 1907). Jahrbuch über die Fortschritte der Mathematik 38, 501.
Delon, F: 1989, ‘Model Theory of Henselian Valued Fields’, in H-D. Ebbinghaus et al. (eds.), Logic Colloquium 87, Elsevier Science Publishers B. V. ( North-Holland ), New York.
Du Bois-Reymond, P.: 1870–71, ‘Sur la grandeur relative des infinis des fonctions’, Annali di matematica pura de applicata 4, 338–353.
Du Bois-Reymond, P.: 1877, ‘Ueber die Paradoxen des Infinitärcalcüls’, Mathematische Annalen 11, 149–167.
Du Bois-Reymond, P.: 1882, Die allgemine Functionentheorie, Tübingen. Dubreil, P.: 1946/1954, Algèbre, Gauthier-Villars, Paris.
Dubreil-Jacotin, M. L., L. Lesieur and R. Croisot: 1953, Leçons Sur La Théorie Des Treillis Des Structures Algébriques Ordonnées Et Des Treillis Géométriques, Gauthier-Villars, Paris.
Ehrlich, P.: 1987, ‘The Absolute Arithmetic and Geometric Continua’, in Arthur Fine and Peter Machamer (eds.), PSA 1986, Volume 2, Philosophy of Science Association, Lansing, MI, pp. 237–246.
Ehrlich, P.: 1988, ‘An Alternative Construction of Conway’s Ordered Field No’, Algebra Universalis 25, 7–16. Errata, p. 233.
Ehrlich, P.: 1989, ‘Absolutely Saturated Models’, Fundamenta Mathematicae 133, 39–46.
Ehrlich, P.: 1992, ‘Universally Extending Arithmetic Continua’, in H. Sinaceur and J. M. Salanskis (eds.), Le Labyrinthe du Continu: Colloque du Cerisy, Springer-Verlag, France, Paris.
Ehrlich, P. (ed.): 1994a, Real Numbers, Generalizations of the Reals, and Theories of Continua, Kluwer Academic Publishers, Dordrecht, The Netherlands.
Ehrlich, P.: 1994b, ‘General Introduction’, In (Ehrlich 1994a, pp. vii-xxxii).
Ehrlich, P.: (Forthcoming), ‘From Completeness to Archimedean Completeness: An Essay in the Foundations of Geometry’, in Boston Studies in the Philosophy of Science, Edited by J. Hintikka.
Endler, O.: 1972. Valuation Theory, Springer-Verlag, New York.
Enriques, F.: 1907, ‘Prinzipien der Geometrie’, Encyklopedia der Mathematischen Wissenschaften III, 1–129.
Erdös, P.: 1956, ‘On the Structure of Ordered Real Vector Spaces’, Publicationes Mathematicae Debrecen 4, 334–343.
Erdös, P., L. Gillman and M. Henriksen: 1955, ‘An Isomorphism Theorem for Real-Closed Fields’, 61, 542–554.
Ersov, J. L.: 1965a, ‘On Elementary Theories of Local Fields’, Algebra i Logika Seminar 4, 5–30.
Ersov, J. L.: 1965b, ‘On the Elementary Theory of Maximal Normed Fields’, Soveit Mathematics Doklady 6, 1390–1393.
Esterle, J.: 1977, ‘Solution d’un Problème d’Erdös, Gillman et Henriksen et Application a l’étude des Homimorphismes de c(K)’, Acta Mathematica Academiae Scientiarum Hungaricae 30, 113–127.
Finsler, P.: 1926, ‘Über die Grundlegung der Mengenlehre’, Mathematische Zeitschrift 25, 683–713.
Finsler, P.: 1927/1928, ‘Erwiderung auf die vorstehende Note des Herrn R. Baer’, Mathematische Zeitschrift 27, 540–542.
Fisher, G. 1981, ‘The Infinite and Infinitesimal Quantities of du Bois-Reymond and their Reception’, Archive for History of Exact Sciences 24, 101–164.
Fisher, G.: 1994, ‘Veronese’s Non-Archimedean Linear Continuum’, in ( Ehrlich, P. 1994a ).
Fleischer, I.: 1981, ‘The Hahn Embedding Theorem: Analysis, Refinements, Proof’, in Algebra Carbondale 1980: Lie Algebras, Group Theory, and Partially Ordered Algebraic Structures; Lecture Notes in Mathematics #848, ed. by R. K. Amayo, Springer-Verlag, Berlin.
