Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-23T07:46:01.626Z Has data issue: false hasContentIssue false
  • English
  • Français

Tarski's System of Geometry

Published online by Cambridge University Press:  15 January 2014

Alfred Tarski  and
Steven Givant
Alfred Tarski
Affiliation:
Department of Mathematics and Computer Science, Mills College, 5000 Macarthur Blvd. Oakland, CA 94613, USAE-mail:givant@mills.edu
Steven Givant
Affiliation:
Department of Mathematics and Computer Science, Mills College, 5000 Macarthur Blvd. Oakland, CA 94613, USAE-mail:givant@mills.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabhäuser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1-dimensional geometry.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 5 , Issue 2 , June 1999 , pp. 175 - 214
Copyright
Copyright © Association for Symbolic Logic 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Bolyai, J., Appendix scientiam spatii absolute veram exhibens: a veritate aut falsitate Axiomatis XI Euclidei (a priori haud unquam decidenda) independentem; adjecta ad casum falsitatis, quadratura circuli geometrica, in F. Bolyai, Tentamen juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris methodo intuitiva, evidentiaque huic propria, introducendi. Cum Appendice triplici, vol. 1, Maros-Vásárhelyini, 1832, German translation in Engel, F. and Stäckel, P., Urkunden zur Geschichte der nichteuklidischen Geometrie, Band 2, Teil 1, 2, B.G. Teubner, Leipzig and Berlin, 1913.Google Scholar
[2] Dedekind, R., Stetigkeit und irrationale Zahlen, F. Vieweg, Braunschweig, 1872, 24 pp.Google Scholar
[3] Enriques, F., Fragen der Elementargeometrie, B.G. Teubner Verlag, Berlin and Leipzig, 1911.Google Scholar
[4] Grochowska, M., Euclidean two dimensional equidistance theory, Demonstratio Mathematica, vol. 17 (1984), pp. 593–607.Google Scholar
[5] Gupta, H. N., Contributions to the axiomatic foundations of geometry, Doctoral dissertation, University of California, Berkeley, 1965, 407 pp.Google Scholar
[6] Kordos, M., On the syntactic form of dimension axiom for affine geometry, Bulletin de l'Academie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 17 (1969), pp. 833–837.Google Scholar
[7] Kordos, M., Szkolna realizacja geometrii jako teorii opartej na aksjomatyce Szmielew i Tarskiego, Wiadomości Matematyczne, vol. 13 (1971), pp. 97–102.Google Scholar
[8] Legendre, A. M., Réflexions sur différentes maniéres de démontrer la théorie des parallèles, Memoires de l'académie royale des sciences de l'institut de France, vol. 12 (1833), pp. 367–410.Google Scholar
[9] Lindenbaum, A. and Tarski, A., Sur l'indépendance des notions primitives dans les systèmes mathématiques, Rocznik Polskiego Towarzystwa Matematycznego (=Annales de la Société Polonaise de Mathématique), vol. 5 (1927), pp. 111–113.Google Scholar
[10] Lorenz, J. F., Grundriss der reinen und angewandten Mathematik, vol. I, C.G. Fleckeisen, Helmsted, 1791–1792.Google Scholar
[11] Makowiecka, H., The theory of bi-proportionality as a geometry, Bulletin de l'Academie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 23 (1975), pp. 657–664.Google Scholar
[12] Makowiecka, H., An elementary geometry in connection with decompositions of a plane, Bulletin de l'Academie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 23 (1975), pp. 665–674.Google Scholar
[13] Makowiecka, H., On primitive notions in n-dimensional elementary geometries, Bulletin de l'Academie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 23 (1975), pp. 675–682.Google Scholar
[14] Makowiecka, H., A general property of ternary relations in elementary geometry, Bulletin de l'Academie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 24 (1976), pp. 163–169.Google Scholar
