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FROM COMPLETENESS TO ARCHIMEDEAN COMPLETENES

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REFERENCES

  • Alling, N.: 1990, 'Pythagorean and Euclidean Ordered Valued Fields'', La Société royale du Canada. Comptes Rendus Mathématique de l'Académie des Sciences (Mathematical Reports) 12(5), 187–92.

    Google Scholar 

  • Baer, R.: 1927, 'Über nicht-archimedisch geordnete Körper', Sitzungsberichte der Heidelberger Akademie der Wissenschaften 8, 3–13.

    Google Scholar 

  • Benhissi, A.: 1990, 'Series FormellesGeneralisees SurUn Corps Pythagoricien',La Société royale du Canada. Comptes Rendus Mathématique des Sciences (Mathematical Reports) 12(5), 193–198.

    Google Scholar 

  • Bernays, P.: 1955, 'Betrachtungen über das Vollständigkeitsaxiom und verwandte Axiom', Mathematische Zeitschrift 63, 219–29.

    Google Scholar 

  • Beth, E. W.: 1966, The Foundations of Mathematics (Revised Edition), Harper & Row, Publishers, New York, NY.

    Google Scholar 

  • Borsuk, K. and W. Szmielew: 1960, Foundations of Geometry, North-Holland Publishing Co., Amsterdam.

    Google Scholar 

  • Bröcker, L.: 1972, 'Über eine Klasse pythagoreischer Körper', Archiv der Mathematik 23, 405–7.

    Google Scholar 

  • Carnap, R. and F. Bachmann: 1936, 'Über Extremalaxiome', Erkenntnis 6, 166–88.

    Google Scholar 

  • 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).

    Google Scholar 

  • Conrad, P.: 1954, 'On Ordered Division Rings'', Proceedings of the AmericanMathematical Society 5, 323–8.

    Google Scholar 

  • Conrad, P., and J. Dauns, 1969, 'An Embedding Theorem for Lattice-ordered Fields',Pacific Journal of Mathematics 30, 385–97.

    Google Scholar 

  • Ehrlich, P. (ed.): 1994, Real Numbers, Generalizations of the Reals, and Theories of Continua Kluwer Academic Publishers, Dordrecht, Holland.

    Google Scholar 

  • Ehrlich, P.: 1994a, 'General Introduction', in (Ehrlich 1994, pp. vii–xxxii).

  • Ehrlich, P.: 1995, 'Hahn's Über die nichtarchimedischen Grössensysteme and the Development of the Modern Theory of Magnitudes and Numbers to Measure Them', in Jaakko Hintikka (ed.), From Dedekind to Gödel, Essays on the Development of the Foundations of Mathematics, Kluwer Academic Publishers, Dordrecht, Holland.

    Google Scholar 

  • Ehrlich, P.: (Forthcoming), 'Dedekind Cuts of Archimedean Complete Ordered Abelian Groups', Algebra Universalis.

  • Eisenhart, L. P.: 1939, Coordinate Geometry, Ginn and Company, Boston.

    Google Scholar 

  • Elman, R. and T.Y. Lam: 1972, 'Quadratic Forms over Formally Real Fields and Pythagorean Fields', American Journal of Mathematics 94, 1155–94.

    Google Scholar 

  • Eves, H.: 1965, A Survey of Geometry, Volume Two. Allyn and Bacon, Inc., Boston.

    Google Scholar 

  • Fuchs, L.: 1963, Partially Ordered Algebraic Systems. Pergamon Press.

  • Hahn, H.: 1907, 'Über die nichtarchimedischen Grössensysteme', Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, Wien, Mathematisch - Naturwissenschaftliche Klasse 116 (Abteilung IIa), 601–655.

    Google Scholar 

  • Henkin, L., P. Suppes, and A. Tarski (eds.): 1959, The Axiomatic Method, North-Holland Publishing Company, Amsterdam.

    Google Scholar 

  • Hilbert, D.: 1899, De </del>Grundlagen der Geometrie. Festschrift zur Feier der Enthüllung des Gauss-Weber Denkals in Göttingen, Teubner, Leipzig.

    Google Scholar 

  • Hilbert, D.: 1900a, 'Über den Zahlbegriff', Jahresbericht der Deutschen Mathematiker - Vereinigung, 8, 180–4. Reprinted in (Hilbert 1930, pp. 241- 246).

    Google Scholar 

  • 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 

  • Hilbert, D.: 1902, Foundations of Geometry. English Translation by E. Townsend of an expanded version of (Hilbert 1899).

  • Hilbert, D.: 1903, Second Edition of (Hilbert 1899).

  • Hilbert, D.: 1904a, '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 

  • Hilbert, D.: 1904b, 'On the Foundations of Logic and Arithmetic', English translation by Beverly Woodward of (Hilbert 1904a) in J. van Heijenoort (ed.), From Frege To Gödel: A Sourcebook in Mathematical Logic, 1879- 1931 (Second Printing), Harvard University Press, Cambridge, Massachusetts, 1971.

