## Bulletin of Symbolic Logic

### Full development of Tarski's geometry of solids

#### 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.

#### Article information

Source
Bull. Symbolic Logic Volume 14, Issue 4 (2008), 481-540.

Dates
First available in Project Euclid: 4 January 2009

http://projecteuclid.org/euclid.bsl/1231081462

Digital Object Identifier
doi:10.2178/bsl/1231081462

Mathematical Reviews number (MathSciNet)
MR2460676

Zentralblatt MATH identifier
1156.03011

#### Citation

Gruszczyński, Rafał; Pietruszczak, Andrzej. Full development of Tarski's geometry of solids. Bull. Symbolic Logic 14 (2008), no. 4, 481--540. doi:10.2178/bsl/1231081462. http://projecteuclid.org/euclid.bsl/1231081462.