Hahn’s Über die Nichtarchimedischen Grössensysteme and the Development of the Modern Theory of Magnitudes and Numbers to Measure Them

  • Philip Ehrlich
Part of the Synthese Library book series (SYLI, volume 251)


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).


Abelian Group American Mathematical Society Modern Theory Isomorphic Copy Embed Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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.CrossRefGoogle Scholar
  2. Ailing, N.: 1962a, ‘On Exponentially Closed Fields’, Proceedings of the American Mathematical Society 13, 706–711.CrossRefGoogle Scholar
  3. 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.Google Scholar
  4. Ailing, N.: 1976, ‘Residue Class Fields of Rings of Continuous Functions’, In Symposia Mathematica, Volume XVII, Academic Press, London, pp. 55–67.Google Scholar
  5. Ailing, N.: 1985, ‘Conway’s Field of Surreal Numbers’, Transactions of the American Mathematical Society 287, 365–386.Google Scholar
  6. Ailing, N.: 1987, Foundations of Analysis over Surreal Number Fields, North-Holland, Amsterdam.Google Scholar
  7. Artin, E.: 1957, Geometric Algebra, Interscience Publishers Inc., New York.Google Scholar
  8. Artin, E.: 1965, The Collected Papers. Edited by S. Lang and J. Tate, Addison, Wesley Publishing Co., Reed.Google Scholar
  9. 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 ).Google Scholar
  10. 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 ).Google Scholar
  11. Ax, J. and S. Kochen: 1965a, ‘Diophantine Problems Over Local Fields: I.’, American Journal of Mathematics 87, 605–630.Google Scholar
  12. Ax, J. and S. Kochen: 1965b, ‘Diophantine Problems Over Local Fields: II.’, American Journal of Mathematics 87, 631–648.CrossRefGoogle Scholar
  13. Ax, J. and S. Kochen: 1966, ‘Diophantine Problems Over Local Fields: III.’, Annals Of Mathematics (Series 2) 83, 437–456.CrossRefGoogle Scholar
  14. Baer, R.: 1927, ‘über nicht-archimedisch geordnete Körper’, Sitzungsberichte der Heidelberger Akademie der Wissenschaften 8, 3–13.Google Scholar
  15. Baer, R.: 1927/1928a, ‘über ein Vollständigkeitsaxiom in der Mengenlehre’, Mathematische Zeitschrift 27, 536–539.Google Scholar
  16. Baer, R.: 1927/1928b, ‘Bemerkungen zu der Erwiderung von Herrn P. Finsler’, Mathematische Zeitschrift 27, 543.Google Scholar
  17. Baer, R.: 1936, ‘The Subgroup of the Elements of Finite Order of an Abelian Group’, Annals of Mathematics 37, 766–781.CrossRefGoogle Scholar
  18. Baer, R.: 1943, Review of (Levi, F. 1942). Mathematical Reviews 4, 192.Google Scholar
  19. Baer, R.: 1955, Review of (Clifford, A. 1954). Zentralblatt für Mathematik und ihre Grenzgebiete 56, 255.Google Scholar
  20. Banaschewski, B.: 1956, ‘Totalgeordnete Moduln’, Archiv der Mathematik 7, 430–440.CrossRefGoogle Scholar
  21. Bergman, G.: 1978, ‘Conjugates and nth roots in Hahn-Laurent group rings’, Bulletin of the Malaysian Mathematical Society (2) 1, 29–41.Google Scholar
  22. 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.Google Scholar
  23. Bettazzi, R.: 1890, Teoria Delle Grandezze. Pisa. Reprinted in Annali Della Università Toscane, Parte Seconda, Scienze Cosmologiche 19 (1893), 1–180.Google Scholar
  24. 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.Google Scholar
  25. Birkhoff, G.: 1942, ‘Lattice Ordered Groups’, Annals of Mathematics 43, 298–331.CrossRefGoogle Scholar
  26. Birkhoff, G.: 1948, Lattice Theory (2nd Revised Edition). American Mathematical Society Colloquium Publications Volume XXV, American Mathematical Society, New York, NY.Google Scholar
  27. Birkhoff, G.: 1967, Lattice Theory (3rd Edition). American Mathematical Society Colloquium Publications Volume X XV, American Mathematical Society, New York, NY.Google Scholar