    Google Scholar 

  • Hilbert, D.: 1909, Third Edition of (Hilbert 1899).

  • Hilbert, D.: 1930, Seventh Edition of (Hilbert 1899).

  • Hilbert, D.: 1956, Eighth Edition with supplements by Paul Bernays of (Hilbert 1899), Teubner, Stuttgart.

    Google Scholar 

  • 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).

  • Hilbert, D. and P. Bernays: 1934, Grundlagen der Mathematik I. Springer Verlag, Berlin.

    Google Scholar 

  • Holland C.: 1963, 'Extensions of Ordered Groups and Sequence Completions', Transactions of the American Mathematical Society 107, 71–82.

    Google Scholar 

  • Kaplansky, I.: 1942, 'Maximal Fields with Valuations', Duke Mathematical Journal 9, 303–21.

    Google Scholar 

  • Krull, W.: 1932, 'Allgemeine Bewertungstheorie', Journal für die Reine and Angewandt Mathematik 167, 160–96.

    Google Scholar 

  • 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 

  • Lam, T. Y.: 1983, Orderings, Valuations and Quadratic Forms, Regional Conference Series in Mathematics #52. American Mathematical Society, Providence, R.I.

    Google Scholar 

  • Martin, G.: 1975, The Foundations of Geometry and the Non-Euclidean Plane. Springer-Verlag, New York.

    Google Scholar 

  • Pejas, W.: 1961, 'Die Modelle desHilbertschen Axiomensystems der absoluten Geometrie', Mathematische Annalen 143, 212–35.

    Google Scholar 

  • Prestel, A.: 1974, 'Euklidsche Geometrie ohne das Axiom von Pasch', Abhandlungen aus dem Mathematischen Seminar der Univeristät Hamburg 41, 82–109.

    Google Scholar 

  • Prestel, A.: 1975, Lectures on Formally Real Fields. (Monografias de Mathemática 22). Instituto de Mathemática Pura e Aplicada. Rio de Janeiro. Reprinted (with corrections) in 1984. Lecture Notes in Mathematics #1093. Springer-Verlag, Berlin.

    Google Scholar 

  • Prestel, A. and M. Ziegler: 1975, 'Erblich euklidische Körper', Journal für die Reine und Angewandt Mathematik 274/275, 196–205.

    Google Scholar 

  • Priess-Crampe, S.: 1973, 'Zum Hahnschen Einbettungssatz für Angeordnete Körper', Archiv der Mathematik 24, 607–14.

    Google Scholar 

  • Priess-Crampe, S.: 1983, Angeordnete Strukturen, Gruppen, Körper, projektive Ebenen. Springer-Verlag, Berlin.

    Google Scholar 

  • Priess-Crampe, S. and R. von Chossy: 1975, 'Ordungsverträgliche Bewertungen eines angeordneten Körpers', Archiv der Mathematik 26, 373–87.

    Google Scholar 

  • 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.

  • Ribenboim, P.: 1990, 'Algebraic Extensions of Fields of Generalized Power Series', La Société royale du Canada. Comptes Rendus Mathématique de l'Académie des Sciences (Mathematical Reports) 12(5), 209–12.

    Google Scholar 

  • Ribenboim, P.: 1992, 'Fields: Algebraically Closed and Others', Manuscripta Mathematica 75, 115–50.

    Google Scholar 

  • Schwabhäuser, W.: 1956, 'Über DieVollständigkeit Der Elementaren Euklidischen Geometrie', Zeitschrift für mathematische Logik and Grundlagen der mathematik 2, 137–65.

    Google Scholar 

  • Schwabhäuser, W., W. Szmielew, and A. Tarski: 1983, Metamathematische Methoden in der Geometrie, Springer-Verlag, Berlin.

    Google Scholar 

  • Szczerba, L.: 1986, 'Tarski and Geometry', The Journal of Symbolic Logic 51, 907–12.

    Google Scholar 

  • Szmielew, W.: 1983, From Affine to Euclidean Geometry, D. Reidel Publishing Company, Dordrecht, Holland.

    Google Scholar 

  • Tarski, A.: 1948/1951,ADecision Method for Elementary Algebra and Geometry, TheRand Corporation, Santa Monica; Second Edition, University of California Press, Berkeley and Los Angeles.

    Google Scholar 

  • Tarski, A.: 1959, 'What is Elementary Geometry?', in (Henkin, Suppes, and Tarski 1959, pp. 16–29).

  • Viola, T.: 1960, 'Sul postulato della completezza lineare dell' Hilbert', Annali Di Matematica Pura Ed Applicata 49, 367–73.

    Google Scholar 

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  1. Department of Philosophy, Ohio University, Athens, OH, 45701, USA

    PHILIP Ehrlich

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Ehrlich, P. FROM COMPLETENESS TO ARCHIMEDEAN COMPLETENES. Synthese 110, 57–76 (1997). https://doi.org/10.1023/A:1004971123925

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