Fraenkel, A. A.: 1928, Einleitung In Die Mengenlehre,Verlag Von Julius Springer, Berlin.
Fraenkel, A. A.: 1976, Abstract Set Theory (Fourth Revised Edition), North-Holland Publishing Company, Amsterdam.
Fuchs, L.: 1963. Partially Ordered Algebraic Systems,Pergamon Press.
Fuchs, L. and L. Salce: 1985, Modules Over Valuation Domains, Marcel Dekker, Inc., New York.
Gabovich, E.: 1976, ‘Fully Ordered Semigroups and Their Applications’, Russian Mathematical Surveys 31, 147–216.
Gemignani, G.: 1962, ‘Digression Sui Campi Ordinati’, Annali Della Scuola Normale Superiore Di Pisa, Scienze Fisiche E Matematiche 16 (Series 3), 143–157.
Gillman, L. and M. Jerison: 1960, Rings of Continuous Functions, D. Van Nostrand Company, Inc., Princeton, New Jersey.
Gleyzal, A.: 1937, ‘Transfinite Real Numbers’, Proceedings of the National Academy of Sciences 23, 581–587.
Goffman, C.: 1974, ‘Completeness of the Real Numbers’, Mathematics Magazine (January-February), 1–8.
Grassmann, H.: 1862, Die Ausdehnungslehre, Leipzig. Reprinted as (Grassmann 1894). For Mark Kormes’ English translation of the relevant material, see A Source Book in Mathematics, edited by D. E. Smith, Dover Publications, New York, 1959, pp. 684–685.
Grassmann, H.: 1894, Gesammealte Mathematische und physike Werke, Volume I, Part II, Leipzig.
Gravett, K.: 1955, ‘Valued Linear Spaces’, Quarterly Journal of Mathematics, Oxford 6, 309–315.
Gravett, K.: 1956, ‘Ordered Abelian Groups’, Quarterly Journal of Mathematics, Oxford 7, 57–63.
Guillaume, M.: 1978, ‘Sur L’Histoirie Des Modeles Non-Standards Et Celle De L’Analyse Non-Standard’, Cashiers Fundamenta Scientiae (Seminaire Sur Les Fondements Des Sciences, Universite Louis Pasteur, Strasbourg) No. 85.
Hahn, H.: 1907, ‘Ober die nichtarchimedischen Grössensysteme’, Sitzungsberichte Kaiserlichen der Akademie Wissenschaften, Wien, Mathematisch–Naturwissenschaftliche Klasse 116 (Abteilung IIa), 601–655.
Hahn, H.: 1980, Empiricism, Logic, and Mathematics: Philosophical Papers. Edited by Brian McGinness with an introduction by Karl Menger, D. Reidel Publishing Company, Dordrecht, Holland.
Hamel, G.: 1905, ‘Eine Basis aller Zahlen und die unstetigen Lösungen der Funcktionalgleichung: f(x + y) = f(x) + f(y)’, Mathematische Annalen 60, 459–462. Hankel, H.: 1867. Theorie der Komplexen Zahlsysteme, Leipzig.
Hausdorff, F.: 1906, ‘Untersuchungen über Ordungtypen’, Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig, Matematisch–Physische Klasse 58, 106–169.
Hausdorff, F.: 1907, ‘Untersuchungen über Ordungtypen’, Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig, Matematisch–Physische Klasse 59, 84–159.
Hausdorff, F.: 1908, ‘Grundzüge einer Theorie der geordneten Mengen’, Mathematische Annalen 65, 435–505.
Hausdorff, F.: 1914, Grundzüge der Mengenlehre,Leipzig.
Hausner, M. and J. G. Wendel: 1952, ‘Ordered Vector Spaces’, Proceedings of the American Mathematical Society 3, 977–982.
Hjelmslev, J.: 1907, ‘Neue Begründung der ebenen Geometrie’, Mathematische Annalen 64, 449–474.
Hessenberg, G.: 1905a, ‘Begründung der elliptischen Geometrie’, Mathematische Annalen 61, 173–184.
Hessenberg, G.: 1905b, ‘Neue Begründung der Sphärik’, Sitzungsberichte der Berliner mathematischen Gesellschaft 4, 69–77.