  28. Blumberg, H. 1920, ‘Hausdorff’s Grundzüge der Mengenlehre’, Bulletin of the American Mathematical Society 27, 116–129.CrossRefGoogle Scholar
  29. 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).Google Scholar
  30. 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.Google Scholar
  31. Carnap, R. and F. Bachmann: 1936, ‘Ober Extremalaxiome’, Erkenntnis 6, 166–188.CrossRefGoogle Scholar
  32. Carnap, R. and F. Bachmann: 1981, ‘On Extremal Axioms’, History and Philosophy of Logic 2, 67–85.CrossRefGoogle Scholar
  33. English translation by H. G. Bohnert of (Carnap and Bachmann 1936 ).Google Scholar
  34. Cavaillès, J.: 1938/1962, Remarques sur la formation de la théorie abstraite des ensem-bles,Paris (1938); Reprinted in Philosophie Mathématique Google Scholar
  35. 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.Google Scholar
  36. Clifford, A. H.: 1958, ‘Totally Ordered Commutative Semigroups’, Bulletin of the American Mathematical Society 64, 305–316.CrossRefGoogle Scholar
  37. Cohen, L. W. and C. Goffman: 1949, ‘The Topology of Ordered Abelian Groups’, Transactions of the American Mathematical Society 67, 310–319.CrossRefGoogle Scholar
  38. Cohen, L. W. and C. Goffman: 1950, ‘On Completeness in the Sense of Archimedes’, American Journal of Mathematics 72, 747–751.CrossRefGoogle Scholar
  39. Conrad, P.: 1953, ‘Embedding Theorems for Abelian Groups with Valuations’, American Journal of Mathematics 75, 1–29.CrossRefGoogle Scholar
  40. Conrad, P.: 1954, ‘On Ordered Division Rings’, Proceedings of the American Mathematical Society 5, 323–328.CrossRefGoogle Scholar
  41. Conrad, P.: 1955, ‘Extensions of Ordered Groups“ Proceedings of the American Mathematical Society 6, 516–528.CrossRefGoogle Scholar
  42. Conrad, P.: 1958, ‘A Note on Valued Linear Spaces’, Proceedings of the American Mathematical Society 9, 646–647.CrossRefGoogle Scholar
  43. Conrad, P.: 1964. Review of (Gemignani, G. 1962). Mathematical Reviews 27, 32.Google Scholar
  44. Conrad, P. and J. Dauns: 1969, ‘An Embedding Theorem for Lattice-Ordered Fields’, Pacific Journal of Mathematics 30, 385–397.Google Scholar
  45. 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.CrossRefGoogle Scholar
  46. Conway, J. H.: 1976. On Numbers and Games, Academic Press.Google Scholar
  47. Dehn, M.: 1900, ‘Die Legendreschen Sätze über die Winkelsumme im Dreieck’, Mathematische Annalen 53, 404–439.CrossRefGoogle Scholar
  48. Dehn, M.: 1905, ‘Ober den inhalt sphärischer Dreiecke’, Mathematische Annalen 60, 166–174.CrossRefGoogle Scholar
  49. Dehn, M.: 1908, Review of (Hahn, H. 1907). Jahrbuch über die Fortschritte der Mathematik 38, 501.Google Scholar
  50. 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.Google Scholar
  51. Du Bois-Reymond, P.: 1870–71, ‘Sur la grandeur relative des infinis des fonctions’, Annali di matematica pura de applicata 4, 338–353.Google Scholar
  52. Du Bois-Reymond, P.: 1877, ‘Ueber die Paradoxen des Infinitärcalcüls’, Mathematische Annalen 11, 149–167.CrossRefGoogle Scholar
  53. Du Bois-Reymond, P.: 1882, Die allgemine Functionentheorie, Tübingen. Dubreil, P.: 1946/1954, Algèbre, Gauthier-Villars, Paris.Google Scholar
  54. 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.Google Scholar
  55. 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.Google Scholar
  56. Ehrlich, P.: 1988, ‘An Alternative Construction of Conway’s Ordered Field No’, Algebra Universalis 25, 7–16. Errata, p. 233.Google Scholar
  57. Ehrlich, P.: 1989, ‘Absolutely Saturated Models’, Fundamenta Mathematicae 133, 39–46.Google Scholar
  58. 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.Google Scholar
  59. Ehrlich, P. (ed.): 1994a, Real Numbers, Generalizations of the Reals, and Theories of Continua, Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar
