Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-21T18:30:12.261Z Has data issue: false hasContentIssue false
  • English
  • Français

Full Development of Tarski's Geometry of Solids

Published online by Cambridge University Press:  15 January 2014

Rafaŀ Gruszczyński  and
Andrzej Pietruszczak
Rafaŀ Gruszczyński
Affiliation:
Chair of Logic, Nicolaus Copernicus University, Toruń, PolandE-mail: [email protected], URL: www.logika.uni.torun.pl, E-mail: [email protected]
Andrzej Pietruszczak
Affiliation:
Chair of Logic, Nicolaus Copernicus University, Toruń, PolandE-mail: [email protected], URL: www.logika.uni.torun.pl, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we give probably an exhaustive analysis of the geometry of solids which was sketched by Tarski in his short paper [20, 21]. We show that in order to prove theorems stated in [20, 21] one must enrich Tarski's theory with a new postulate asserting that the universe of discourse of the geometry of solids coincides with arbitrary mereological sums of balls, i.e., with solids. We show that once having adopted such a solution Tarski's Postulate 4 can be omitted, together with its versions 4′ and 4″. We also prove that the equivalence of postulates 4, 4′ and 4″ is not provable in any theory whose domain contains objects other than solids. Moreover, we show that the concentricity relation as defined by Tarski must be transitive in the largest class of structures satisfying Tarski's axioms.

We build a model (in three-dimensional Euclidean space) of the theory of so called T*-structures and present the proof of the fact that this is the only (up to isomorphism) model of this theory.

Moreover, we propose different categorical axiomatizations of the geometry of solids. In the final part of the paper we answer the question concerning the logical status (within the theory of T*-structures) of the definition of the concentricity relation given by Tarski.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 14 , Issue 4 , December 2008 , pp. 481 - 540
Copyright
Copyright © Association for Symbolic Logic 2008

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] Bennett, Brandon, A categorical axiomatization of region-based geometry, Fundamenta Informaticae, vol. 46 (2001), no. 1-2, pp. 145–158.Google Scholar
[2] Bennett, Brandon et al., A foundation for region-based geometry, Proceedings of ECAI-2000 (Horn, W., editor), IOS Press, 2000, pp. 204–208.Google Scholar
[3] Biactno, Loredana and Gerla, Giangiacomo, Connection structures, Notre Dame Journal of Formal Logic, vol. 33 (1991), no. 2, pp. 242–247.Google Scholar
[4] Biactno, Loredana and Gerla, Giangiacomo, Connection structures: Grzegorczyk's and Whitehead's definitions of point, Notre Dame Journal of Formal Logic, vol. 37 (1996), no. 3, pp. 431–439.Google Scholar
[5] Borsuk, Karol and Szmielew, Wanda, Foundations of geometry, Euclidean and Bolyai-Lobachevskian geometry, projective geometry, North Holand Publishing Company, Amsterdam, 1960.Google Scholar
[6] Gerla, Giangiacomo, Pointless geometries, Handbook of incidence geometry, Elsevier Science B.V., 1995, pp. 1015–1331.Google Scholar
[7] Gruszczyński, Rafał and Pietruszczak, Andrzej, Pieri's structures, Fundamenta Informaticae, vol. 81 (2007), no. 1-3, pp. 139–154.Google Scholar
[8] Grzegorczyk, Andrzej, Axiomatizability of geometry without points, Synthese, vol. 12 (1960), pp. 228–235.Google Scholar
[9] Hilbert, David, Foundations of geometry, The Open Court Publishing Company, La Salle, 1950.Google Scholar
[10] Jaśkowski, Stanisław, Une modification des définitions fundamentales de la géometrie des corps de M. A. Tarski, Annales de la Sociéte Polonaise de Mathématique, vol. XXI (1948), pp. 298–301.Google Scholar
[11] Jaśkowski, Stanisław, Geometria brył, Matematyka, vol. 1 (1949), no. 3, pp. 1–6.Google Scholar
[12] Koppelberg, Sabine, General theory of boolean algebras, Handbook of Boolean algebras (Monk, J. Donald and Bonnet, Robert, editors), vol. 1, North Holland, Amsterdam, 1989.Google Scholar
[13] Leśniewski, Stanisław, On the foundations of mathematics, Collected works (Surma, S.J. et al., editors), vol. I, Nijhoff International Philosophy Series, no. 44, Kluwer Academic Publishers, Dordrecht, 1991, pp. 174–382.Google Scholar
[14] Marchisotto, Elena Anne and Smith, James T. (editors), The legacy of Mario Pieri in geometry and arithmetic, Birkhäuser, Boston, 2007.Google Scholar
[15] Pieri, Mario, La geometria elementare istituita sulle nozioni “punto” è “sfera”, Matematica e di Fisica della Società Italiana delle Scienze, vol. 15 (1908), pp. 345–450.Google Scholar
[16] Pietruszczak, Andrzej, Metamereologia, Nicolaus Copernicus University Press, Toruń, 2000.Google Scholar
[17] Pietruszczak, Andrzej, Pieces of mereology, Logic and Logical Philosophy, vol. 14 (2005), no. 2, pp. 345–450.Google Scholar
[18] Roeper, Peter, Region-based topology, Journal of Philosophical Logic, vol. 26 (1997), pp. 251–309.CrossRefGoogle Scholar
[19] Schwabhäuser, Wolfram, Szmielew, Wanda, and Tarski, Alfred, Metamathematische Methoden in der Geometrie, Springer-Verlag, Berlin, 1983.Google Scholar
[20] Tarski, Alfred, Les fondements de la géometrié de corps, Ksiega Pamiatkowa Pierwszego Polskiego Zjazdu Matematycznego, supplement to Annales de la Societé Polonaise de Mathématique , Kraków, 1929, pp. 29–33.Google Scholar
[21] Tarski, Alfred, Foundations of the geometry of solids, Logic, semantics, metamathematics, papers from 1923 to 1938, Clarendon Press, Oxford, 1956, pp. 24–29.Google Scholar
[22] Tarski, Alfred, On the foundations of Boolean algebra, Logic, semantics, metamathematics, papers from 1923 to 1938, Clarendon Press, Oxford, 1956, pp. 320–341.Google Scholar
[23] Tarski, Alfred and Givant, Steven, Tarski's system of geometry, this Bulletin, vol. 5 (1999), no. 2, pp. 175–214.Google Scholar