Hessenberg, G.: 1905c, ‘Beweis des Desarguesschen Satzes aus dem Pascalschen’, Mathematische Annalen 61, 161–172.
Hessenberg, G.: 1930, Grundlagen der Geometrie,Berlin.
Higman, G.: 1952, ‘Ordering by Divisibility in Abstract Algebras’, Proceedings of the London Mathematical Society 2, 326–336.
Hilbert, D.: 1899, Grundlagen der Geometrie, Festschrift zur Feier der Enthüllung des Gauss-Weber Denkals in Göttingen, Teubner, Leipzig.
Hilbert, D.: 1900a, ‘über den Zahlbegriff’, Jahresbericht der Deutschen Mathematiker — Vereinigung 8, 180–184. Reprinted in (Hilbert 1909, pp. 256–262; Hilbert 1930, pp. 241–246 ).
Hilbert, D.: 1900b, Les principes Fondamentaux de la géometrie, Gautier-Villars, Paris. French translation by L. Laugel of an expanded version of (Hilbert 1899). Reprinted in Annales Scientifiques de L’école Normal Supérieure 17 (1900), 103–209.
Hilbert, D.: 1900c, ‘Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Kongress zu Paris. 1900’, Nachrichten, Akademie der Wissenschaften, Göttingen, 253–297. Reprinted in (Hilbert 1909, pp. 263–279 );
Hilbert 1930, pp. 247–261). English Translation in Bulletin of the American Mathematical Society 8 (1902), 437–479.
Reprinted in Mathematical Developments Arising From Hilbert’s Problems, Proceedings of Symposia in Pure Mathematics Volume XXVII, Part I, American Mathematical Society, Providence, RI, 1976.
Hilbert, D.: 1902. Foundations of Geometry, English Translation by E. Townsend of an expanded version of (Hilbert 1899 ).
Hilbert, D.: 1903a, Second Edition of (Hilbert 1899 ).
Hilbert, D.: 1903b, ‘Über den Satz von der Gleichheit der Basiswinkel im gleichschenkligen Dreiech’, Proceedings of the London Mathematical Society 35, 50–68.
Reprinted with revisions in (Hilbert 1903a, pp. 88–107; 1930, pp. 133–158). For an English translation of the revised text, see (Hilbert 1971, pp. 113–132).
Hilbert, D.: 1903c, ‘Neue Begründung der Bolyai-Lobatschefskyschen Geometrie’, Mathematische Annalen 57, 137–150.
Reprinted as an appendix in (Hilbert 1903a ). For an English translation, see (Hilbert 1971, pp. 133–149 ).
Hilbert, D.: 1904, ‘Über die Grundlagen der Logik und Arithmetik’, in Verhandlungen des Dritten Internationalen Mathematiker — Kongresses in Heidelberg vom 8. bis 13. August 1904, Teubner, Leipzig, 1905.
For an English translation of (Hilbert 1904), see From Frege To Gödel: A Sourcebook in Mathematical Logic, 1879–1931 (Second Printing), edited by J. van Heijenoort, Harvard University Press, Cambridge, Massachusetts, 1971, pp. 129–138.
Hilbert D.: 1909, Third Edition of (Hilbert 1899).
Hilbert D.: 1930, Seventh Edition of (Hilbert 1899).
Hilbert, D.: 1971, Foundations of Geometry, Open Court, LaSalle, Illinois. English translation by Leo Unger of Paul Bernays’ Revised and Enlarged Tenth Edition of (Hilbert 1899 ).
Hölder, 0.: 1901, ‘Der Quantität und die Lehre vom Mass’, Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig, Matematisch — Physische Classe 53, 1–64.
Holland, C.: 1963, ‘Extensions of Ordered Groups and Sequence Completions’, Transactions of the American Mathematical Society 107, 71–82.
Huntington, E. V.: 1902, ‘A Complete Set of Postulates for the Theory of Absolute Continuous Magnitude’, Transactions of the American Mathematical Society 3, 264–279.
Hüper, H.: 1977, ‘Ober ordnungsverträglich bewertete, angeordnete Körper’, Dissertation München. (A detailed account with proofs of Hüper’s results is contained in PriessCrampe 1983, Chapter III, Section 3.)
