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December 2008 Full development of Tarski's geometry of solids
Rafał Gruszczyński, Andrzej Pietruszczak
Bull. Symbolic Logic 14(4): 481-540 (December 2008). DOI: 10.2178/bsl/1231081462

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.

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Rafał Gruszczyński. Andrzej Pietruszczak. "Full development of Tarski's geometry of solids." Bull. Symbolic Logic 14 (4) 481 - 540, December 2008. https://doi.org/10.2178/bsl/1231081462

Information

Published: December 2008
First available in Project Euclid: 4 January 2009

zbMATH: 1156.03011
MathSciNet: MR2460676
Digital Object Identifier: 10.2178/bsl/1231081462

Rights: Copyright © 2008 Association for Symbolic Logic

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Vol.14 • No. 4 • December 2008
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