We give necessary and sufficient conditions on a non-oscillatory curve in an o-minimal structure such that, for any bounded definable function, there exists a definable closed set containing an initial segment of the curve on which the function is continuous. This question is translated into one on types: What are the conditions on an n-type such that, for any bounded definable function, there ...

Uniformization, choice functions and well orders in the class of trees. ABSTRACT The monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have a definable choice function (by a monadic formula with parameters)? A natural dichotomy arises where the trees that fall in the fir...

Let G be a compact definable C group and 1 ≤ r < ∞. Let X be an affine definable CG manifold and X1, . . . , Xk definable CG submanifolds of X such that X1, . . . , Xk are in general position in X. Suppose that f : X → R is a G invariant proper surjective submersive definable C function such that for every 1 ≤ i1 < · · · < is ≤ k, f |Xi1 ∩ · · · ∩ Xis : Xi1 ∩ · · · ∩ Xis → R is a proper surject...

Let R be an expansion of the ordered real additive group. Then one of the following holds: either every continuous function [0, 1] → R definable in R is C2 on an open dense subset of [0, 1], or every C2 function [0, 1] → R definable in R is affine, or every continuous function [0, 1] → R is definable in R. If R is NTP2 or more generally does not interpret a model of the monadic second order the...

Let M be an o-minimal expansion of a real closed field R. We define the notion of a lattice in a locally definable group and then prove that every connected, definably generated subgroup of 〈R,+〉 contains a definable generic set and therefore admits a lattice. The goal of this note is to re-formulate some problems which appeared in [4], introduce the notion of a lattice in a locally definable g...

We consider the logical theory of the monoid of subsets of N endowed solely with addition lifted to sets: no other set theoretical predicate or function, no constant (contrarily to previous work by Jeż and Okhotin cited below). We prove that the class of true Σ5 formulas is undecidable and that the whole theory is recursively isomorphic to second-order arithmetic. Also, each ultimately periodic...