Iwasawa, K.: 1948, ‘On Linearly Ordered Groups’, Journal of the Mathematical Society of Japan 1, 1–9.
Kaplansky, I.: 1942, ‘Maximal Fields with Valuations’, Duke Mathematical Journal 9, 303–321.
Kaplansky, I.: 1949. Review of (Mal’cev 1948), Mathematical Reviews 10, 8.
Klein, F.: 1908/1939, Elementary Mathematics From An Advanced Standpoint, Arithmetic
Algebra, Analysis, Translated from the German by E. R. Hedrick and C. A. Noble, Dover Publications, New York.
Kochen, S.: 1975, ‘The Model Theory Of Local Fields’, in G. H. Müller, A. Oberschelp and K. Potthoff (eds.), Logic Conference, Kiel 1974. Lecture Notes in Mathematics #499, Springer-Verlag, Berlin.
Kokorin, A. I. and V. M. Kopytov: 1974, Fully Ordered Groups, John Wiley & Sons, New York-Toronto.
Krull, W.: 1932, ‘Allegemeine Bewertungstheorie’, Journal fiir die Reine and Angewandt Mathematik 167, 160–196.
Krull, W.: 1955. Review of (Gravett, K. 1955), Zentralblatt für Mathematik and ihre Grenzgebiete 67, 266.
Krull, W.: 1956. Review of (Gravett, K. 1956), Zentralblatt für Mathematik and ihre Grenzgebiete 74, 21.
Krull, W.: 1957–71, ‘Review of (Ribenboim, P. 1958)’, Zentralblatt für Mathematik and ihre Grenzgebiete 95, 259.
Krull, W.: 1970, ‘Felix Hausdorff 1868–1942’, in Bonner Gelehrte. Beiträge zur Geschichte der Wissenschaften in Bonn, Volume 9, pp. 54–69.
Lam, T. Y.: 1980, ‘The Theory of Ordered Fields’, in B. McDonald (ed.), Ring Theory and Algebra III: Proceedings of the Third Oklahoma Conference, Marcel Dekker, Inc., New York.
Lam, T. Y.: 1983, Orderings, Valuations and Quadratic Forms. Regional Conference
Series in Mathematics #52. American Mathematical Society, Providence, R. I.
Lam, T. Y.: 1991, A First Course in Noncommutative Rings,Springer-Verlag, New York.
Lang, S.: 1953, ‘The Theory of Real Places’, Annals of Mathematics 57, 387–391.
Laugwitz, D.: 1975, ‘Tullio Levi-Civita’s Work on Non-Archimedean Structures (With an Appendix: Properties of Levi-Civita Fields)’, in Tullio Levi-Civita Convegno Internazionale Celebrativo Del Centenario Della Nascita, Accademia Nazionale Dei Lincei, Rome, Atti Dei Convegni Lincei 8, 297–312.
Laugwitz, D.: 1986. Zahlen and Kontinuum, Eine Einführung in die Infinitesimalmathematik, Bibliographisches Institut, Mannheim.
Levi, F. W.: 1942, ‘Ordered Groups’, Proceedings of the Indian Academy of Sciences (Section A) 16, 256–263.
Levi-Civita, T.: 1893, ‘Sugli infiniti ed infinitesimi attuali quali elementi analitici’, Atti del Reale Instituto Veneto di Scienze Lettre ed Arti, Venezia (7) 4 (1892–93), 1765–1815. Reprinted in (Levi-Civita 1954, pp. 1–39 ).
Levi-Civita, T.: 1898, ‘Sui Numeri Transfiniti’, Atti della Reale Accademia dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti, Roma Serie Va, 7, 91–96, 113–121. Reprinted in (Levi-Civita 1954, pp. 315–329 ).
Levi-Civita, T.: 1954. Tullio Levi-Civita, Opere Matematiche, Memorie e Note, Volume primo 1893–1900, edited by Nicola Zanichelli, Bologna.
Loonstra, F.: 1950, ‘The Classes of Ordered Groups’, in Proceedings of the International Congress of Mathematicians, Volume I, Cambridge, pp. 312–313.
Mac Lane, S.: 1939, ‘The Universality of Formal Power Series Fields’, Bulletin of the American Mathematical Society 45, 888–890.
Mal’cev, A. L.: 1948, ‘On Embedding of Group Algebras in a Division Algebra’, (Russian) Doklady Akademii Nauk SSSR (N.S.) 60, 1499–1501.
