Undecidable first-order theories of affine geometries (Q2871479)

In this paper, the authors refute a conjecture formulated by \textit{M. Aiello} and \textit{J. van Benthem} [J. Appl. Non-Class. Log. 12, No. 3--4, 319--363 (2002; Zbl 1185.03060)], by showing that, for \(n \geq 2\), the first-order theory of the class of all expansions of \(({\mathbb R}^n, \beta)\), where \(\beta\) is the ternary betweenness predicate, with a single unary predicate is highly undecidable (\(\Pi^1_1\)-hard). What's more, for a wide class of substructures \((T, \beta)\) of \(({\mathbb R}^n, \beta)\) with \(n\geq 2\) (including \(({\mathbb Q}^2 , \beta)\) and the real unit square \(([0, 1]^2 , \beta)\)), the following hold: (1) The first-order theory of the class of expansions of \((T, \beta)\) with a single unary predicate is \(\Sigma_{1}^0\)-hard; (2) The first-order theory of the class of expansions of \((T, \beta)\) with a single finite unary predicate is \(\Pi_{1}^0\) -hard.NEWLINENEWLINEThe paper also treats the expressiveness of universal weak monadic second order logic and universal monadic second order logic in general and over expansions of \(({\mathbb R}^n, \beta)\).