[15] Makowiecka, H., On minimal systems of primitives in elementary Euclidean geometry, Bulletin de l'Academie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 25 (1977), pp. 269–277.Google Scholar
[16] Mollerup, J., Die Beweise der ebenen Geometrie ohne Benutzung der Gleichheit und Ungleichheit der Winkel, Mathematische Annalen, vol. 58 (1903), pp. 479–496.CrossRefGoogle Scholar
[17] Moore, E. H., On the projective axioms of geometry, Transactions of the American Mathematical Society, vol. 3 (1902), pp. 142–158.Google Scholar
[18] Moszyńska, M., Review of Schwabhäuser-Szmielew-Tarski [29], this Journal, vol. 51 (1986), pp. 1073–1075.Google Scholar
[19] Mycielski, J. and Świerczkowski, S., On the Lebesgue measurability and the axiom of determinateness, Fundamenta Mathematicae, vol. 54 (1964), pp. 67–71.CrossRefGoogle Scholar
[20] Pambuccian, V., Simple axiom systems for Euclidean geometry, Mathematical Chronicle, vol. 18 (1989), pp. 63–74.Google Scholar
[21] Pambuccian, V., On the simplicity of an axiom system for plane Euclidean geometry, Demonstratio Mathematica, vol. 30 (1997), pp. 509–512.Google Scholar
[22] Pasch, M., Vorlesungen über neuere Geometrie, B.G. Teubner, Leipzig, 1882, 201 pp.Google Scholar
[23] Pieri, M., La geometria elementare istituita sulle nozioni ‘punto’ é ‘sfera’, Memorie di Matematica e di Fisica della Società Italiana delle Scienze, vol. 15 (1908), pp. 345–450.Google Scholar
[24] Richter, G. and Schnabel, R., Euklidische Räume, Journal of Geometry, vol. 58 (1997), pp. 164–178.Google Scholar
[25] Robinson, R. M., Binary relations as primitive notions in elementary geometry, The axiomatic method, with special reference to geometry and physics (Henkin, L., Suppes, P., and Tarski, A., editors), North-Holland Publishing Company, Amsterdam, 1959, pp. 68–85.Google Scholar
[26] Saccheri, G., Euclides ab omnis naevo vindicatus: sive conatus geometricus quo stabiliuntur prima ipsa universae geometricae principia, Paolo Antonio Montano, Milano, 1733.Google Scholar
[27] Schnabel, R., Euklidische Geometrie, Habilitation, Kiel, 1978, iii+111 pp.Google Scholar
[28] Schur, F., Grundlagen der Geometrie, B.G. Teubner Verlag, Berlin and Leipzig, 1909, x+192 pp.Google Scholar
[29] Schwabhäuser, W., Szmielew, W., and Tarski, A., Metamathematische Methoden in der Geometrie, Hochschultext, Springer-Verlag, Berlin, 1983, viii+482 pp.Google Scholar
[30] Scott, D., Dimension in elementary Euclidean geometry, The axiomatic method, with special reference to geometry and physics (Henkin, L., Suppes, P., and Tarski, A., editors), North-Holland Publishing Company, Amsterdam, 1959, pp. 53–67.Google Scholar
[31] Sierpiński, W., Cardinal and ordinal numbers, Państwowe Wydawnictwo Naukowe, Warszawa, 1958, 491 pp.Google Scholar
[32] Solovay, R. M., A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970), pp. 1–56.CrossRefGoogle Scholar
[33] Szczerba, L. W., Independence of Pasch's axiom, Bulletin de l'Academie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 18 (1970), pp. 491–498.Google Scholar
[34] Szczerba, L. W. and Tarski, A., Metamathematical properties of some affine geometries, Proceedings of the 1964 International Congress for Logic, Methodology and Philosophy of Science (Bar-Hillel, Y., editor), Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Company, Amsterdam, 1965, pp. 166–178.Google Scholar
[35] Szczerba, L. W. and Tarski, A., Metamathematical discussion of some affine geometries, Fundamenta Mathematicae, vol. 104 (1979), pp. 155–192.Google Scholar
[36] Szmielew, W., Elementary properties of Abelian groups, Fundamenta Mathematicae, vol. 41 (1955), pp. 203–271.CrossRefGoogle Scholar
[37] Szmielew, W., Some metamathematical problems concerning elementary hyperbolic geometry, The axiomatic method, with special reference to geometry and physics (Henkin, L., Suppes, P., and Tarski, A., editors), North-Holland Publishing Company, Amsterdam, 1959, pp. 30–52.Google Scholar