Mal’cev, A. L.: 1949, ‘On Ordered Groups’, (Russian) Izvestiia Akademii Nauk. SSSR (Seriia Matematicheskaia) 13, 473–482.
Mayrhofer, K.: 1934, K.: 1934, ‘Hans Hahn’, Monatschefte fir Mathematik und Physik 41, 22 1238.
Moore, G.: 1982, Zermelo’s Axiom of Choice: Its Origins Development and Influence, Springer-Verlag, New York.
Mourgues, M. H. and J. P. Ressayre: 1992, ‘Tout corps réel clos possède une partie entière’, Compte Rendus de l’Académie des Sciences, Paris, (Série 1, Mathématique) 314, 813–816.
Mourgues, M. H. and J. P. Ressayre: 1993, ‘Every Real Closed Field Has An Integer Part’, The Journal of Symbolic Logic 58, 641–647. (This is a revised English translation of (Mourgues and Ressayre 1992)).
Mura, R. B. and A. Rhemtulla: 1977, Orderable Groups, Marcel Dekker, Inc., New York. Neumann, B.: 1949, ‘On Ordered Division Rings’, Transactions of the American Mathematical Society 66, 202–252.
Neumann, H.: 1954, Review of (Conrad, P. 1953). Zentralblatt für Mathematik und ihre Grenzgebiete 50 23.
Ostrowski, A.: 1935, ‘Untersuchungen zur arithmetischen Theorie der Körper’
Mathematische Zeitzchrift 39 269–404. Reprinted in (Ostrowski 1983, pp. 336–485). Ostrowski, A.: 1983, Alexander Ostrowski: Collected Mathematical Papers, Volume 2,Birkhäuser Verlag, Basel.
Pickert, G.: 1953, Review of (Hausner, M. and Wendel, J. 1952), Zentralblatt für Mathematik und ihre Grenzgebiete 48, 87.
Poincaré, H.: 1905, ‘Les Mathématiques et la logique’, Revue de métaphysique et de morale 13, 815–835.
Prestel, A.: 1984, Lectures on Formally Real Fields, Lecture Notes in Mathematics #1093, Springer-Verlag, Berlin.
Priess-Crampe, S.: 1973, ‘Zum Hahnschen Einbettungssatz für Angeordnete Körper’, Archiv der Mathematik 24, 607–614.
Priess-Crampe, S.: 1983, Angeordnete Strukturen, Gruppen, Körper, projektive Ebenen, Springer-Verlag, Berlin.
Priess-Crampe, S. and R. von Chossy: 1975, ‘Ordungsverträgliche Bewertungen eines angeordneten Körpers’, Archiv der Mathematik 26, 373–387.
Rayner, F. J.: 1976, ‘Ordered Fields’, in Seminaire de Théorie des Nombres 1975–76. (Univ. Bordeaux I. Talence). Exp. No. 1, pp. 1–8. Lab. Théorie des Nombres, Centre Nat. Recherche Sci., Talence.
Rédei, L.: 1954/1967, Algebra, Volume 1, Pergamon Press, Oxford.
Redfield, R. H.: 1986, ‘Embeddings Into Power Series Rings’, Manuscripta Mathematica 56, 247–268.
Redfield, R. H.: 1989a, ‘Banaschewski Functions and Ring-Embeddings’, in J. Martinez (ed.), Ordered Algebraic Structures, Kluwer Academic Publishers, The Netherlands.
Redfield, R. H.: 1989b, ‘Non-Embeddable O-Rings’, Communications in Algebra 17, 59–71.
Ribenboim, P.: 1958, ‘Sur les groupes totalement ordonnés et l’arithmétique des anneaux de valuation’, Summa Brasilliensis Mathematicae 4, 1–64.
Ribenboim, P.: 1964. Théorie des Groupes Ordonnés, Universidad Nacional Del Sur, Bahia Blanca.
Robinson, A.: 1961, ‘Non-standard Analysis’, Proceedings of the Royal Academy of Sciences, Amsterdam (Series A) 64, 432–440. Reprinted in (Robinson 1979, pp. 3–11 ).
Robinson, A.: 1966, Non-standard Analysis, North-Holland Publishing Company, Amsterdam.