  60. Ehrlich, P.: 1994b, ‘General Introduction’, In (Ehrlich 1994a, pp. vii-xxxii).Google Scholar
  61. 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.Google Scholar
  62. Endler, O.: 1972. Valuation Theory, Springer-Verlag, New York.CrossRefGoogle Scholar
  63. Enriques, F.: 1907, ‘Prinzipien der Geometrie’, Encyklopedia der Mathematischen Wissenschaften III, 1–129.Google Scholar
  64. Erdös, P.: 1956, ‘On the Structure of Ordered Real Vector Spaces’, Publicationes Mathematicae Debrecen 4, 334–343.Google Scholar
  65. Erdös, P., L. Gillman and M. Henriksen: 1955, ‘An Isomorphism Theorem for Real-Closed Fields’, 61, 542–554.Google Scholar
  66. Ersov, J. L.: 1965a, ‘On Elementary Theories of Local Fields’, Algebra i Logika Seminar 4, 5–30.Google Scholar
  67. Ersov, J. L.: 1965b, ‘On the Elementary Theory of Maximal Normed Fields’, Soveit Mathematics Doklady 6, 1390–1393.Google Scholar
  68. 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.Google Scholar
  69. Finsler, P.: 1926, ‘Über die Grundlegung der Mengenlehre’, Mathematische Zeitschrift 25, 683–713.CrossRefGoogle Scholar
  70. Finsler, P.: 1927/1928, ‘Erwiderung auf die vorstehende Note des Herrn R. Baer’, Mathematische Zeitschrift 27, 540–542.Google Scholar
  71. Fisher, G. 1981, ‘The Infinite and Infinitesimal Quantities of du Bois-Reymond and their Reception’, Archive for History of Exact Sciences 24, 101–164.CrossRefGoogle Scholar
  72. Fisher, G.: 1994, ‘Veronese’s Non-Archimedean Linear Continuum’, in ( Ehrlich, P. 1994a ).Google Scholar
  73. 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.Google Scholar
  74. Fraenkel, A. A.: 1928, Einleitung In Die Mengenlehre,Verlag Von Julius Springer, Berlin.Google Scholar
  75. Fraenkel, A. A.: 1976, Abstract Set Theory (Fourth Revised Edition), North-Holland Publishing Company, Amsterdam.Google Scholar
  76. Fuchs, L.: 1963. Partially Ordered Algebraic Systems,Pergamon Press.Google Scholar
  77. Fuchs, L. and L. Salce: 1985, Modules Over Valuation Domains, Marcel Dekker, Inc., New York.Google Scholar
  78. Gabovich, E.: 1976, ‘Fully Ordered Semigroups and Their Applications’, Russian Mathematical Surveys 31, 147–216.CrossRefGoogle Scholar
  79. Gemignani, G.: 1962, ‘Digression Sui Campi Ordinati’, Annali Della Scuola Normale Superiore Di Pisa, Scienze Fisiche E Matematiche 16 (Series 3), 143–157.Google Scholar
  80. Gillman, L. and M. Jerison: 1960, Rings of Continuous Functions, D. Van Nostrand Company, Inc., Princeton, New Jersey.Google Scholar
  81. Gleyzal, A.: 1937, ‘Transfinite Real Numbers’, Proceedings of the National Academy of Sciences 23, 581–587.CrossRefGoogle Scholar
  82. Goffman, C.: 1974, ‘Completeness of the Real Numbers’, Mathematics Magazine (January-February), 1–8.Google Scholar
  83. 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.Google Scholar
  84. Grassmann, H.: 1894, Gesammealte Mathematische und physike Werke, Volume I, Part II, Leipzig.Google Scholar
  85. Gravett, K.: 1955, ‘Valued Linear Spaces’, Quarterly Journal of Mathematics, Oxford 6, 309–315.CrossRefGoogle Scholar
  86. Gravett, K.: 1956, ‘Ordered Abelian Groups’, Quarterly Journal of Mathematics, Oxford 7, 57–63.CrossRefGoogle Scholar
  87. 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.Google Scholar
  88. Hahn, H.: 1907, ‘Ober die nichtarchimedischen Grössensysteme’, Sitzungsberichte Kaiserlichen der Akademie Wissenschaften, Wien, Mathematisch–Naturwissenschaftliche Klasse 116 (Abteilung IIa), 601–655.Google Scholar
  89. 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.Google Scholar
  90. 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.Google Scholar
