Abstract Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove that a definable group in an o-minimal structure is a product of cohesive orthogonal subsets. If the group has dimension one, or it is definably simple, then it is itself cohesive. As an application, we show that an abelian group definable in the disjoint union of finitely many o-minimal structures is a quotient, by a discrete normal subgroup, of a direct product of locally definable groups in the single structures.

Abstract Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$ , the identity function ${\mathbf {x}}$ , and such that whenever f and g are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below $2^{2^{\mathbf {x}}}$ . Here we prove that the set of asymptotic classes within any Archimedean class of Skolem functions has order type $\omega $ . As a consequence we obtain, for each positive integer n, an upper bound for the fragment below $2^{n^{\mathbf {x}}}$ . We deduce an epsilon-zero upper bound for the fragment below $2^{{\mathbf {x}}^{\mathbf {x}}}$ , improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway’s surreal number for asymptotic calculations.

Abstract By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.

Abstract By recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal “infinitesimal subgroup” G00 such that the quotient G/G00, equipped with the “logic topology”, is a compact (real) Lie group. Our first result is that the functor G ↦ G/G00 sends exact sequences of definably compact groups into exact sequences of Lie groups. We then study the connections between the Lie group G/G00 and the o-minimal spectrum of G. We prove that G/G00 is a topological quotient of . We thus obtain a natural homomorphism Ψ* from the cohomology of G/G00 to the (Čech-)cohomology of . We show that if G00 satisfies a suitable contractibility conjecture then is acyclic in Čech cohomology and Ψ is an isomorphism. Finally we prove the conjecture in some special cases.

Abstract LetMbe an o-minimal expansion of an ordered field. Letφbe a formula in the language of ordered domains. In this note we establish some topological properties which are transferred fromφMtoφRand vice versa. Then, we apply these transfer results to give a new proof of a result ofM. Edmundo—based on the work of A. Strzebonski—showing the existence of torsion points in any definably compact group defined in an o-minimal expansion of an ordered field.

Abstract Models of normal open induction are those normal discretely ordered rings whose nonnegative part satisfy Peano's axioms for open formulas in the language of ordered semirings. (Where normal means integrally closed in its fraction field.) In 1964 Shepherdson gave a recursive nonstandard model of open induction. His model is not normal and does not have any infinite prime elements. In this paper we present a recursive nonstandard model of normal open induction with an unbounded set of infinite prime elements.

Abstract PA is Peano arithmetic. The formula InterpPA(α, β) is a formalization of the assertion that the theory PA + α interprets the theory PA + β (the variables α and β are intended to range over codes of sentences of PA). We extend Solovay's modal analysis of the formalized provability predicate of PA, PrPA(x), to the case of the formalized interpretability relation InterpPA(x, y). The relevant modal logic, in addition to the usual provability operator ‘□’, has a binary operator ‘⊳’ to be interpreted as the formalized interpretability relation. We give an axiomatization and a decision procedure for the class of those modal formulas that express valid interpretability principles (for every assignment of the atomic modal formulas to sentences of PA). Our results continue to hold if we replace the base theory PA with Zermelo-Fraenkel set theory, but not with Gödel-Bernays set theory. This sensitivity to the base theory shows that the language is quite expressive. Our proof uses in an essential way earlier work done by A. Visser, D. de Jongh, and F. Veltman on this problem.