Robinson, A.: 1979, Selected Papers of Abraham Robinson, Volume 2: Nonstandard Analysis and Philosophy, edited with introductions by W. A. J. Luxemburg and S. Körner, Yale University Press, New Haven.
Satyanarayana, M.: 1979, Positively Ordered Semigroups, Marcel Dekker, Inc., New York. Schilling, O. F. G.: 1937, ‘Arithmetic in Fields of Formal Power Series in Several Variables’, Annals of Mathematics 38, 551–576.
Schilling, O. F. G.: 1945, ‘Noncommutative Valuations’, Bulletin of the American Mathematical Society 51, 297–304.
Schilling, O. F. G.: 1950, The Theory of Valuations, Mathematical Surveys I V, American Mathematical Society, New York.
Schmieden, C. and D. Laugwitz: 1958, ‘Eine Erweiterung der Infinitesimalrechnung’, Mathematische Zeitschrift 69, 1–39.
Schoenflies, A.: 1906, ‘über die Möglichkeit einer projektiven Geometrie bei trans-finiter (nicht archimedischer) Massbestimmung’, Jahresbericht der Deutschen Mathematiker-Vereinigung 15, 26–47.
Schoenflies, A.: 1908, ‘Die Entwickelung Der Lehre Von Den Punktmannigfaltigkeiten, Zweiter Teil’, Jahresbericht der Deutschen Mathematiker-Vereinigung, Ergänzungsband 2, 1–331. Reprinted as a separate volume by Druck Und Verlag Von B. G. Teubner, Leipzig.
Schur, F.: 1899, ‘Ueber den Fundamentalsatz der projectiven Geometrie’, Mathematische Annalen 51, 401–409.
Schur, F.: 1902, ‘Ueber die Grundlagen der Geometrie’, Mathematische Annalen 55, 401–409.
Schur, F.: 1903, ‘Zur Proportionslehre’, Mathematische Annalen 57, 205–208.
Schur, F.: 1904, ‘Zur Bolyai-Lobatschefskijschen Geometrie’, Mathematische Annalen 59, 314–320.
Schur, F.: 1909, Grundlagen der Geometrie, Druck und Verlag Von B. G. Teubner, Leipzig und Berlin.
Schwartz, N.: 1978, ‘rlpStrukturen’, Mathematische Zeitschrift 158, 147–155.
Sinaceur, H.: 1989, ‘Une origin du concept d’analyse non-standard’, in H. Barreau and
J. Harthong (eds.), La Mathèmatique Non Standard, Historie-Philosophie Dossier
Scientifique,Éditions Du Centre National De La Recherche Scientifique, Paris. Sinaceur, H.: 1991, Corps et Modèles: Essai sur l’historie de l’algèbre réelle,Librairie Philosophique J. Vrin, Paris.
Steinitz, E.: 1910, ‘Algebraische Theorie der Körper’, Journal für die Reine und Angewandt Mathematik 137, 167–309.
Stolz, 0.: 1881, ‘B. Bolzano’s Bedeutung in der Geschichte der Infinitesimalrechnung’, Mathematische Annalen 18, 255–279.
Stolz, 0.: 1882, ‘Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes’
Berichte des Naturwissenschaftlich-Medizinischen, Vereines in Innsbruck 12 74–89. Stolz, 0.: 1883, ‘Zur Gometrie der Alten, insbesondere über ein Axiom des Archimedes’
Mathematische Annalen 22 504–519. (This is a revised version of (Stolz 1882)). Stolz, 0.: 1884, ‘Die unendlich kleinen Grössen’, Berichte des Naturwissenschaftlich-
Medizinischen, Vereines in Innsbruck 14 21–43.
Stolz, 0.: 1885. Vorlesungen über Allgemeine Arithmetik; Erster Theil: Allgemeines und Arithhmetik der Reelen Zahlen, Teubner, Leipzig.
Stolz, 0.: 1886, Vorlesungen über Allgemeine Arithmetik; Zweiter Theil: Arithhmetik der Complexen Zahlen, Teubner, Leipzig.
Stolz, 0.: 1891, ‘Veber das Axiom des Archimedes’, Mathematische Annalen 39, 107112.
Stolz, O. and J. A. Gmeiner: 1902, Theoretische Arithmetik, Teubner, Leipzig.