  91. Hausdorff, F.: 1906, ‘Untersuchungen über Ordungtypen’, Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig, Matematisch–Physische Klasse 58, 106–169.Google Scholar
  92. Hausdorff, F.: 1907, ‘Untersuchungen über Ordungtypen’, Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig, Matematisch–Physische Klasse 59, 84–159.Google Scholar
  93. Hausdorff, F.: 1908, ‘Grundzüge einer Theorie der geordneten Mengen’, Mathematische Annalen 65, 435–505.CrossRefGoogle Scholar
  94. Hausdorff, F.: 1914, Grundzüge der Mengenlehre,Leipzig.Google Scholar
  95. Hausner, M. and J. G. Wendel: 1952, ‘Ordered Vector Spaces’, Proceedings of the American Mathematical Society 3, 977–982.CrossRefGoogle Scholar
  96. Hjelmslev, J.: 1907, ‘Neue Begründung der ebenen Geometrie’, Mathematische Annalen 64, 449–474.CrossRefGoogle Scholar
  97. Hessenberg, G.: 1905a, ‘Begründung der elliptischen Geometrie’, Mathematische Annalen 61, 173–184.CrossRefGoogle Scholar
  98. Hessenberg, G.: 1905b, ‘Neue Begründung der Sphärik’, Sitzungsberichte der Berliner mathematischen Gesellschaft 4, 69–77.Google Scholar
  99. Hessenberg, G.: 1905c, ‘Beweis des Desarguesschen Satzes aus dem Pascalschen’, Mathematische Annalen 61, 161–172.CrossRefGoogle Scholar
  100. Hessenberg, G.: 1930, Grundlagen der Geometrie,Berlin.Google Scholar
  101. Higman, G.: 1952, ‘Ordering by Divisibility in Abstract Algebras’, Proceedings of the London Mathematical Society 2, 326–336.CrossRefGoogle Scholar
  102. Hilbert, D.: 1899, Grundlagen der Geometrie, Festschrift zur Feier der Enthüllung des Gauss-Weber Denkals in Göttingen, Teubner, Leipzig.Google Scholar
  103. 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 ).Google Scholar
  104. 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.Google Scholar
  105. 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 );Google Scholar
  106. Hilbert 1930, pp. 247–261). English Translation in Bulletin of the American Mathematical Society 8 (1902), 437–479.CrossRefGoogle Scholar
  107. 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.Google Scholar
  108. Hilbert, D.: 1902. Foundations of Geometry, English Translation by E. Townsend of an expanded version of (Hilbert 1899 ).Google Scholar
  109. Hilbert, D.: 1903a, Second Edition of (Hilbert 1899 ).Google Scholar
  110. Hilbert, D.: 1903b, ‘Über den Satz von der Gleichheit der Basiswinkel im gleichschenkligen Dreiech’, Proceedings of the London Mathematical Society 35, 50–68.CrossRefGoogle Scholar
  111. 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).Google Scholar
  112. Hilbert, D.: 1903c, ‘Neue Begründung der Bolyai-Lobatschefskyschen Geometrie’, Mathematische Annalen 57, 137–150.CrossRefGoogle Scholar
  113. Reprinted as an appendix in (Hilbert 1903a ). For an English translation, see (Hilbert 1971, pp. 133–149 ).Google Scholar
  114. 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.Google Scholar
  115. 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.Google Scholar
  116. Hilbert D.: 1909, Third Edition of (Hilbert 1899).Google Scholar
  117. Hilbert D.: 1930, Seventh Edition of (Hilbert 1899).Google Scholar
  118. 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 ).Google Scholar
  119. 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.Google Scholar
  120. Holland, C.: 1963, ‘Extensions of Ordered Groups and Sequence Completions’, Transactions of the American Mathematical Society 107, 71–82.CrossRefGoogle Scholar
  121. 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.CrossRefGoogle Scholar
  122. 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.)Google Scholar
  123. Iwasawa, K.: 1948, ‘On Linearly Ordered Groups’, Journal of the Mathematical Society of Japan 1, 1–9.CrossRefGoogle Scholar
  124. Kaplansky, I.: 1942, ‘Maximal Fields with Valuations’, Duke Mathematical Journal 9, 303–321.CrossRefGoogle Scholar
