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

• Authors: Sitek Grzegorz
• Languages of publication: EN

Abstracts

EN

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

EN

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

• Publisher: Uniwersytet Mikołaja Kopernika (Nicolaus Copernicus University in Toruń)
• Journal: Logic and Logical Philosophy
• Year: 2017
• Volume: 26
• Issue: 4
• Pages: 531–562

Dates

published

2017-12-15

Contributors

author

Sitek Grzegorz

• Department of Logic, Faculty of Humanities, Nicolaus Copernicus University in Toruń, Poland
• Institute of Philosophy, Faculty of Philology and History, Jan Długosz University in Częstochowa, Poland

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].

Identifiers

DOI

10.12775/LLP.2017.016