Stroyan, K. and W. Luxemburg: 1976, Introduction To The Theory Of Infnitesimals, Academic Press, New York.
Temple, G.: 1981, 100 Years of Mathematics, Springer-Verlag, New York.
Thomae, J.: 1870, Abriss einer Theorie der complexen Functionen und der Thetafunctionen einer Veränderlichen, Nebert, Halle.
Thomae, J.: 1880, Elementare Theorie der analytischen Functionen einer complexen Veränderlichen, Nebert, Halle.
Trias I Pairó, J.: 1984, ‘Sistemes Algèbrics Ordenats: Aproximació Històrica’, Butlleti’ de la Societat Catalana de Ciencies Fisiyues, Quimiques i Matematiques. Barcelona 2, 39–57.
Vahlen, K. Th.: 1905, Abstrackte Geometrie, B. G. Teubner, Leipzig.
Vahlen, K. Th.: 1907, Über nicht-archimedische Algebra’, Jahresbericht der Deutschen Mathematiker-Vereinigung 16, 409–421.
van der Waerden, B. L.: 1930, Moderne Algebra I, Julius Springer, Berlin.
Veronese, G.: 1889, ‘Il continuo rettilineo e l’assioma V d’Archimede’, Atti Della R. Accademia Dei Lincei, Memorie (Della Classe Di Scienze Fisiche, Matematiche E Naturali) Roma 6, 603–624.
Veronese, G.: 1891, Fondamenti di Geometria,Padova.
Veronese, G.: 1894, Grundzüge der Geometrie,Leipzig.
Veronese, G.: 1909/1994, ‘On Non-Archimedean Geometry’, in (Ehrlich, P. 1994), pp. 169–187. (This is a translation by M. Marion with editorial notes by Philip Ehrlich of ‘La geometria non-Archimedea’, Atti del IV Congresso Internazionale dei
Matematici (Roma 6–11 Aprile 1908) Vol. I, Rome, 1909, pp. 197–208.) Vivanti, G.: 1891, ‘Sull’infinitesimo attuale’, Rivista di Matematica 1, 135–153.
Wang, H.: 1987, Reflections on Kurt Glide!, A Bradford Book, The MIT Press, Cambridge Massachusetts.
Warner, S.: 1990, Modern Algebra, Dover Publications, New York. Reprinted with corrections: Modern Algebra, Prentice-Hall, Inc., Englewoods
New Jersey, 1965. Weispfenning, V.: 1971, ‘On the Elementary Theory of Hensel Fields’, Doctoral Dissertation, Heidelberg.
Weispfenning, V.: 1976, ‘On the Elementary Theory of Hensel Fields’, Annals of Mathematical Logic 10, 59–93.
Weispfenning, V.: 1984, ‘Quantifier Elimination and Decision Procedures for Valued Fields’, in G. H. Müller and M. M. Richter (eds.), Models and Sets. Lecture Notes in Mathematics #1103, Springer-Verlag, Berlin.
Wolfenstein, S.: 1966, ‘Sur les groupes réticulés archimédiennement complets’, Compte Rendus Hebdomadaires des Séances de l’Académie des Sciences, Paris, ( Série A, Mathématiques ) 262, 813–816.
Zelinsky, D.: 1948, ‘Nonassociative Valuations’, Bulletin of the American Mathematical Society 54, 175–183.
Zermelo, E.: 1904, ‘Beweis, dass jede Menge wohlgeordnet werden kann’, Mathematische Annalen 59, 514–516. For an English translation of (Zermelo 1904), see From Frege To Gödel: A Sourcebook in Mathematical Logic, 1879–1931 (Second Printing), edited by J. van Heijenoort, Harvard University Press, Cambridge, Massachusetts, 1971, pp. 183–198.
Ziegler, M.: 1972, Die elementare Theorie der Hensel Körper, Doctoral Dissertation, Köln.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Ehrlich, P. (1995). Hahn’s Über die Nichtarchimedischen Grössensysteme and the Development of the Modern Theory of Magnitudes and Numbers to Measure Them. In: Hintikka, J. (eds) From Dedekind to Gödel. Synthese Library, vol 251. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8478-4_8
Download citation
DOI: https://doi.org/10.1007/978-94-015-8478-4_8
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4554-6
Online ISBN: 978-94-015-8478-4
eBook Packages: Springer Book Archive