  125. Kaplansky, I.: 1949. Review of (Mal’cev 1948), Mathematical Reviews 10, 8.Google Scholar
  126. Klein, F.: 1908/1939, Elementary Mathematics From An Advanced Standpoint, Arithmetic Google Scholar
  127. Algebra, Analysis, Translated from the German by E. R. Hedrick and C. A. Noble, Dover Publications, New York.Google Scholar
  128. 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.Google Scholar
  129. Kokorin, A. I. and V. M. Kopytov: 1974, Fully Ordered Groups, John Wiley & Sons, New York-Toronto.Google Scholar
  130. Krull, W.: 1932, ‘Allegemeine Bewertungstheorie’, Journal fiir die Reine and Angewandt Mathematik 167, 160–196.Google Scholar
  131. Krull, W.: 1955. Review of (Gravett, K. 1955), Zentralblatt für Mathematik and ihre Grenzgebiete 67, 266.Google Scholar
  132. Krull, W.: 1956. Review of (Gravett, K. 1956), Zentralblatt für Mathematik and ihre Grenzgebiete 74, 21.Google Scholar
  133. Krull, W.: 1957–71, ‘Review of (Ribenboim, P. 1958)’, Zentralblatt für Mathematik and ihre Grenzgebiete 95, 259.Google Scholar
  134. Krull, W.: 1970, ‘Felix Hausdorff 1868–1942’, in Bonner Gelehrte. Beiträge zur Geschichte der Wissenschaften in Bonn, Volume 9, pp. 54–69.Google Scholar
  135. 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.Google Scholar
  136. Lam, T. Y.: 1983, Orderings, Valuations and Quadratic Forms. Regional ConferenceGoogle Scholar
  137. Series in Mathematics #52. American Mathematical Society, Providence, R. I.Google Scholar
  138. Lam, T. Y.: 1991, A First Course in Noncommutative Rings,Springer-Verlag, New York.Google Scholar
  139. Lang, S.: 1953, ‘The Theory of Real Places’, Annals of Mathematics 57, 387–391.CrossRefGoogle Scholar
  140. 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.Google Scholar
  141. Laugwitz, D.: 1986. Zahlen and Kontinuum, Eine Einführung in die Infinitesimalmathematik, Bibliographisches Institut, Mannheim.Google Scholar
  142. Levi, F. W.: 1942, ‘Ordered Groups’, Proceedings of the Indian Academy of Sciences (Section A) 16, 256–263.Google Scholar
  143. 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 ).Google Scholar
  144. 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 ).Google Scholar
  145. Levi-Civita, T.: 1954. Tullio Levi-Civita, Opere Matematiche, Memorie e Note, Volume primo 1893–1900, edited by Nicola Zanichelli, Bologna.Google Scholar
  146. Loonstra, F.: 1950, ‘The Classes of Ordered Groups’, in Proceedings of the International Congress of Mathematicians, Volume I, Cambridge, pp. 312–313.Google Scholar
  147. Mac Lane, S.: 1939, ‘The Universality of Formal Power Series Fields’, Bulletin of the American Mathematical Society 45, 888–890.CrossRefGoogle Scholar
  148. Mal’cev, A. L.: 1948, ‘On Embedding of Group Algebras in a Division Algebra’, (Russian) Doklady Akademii Nauk SSSR (N.S.) 60, 1499–1501.Google Scholar
  149. Mal’cev, A. L.: 1949, ‘On Ordered Groups’, (Russian) Izvestiia Akademii Nauk. SSSR (Seriia Matematicheskaia) 13, 473–482.Google Scholar
  150. Mayrhofer, K.: 1934, K.: 1934, ‘Hans Hahn’, Monatschefte fir Mathematik und Physik 41, 22 1238.Google Scholar
  151. Moore, G.: 1982, Zermelo’s Axiom of Choice: Its Origins Development and Influence, Springer-Verlag, New York.CrossRefGoogle Scholar
  152. 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.Google Scholar
  153. 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)).Google Scholar
  154. 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.Google Scholar
  155. Neumann, H.: 1954, Review of (Conrad, P. 1953). Zentralblatt für Mathematik und ihre Grenzgebiete 50 23.Google Scholar
  156. Ostrowski, A.: 1935, ‘Untersuchungen zur arithmetischen Theorie der Körper’Google Scholar
  157. 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.Google Scholar
