Authors: Antti Kuusisto ; Jeremy Meyers ; Jonni Virtema

Tarski initiated a logic-based approach to formal geometry that studies first-order structures with a ternary betweenness relation \beta, and a quaternary equidistance relation \equiv. Tarski established, inter alia, that the first-order (FO) theory of (R^2,\beta,\equiv) is decidable. Aiello and van Benthem (2002) conjectured that the FO-theory of expansions of (R^2,\beta) with unary predicates is decidable. We refute this conjecture by showing that for all n>1, the FO-theory of the class of expansions of (R^2,\beta) with just one unary predicate is not even arithmetical. We also define a natural and comprehensive class C of geometric structures (T,\beta), and show that for each structure (T,\beta) in C, the FO-theory of the class of expansions of (T,\beta) with a single unary predicate is undecidable. We then consider classes of expansions of structures (T,\beta) with a restricted unary predicate, for example a finite predicate, and establish a variety of related undecidability results. In addition to decidability questions, we briefly study the expressivities of universal MSO and weak universal MSO over expansions of (R^n,\beta). While the logics are incomparable in general, over expansions of (R^n,\beta), formulae of weak universal MSO translate into equivalent formulae of universal MSO.

Comment: 23 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1208.4930

• 03B25 - Decidability of theories and sets of sentences
• 03B30 - Foundations of classical theories (including reverse mathematics)
• 03D35 - Undecidability and degrees of sets of sentences
• 51E15 - Finite affine and projective planes (geometric aspects)
• 51G05 - Ordered geometries (ordered incidence structures, etc.)

• 1 zbMATH Open