The Notion of the Diameter of Mereological Ball in Tarski's Geometry of Solids

Grzegorz Sitek

DOI: http://dx.doi.org/10.12775/LLP.2017.016

Abstract


In  the paper "Full development of Tarski's geometry of solids" Gruszczyński and Pietruszczak have obtained the full development of Tarski’s geometry of solids that was sketched in [14, 15]. In this paper 1 we introduce in Tarski’s theory the notion of congruence of mereological balls and then the notion of diameter of mereological ball. We prove many facts about these new concepts, e.g., we give a characterization of mereological balls in terms of its center and diameter and we prove that the set of all diameters together with the relation of inequality of diameters is the dense linearly ordered set without the least and the greatest element.

Keywords


Tarski’s geometry of solids; mereology; diameter of mereological ball; congruence of mereological balls; point-free geometry

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References


Borsuk, K., and W. Szmielew, Foundations of geometry: Euclidean and Bolyai-Lobachevskian Geometry, Projective Geometry, North-Holland Publishing Company, Amsterdam, 1960.

Gruszczyński, R., and A. Pietruszczak, “Pieri’s structures”, Fundamenta Informaticae 81, 1–3 (2007): 1–16.

Gruszczyński, R., and A. Pietruszczak, “Full development of Tarski’s geometry of solids”, The Bulletin of Symbolic Logic 14, 4 (2008): 481–540. DOI: 10.2178/bsl/1231081462

Gruszczyński, R., and A. Pietruszczak, “Space, points and mereology. On foundations of point-free Euclidean geometry”, Logic and Logical Philosophy 18, 2 (2009): 145–188. DOI: 10.12775/LLP.2009.009

Gruszczyński, R., and A. Pietruszczak, “How to define a mereological (collective) set”, Logic and Logical Philosophy 19, 4 (2010): 309–328. DOI: 10.12775/LLP.2010.011

Gruszczyński, R., and A.C. Varzi, “Mereology then and now”, Logic and Logical Philosophy 24, 4 (2015): 409–427. DOI: 10.12775/LLP.2015.024

Leśniewski, S., “O podstawach matematyki”, Przegląd Filozoficzny XXX–XXXIV (1927–1931): 164–206, 261–291, 60–101, 77–105, 142–170.

Leśniewski, S., “On the foundations of mathematics”, pages 174–382 in Collected works, S.J. Surma et al (eds.), vol. I, Nijhoff International Philosophy Series, no. 44, Kluwer Academic Publishers, Dordrecht, 1991. English version of [6].

Pietruszczak, A., Metamereologia (Metamereology), Nicolaus Copernicus University Press, Toruń, 2000.

Pietruszczak, A., “Pieces of mereology”, Logic and Logical Philosophy 14, 2 (2005): 211–234. DOI: 10.12775/LLP.2005.014

Pietruszczak, A., Podstawy teorii części (Foundations of the theory of parthood), Nicolaus Copernicus University Scientific Publishing House, Toruń, 2013.

Pietruszczak, A., “A general concept of being a part of a whole”, Notre Dame Journal of Formal Logic 55, 3 (2014): 359–381. DOI: 10.1215/00294527-2688069

Pietruszczak, A., “Classical mereology is not elementarily axiomatizable”, Logic and Logical Philosophy 24, 4 (2015): 485–498. DOI: 10.12775/LLP.2015.017

Sitek, G., “Konstrukcje nowych pojęć w Tarskiego geometrii brył i ich zastosowanie w metaarytmetyce”, PhD thesis, Nicolaus Copernicus University in Toruń, 2016.

Tarski, A., “Les fondements de la géometrié de corps”, pages 29–33 in Księga Pamiątkowa Pierwszego Polskiego Zjazdu Matematycznego, supplement to Annales de la Societé Polonaise de Mathématique, Kraków, 1929.

Tarski, A., “Fundations of the geometry of solids”, pages 24–29 in Logic, Semantics, Metamathematics. Papers from 1923 to 1938, J.H. Woodger (ed.), Clarendon Press, Oxford, 1956. English version of [15].








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