  158. Pickert, G.: 1953, Review of (Hausner, M. and Wendel, J. 1952), Zentralblatt für Mathematik und ihre Grenzgebiete 48, 87.Google Scholar
  159. Poincaré, H.: 1905, ‘Les Mathématiques et la logique’, Revue de métaphysique et de morale 13, 815–835.Google Scholar
  160. Prestel, A.: 1984, Lectures on Formally Real Fields, Lecture Notes in Mathematics #1093, Springer-Verlag, Berlin.Google Scholar
  161. Priess-Crampe, S.: 1973, ‘Zum Hahnschen Einbettungssatz für Angeordnete Körper’, Archiv der Mathematik 24, 607–614.CrossRefGoogle Scholar
  162. Priess-Crampe, S.: 1983, Angeordnete Strukturen, Gruppen, Körper, projektive Ebenen, Springer-Verlag, Berlin.Google Scholar
  163. Priess-Crampe, S. and R. von Chossy: 1975, ‘Ordungsverträgliche Bewertungen eines angeordneten Körpers’, Archiv der Mathematik 26, 373–387.Google Scholar
  164. 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.Google Scholar
  165. Rédei, L.: 1954/1967, Algebra, Volume 1, Pergamon Press, Oxford.Google Scholar
  166. Redfield, R. H.: 1986, ‘Embeddings Into Power Series Rings’, Manuscripta Mathematica 56, 247–268.CrossRefGoogle Scholar
  167. Redfield, R. H.: 1989a, ‘Banaschewski Functions and Ring-Embeddings’, in J. Martinez (ed.), Ordered Algebraic Structures, Kluwer Academic Publishers, The Netherlands.Google Scholar
  168. Redfield, R. H.: 1989b, ‘Non-Embeddable O-Rings’, Communications in Algebra 17, 59–71.CrossRefGoogle Scholar
  169. Ribenboim, P.: 1958, ‘Sur les groupes totalement ordonnés et l’arithmétique des anneaux de valuation’, Summa Brasilliensis Mathematicae 4, 1–64.Google Scholar
  170. Ribenboim, P.: 1964. Théorie des Groupes Ordonnés, Universidad Nacional Del Sur, Bahia Blanca.Google Scholar
  171. 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 ).Google Scholar
  172. Robinson, A.: 1966, Non-standard Analysis, North-Holland Publishing Company, Amsterdam.Google Scholar
  173. 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.Google Scholar
  174. 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.Google Scholar
  175. Schilling, O. F. G.: 1945, ‘Noncommutative Valuations’, Bulletin of the American Mathematical Society 51, 297–304.CrossRefGoogle Scholar
  176. Schilling, O. F. G.: 1950, The Theory of Valuations, Mathematical Surveys I V, American Mathematical Society, New York.Google Scholar
  177. Schmieden, C. and D. Laugwitz: 1958, ‘Eine Erweiterung der Infinitesimalrechnung’, Mathematische Zeitschrift 69, 1–39.CrossRefGoogle Scholar
  178. Schoenflies, A.: 1906, ‘über die Möglichkeit einer projektiven Geometrie bei trans-finiter (nicht archimedischer) Massbestimmung’, Jahresbericht der Deutschen Mathematiker-Vereinigung 15, 26–47.Google Scholar
  179. 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.Google Scholar
  180. Schur, F.: 1899, ‘Ueber den Fundamentalsatz der projectiven Geometrie’, Mathematische Annalen 51, 401–409.CrossRefGoogle Scholar
  181. Schur, F.: 1902, ‘Ueber die Grundlagen der Geometrie’, Mathematische Annalen 55, 401–409.Google Scholar
  182. Schur, F.: 1903, ‘Zur Proportionslehre’, Mathematische Annalen 57, 205–208.CrossRefGoogle Scholar
  183. Schur, F.: 1904, ‘Zur Bolyai-Lobatschefskijschen Geometrie’, Mathematische Annalen 59, 314–320.CrossRefGoogle Scholar
  184. Schur, F.: 1909, Grundlagen der Geometrie, Druck und Verlag Von B. G. Teubner, Leipzig und Berlin.Google Scholar
  185. Schwartz, N.: 1978, ‘rlpStrukturen’, Mathematische Zeitschrift 158, 147–155.CrossRefGoogle Scholar
  186. Sinaceur, H.: 1989, ‘Une origin du concept d’analyse non-standard’, in H. Barreau andGoogle Scholar
  187. J. Harthong (eds.), La Mathèmatique Non Standard, Historie-Philosophie Dossier Google Scholar
  188. 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.Google Scholar
