Interpretations of Euclidean geometry
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- by S. Świerczkowski
- Trans. Amer. Math. Soc. 322 (1990), 315-328
- DOI: https://doi.org/10.1090/S0002-9947-1990-0982234-9
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Abstract:
Following Tarski, we view $n$-dimensional Euclidean geometry as a first-order theory ${E_n}$ with an infinite set of axioms about the relations of betweenness (among points on a line) and equidistance (among pairs of points). We show that for $k < n$, ${E_n}$ does not admit a $k$-dimensional interpretation in the theory RCF of real closed fields, and we deduce that ${E_n}$ cannot be interpreted $r$-dimensionally in ${E_s}$, when $r \cdot s < n$.References
- Maurice Boffa, Élimination des quantificateurs en algèbre, Bull. Soc. Math. Belg. Sér. B 32 (1980), no. 1, 107–133 (French). MR 670828 C. C. Chang and H. J. Keisler, Model theory, North-Holland, Amsterdam, 1973.
- Michel Coste, Ensembles semi-algébriques, Real algebraic geometry and quadratic forms (Rennes, 1981) Lecture Notes in Math., vol. 959, Springer, Berlin-New York, 1982, pp. 109–138 (French). MR 683131 L. van den Dries, Definable sets in $0$-minimal structures, Lectures at University of Konstanz, spring 1985.
- Lou van den Dries, A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bull. Amer. Math. Soc. (N.S.) 15 (1986), no. 2, 189–193. MR 854552, DOI 10.1090/S0273-0979-1986-15468-6
- Lou van den Dries, Alfred Tarski’s elimination theory for real closed fields, J. Symbolic Logic 53 (1988), no. 1, 7–19. MR 929371, DOI 10.2307/2274424
- Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, N.J., 1952. MR 0050886, DOI 10.1515/9781400877492
- G. Kreisel and J.-L. Krivine, Elements of mathematical logic. Model theory, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1967. MR 0219380
- Heisuke Hironaka, Triangulations of algebraic sets, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 165–185. MR 0374131
- S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 449–474. MR 173265
- R. Montague and R. L. Vaught, A note on theories with selectors, Fund. Math. 47 (1959), 243–247. MR 124204, DOI 10.4064/fm-47-2-243-247
- Jan Mycielski, A lattice of interpretability types of theories, J. Symbolic Logic 42 (1977), no. 2, 297–305. MR 505480, DOI 10.2307/2272134 J. Mycielski, P. Pudlák, and A. S. Stern, A lattice of chapters of mathematics, Mem. Amer. Math. Soc. (to appear).
- A. Seidenberg, A new decision method for elementary algebra, Ann. of Math. (2) 60 (1954), 365–374. MR 63994, DOI 10.2307/1969640 L. W. Szczerba, Dimensions of interpretations (abstract), J. Symbolic Logic 49 (1984), 677.
- Alfred Tarski, A decision method for elementary algebra and geometry, University of California Press, Berkeley-Los Angeles, Calif., 1951. 2nd ed. MR 0044472, DOI 10.1525/9780520348097
- Alfred Tarski, What is elementary geometry?, Proceedings of an International Symposium held at the Univ. of Calif., Berkeley, Dec. 26, 1957-Jan. 4, 1958, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1959, pp. 16–29. The axiomatic method; With special reference to geometry and Physics; Edited by L. Henkin, P. Suppes and A. Tarski. MR 0106185
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 322 (1990), 315-328
- MSC: Primary 03F25; Secondary 03C65, 51M99
- DOI: https://doi.org/10.1090/S0002-9947-1990-0982234-9
- MathSciNet review: 982234