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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]
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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

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