  189. Steinitz, E.: 1910, ‘Algebraische Theorie der Körper’, Journal für die Reine und Angewandt Mathematik 137, 167–309.Google Scholar
  190. Stolz, 0.: 1881, ‘B. Bolzano’s Bedeutung in der Geschichte der Infinitesimalrechnung’, Mathematische Annalen 18, 255–279.CrossRefGoogle Scholar
  191. Stolz, 0.: 1882, ‘Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes’Google Scholar
  192. Berichte des Naturwissenschaftlich-Medizinischen, Vereines in Innsbruck 12 74–89. Stolz, 0.: 1883, ‘Zur Gometrie der Alten, insbesondere über ein Axiom des Archimedes’Google Scholar
  193. Mathematische Annalen 22 504–519. (This is a revised version of (Stolz 1882)). Stolz, 0.: 1884, ‘Die unendlich kleinen Grössen’, Berichte des Naturwissenschaftlich- Google Scholar
  194. Medizinischen, Vereines in Innsbruck 14 21–43.Google Scholar
  195. Stolz, 0.: 1885. Vorlesungen über Allgemeine Arithmetik; Erster Theil: Allgemeines und Arithhmetik der Reelen Zahlen, Teubner, Leipzig.Google Scholar
  196. Stolz, 0.: 1886, Vorlesungen über Allgemeine Arithmetik; Zweiter Theil: Arithhmetik der Complexen Zahlen, Teubner, Leipzig.Google Scholar
  197. Stolz, 0.: 1891, ‘Veber das Axiom des Archimedes’, Mathematische Annalen 39, 107112.Google Scholar
  198. Stolz, O. and J. A. Gmeiner: 1902, Theoretische Arithmetik, Teubner, Leipzig.Google Scholar
  199. Stroyan, K. and W. Luxemburg: 1976, Introduction To The Theory Of Infnitesimals, Academic Press, New York.Google Scholar
  200. Temple, G.: 1981, 100 Years of Mathematics, Springer-Verlag, New York.Google Scholar
  201. Thomae, J.: 1870, Abriss einer Theorie der complexen Functionen und der Thetafunctionen einer Veränderlichen, Nebert, Halle.Google Scholar
  202. Thomae, J.: 1880, Elementare Theorie der analytischen Functionen einer complexen Veränderlichen, Nebert, Halle.Google Scholar
  203. 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.Google Scholar
  204. Vahlen, K. Th.: 1905, Abstrackte Geometrie, B. G. Teubner, Leipzig.Google Scholar
  205. Vahlen, K. Th.: 1907, Über nicht-archimedische Algebra’, Jahresbericht der Deutschen Mathematiker-Vereinigung 16, 409–421.Google Scholar
  206. van der Waerden, B. L.: 1930, Moderne Algebra I, Julius Springer, Berlin.Google Scholar
  207. 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.Google Scholar
  208. Veronese, G.: 1891, Fondamenti di Geometria,Padova.Google Scholar
  209. Veronese, G.: 1894, Grundzüge der Geometrie,Leipzig.Google Scholar
  210. 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 Google Scholar
  211. 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.Google Scholar
  212. Wang, H.: 1987, Reflections on Kurt Glide!, A Bradford Book, The MIT Press, Cambridge Massachusetts.Google Scholar
  213. Warner, S.: 1990, Modern Algebra, Dover Publications, New York. Reprinted with corrections: Modern Algebra, Prentice-Hall, Inc., EnglewoodsGoogle Scholar
  214. New Jersey, 1965. Weispfenning, V.: 1971, ‘On the Elementary Theory of Hensel Fields’, Doctoral Dissertation, Heidelberg.Google Scholar
  215. Weispfenning, V.: 1976, ‘On the Elementary Theory of Hensel Fields’, Annals of Mathematical Logic 10, 59–93.CrossRefGoogle Scholar
  216. 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.Google Scholar
  217. 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.Google Scholar
  218. Zelinsky, D.: 1948, ‘Nonassociative Valuations’, Bulletin of the American Mathematical Society 54, 175–183.CrossRefGoogle Scholar
  219. 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.Google Scholar
  220. Ziegler, M.: 1972, Die elementare Theorie der Hensel Körper, Doctoral Dissertation, Köln.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Philip Ehrlich
    • 1
  1. 1.Ohio UniversityAthensUSA

Personalised recommendations