[38] Szmielew, W., From affine to Euclidean geometry: An axiomatic approach, translated by Moszyńska, M., PWN-Polish Scientific Publishers, Warszawa, and D. Reidel Publishing Company, Dordrecht, 1983, xiii+194 pp.; this is an English translation of Od geometrii afinicznej do Euklidesowej—Rozważania nad aksjomatyka̧, edited and prepared for publication by M. Moszyńska, Państwowe Wydawnictwo Naukowe, Warszawa, 1981, 172 pp.Google Scholar
[39] Tarski, A., Sur les ensembles définissables de nombres réels, I, Fundamenta Mathematicae, vol. 17 (1931), pp. 210–239; an English translation appeared as Article VI in [43].Google Scholar
[40] Tarski, A., Einige methodologische Untersuchungen über die Definierbarkeit der Begriffe, Erkenntnis, vol. 5 (1935), pp. 80–100; this is a German translation of Z badań metodologicznych nad definjowalnościa̧ terminów, Przegla̧d Filozoficzny (=Revue Philosophique), vol. 37 (1934), pp. 438–460; an English translation appears as Article X in Tarski [43].CrossRefGoogle Scholar
[41] Tarski, A., A decision method for elementary algebra and geometry, prepared for publication by J. C. C. McKinsey, U.S. Air Force Project RAND, R-109, the RAND Corporation, Santa Monica, 1948, iv+60 pp.; a second, revised edition was published by the University of California Press, Berkeley and Los Angeles, CA, 1951, iii+63 pp.Google Scholar
[42] Tarski, A., A general theorem concerning primitive notions of Euclidean geometry, Indagationes Mathematicae, vol. 18 (= Koninklijkle Nederlandse Akademie van Wetenschappen, Proceedings, Series A, Mathematical Sciences, vol. 59) (1956), pp. 468–474.Google Scholar
[43] Tarski, A., Logic, semantics, metamathematics: Papers from 1923 to 1938 (translated by Woodger, J. H.), Clarendon Press, Oxford, 1956, xiv+471 pp.Google Scholar
[44] Tarski, A., What is elementary geometry?, The axiomatic method, with special reference to geometry and physics (Henkin, L., Suppes, P., and Tarski, A., editors), North-Holland Publishing Company, Amsterdam, 1959, pp. 16–29.Google Scholar
[45] Tarski, A., The completeness of elementary algebra and geometry, Institut Blaise Pascal, Paris, 1967, iv+50 pp.Google Scholar
[46] Tarski, A., Alfred Tarski: Collected papers (Givant, S. R. and McKenzie, R. N., editors), Birkhäuser, Basel, 1986, vol. 1, xvi+659 pp.; vol. 2, xvi+69 pp.; vol. 3, xvi+682 pp.; vol. 4, xvi+757 pp.Google Scholar
[47] Tarski, A., Feldman, N., Green, T., Hanf, W., Henkin, L., Isard, S., and Kostinsky, A., Basic research in the foundations of mathematics, Report for the period 07 1, 1964–June 30, 1968 of a project supported by National Science Foundation Grants GP-1395, GP-4608, and GP-6232X, Department of Mathematics, University of California, Berkeley, CA, mimeograph, 1970, ii+53 pp.Google Scholar
[48] Tarski, A., Gaifman, H., and Lévy, A., Basic research in the foundations of mathematics, Report for the period 07 1, 1959–June 30, 1961 of a project supported by National Science Foundation Grants G6693 and G14006, Department of Mathematics, University of California, Berkeley, CA, mimeograph, 1962, 33 pp.Google Scholar
[49] Tarski, A., Henkin, L., and Rupley, W., Basic research in the foundations of mathematics, Report for the period 07 1, 1961–June 30, 1964 of a project supported by National Science Foundation Grants G6693, G14006, G19673, and GP-1395, Department of Mathematics, University of California, Berkeley, CA, mimeograph, 1965, 80 pp.Google Scholar
[50] Veblen, O., A system of axioms for geometry, Transactions of the American Mathematical Society, vol. 5 (1904), pp. 343–384.Google Scholar
[51] Veblen, O., The foundations of geometry, Monographs on topics of modern mathematics, relevant to the elementary field (Young, J. W. A., editor), Longsman, Green, and Company, New York, 1914, pp. 1–51.Google Scholar