Related papers: Externally definable sets and dependent pairs

Let $k,\ell\geq 2$ be two multiplicatively independent integers. Cobham's famous theorem states that a set $X\subseteq \mathbb{N}$ is both $k$-recognizable and $\ell$-recognizable if and only if it is definable in Presburger arithmetic.…

We describe the ind- and pro- categories of the category of definable sets, in some first order theory, in terms of points in a sufficiently saturated model.

The following conjecture is due to Shelah-Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non-trivial definable henselian valuation, in the language of rings. We specialise this conjecture to…

A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V=HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are…

The general theory of Lyapunov's stability of first-order differential inclusions in Hilbert spaces has been studied by the authors in a previous work. This new contribution focuses on the natural case when the maximally monotone operator…

We consider definable topological dynamics for $NIP$ groups admitting certain decompositions in terms of specific classes of definably amenable groups. For such a group, we find a description of the Ellis group of its universal definable…

In this paper we introduce the notion of existentially closed Leibniz algebras. Then we use HNN-extensions of Leibniz algebras in order to prove an embedding theorem.

For a NIP theory $T$, a sufficiently saturated model $\mathfrak{C}$ of $T$, and an invariant (over some small subset of $\mathfrak{C}$) global type $p$, we prove that there exists a finest relatively type-definable over a small set of…

Let $\cal R$ be an ordered vector space over an ordered division ring. We prove that every definable set $X$ is a finite union of relatively open definable subsets which are definably simply-connected, settling a conjecture from [5]. The…

We consider implicit definability of the standard part {0,1,...} in nonstandard models of Peano arithmetic (PA), and we ask whether there is a model of PA in which the standard part is implicitly definable. In section 1, we define a certain…

The classes stable, simple and NSOP$_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one…

This paper is a sequel to [1] and considers definability in differential-henselian monotone fields with c-map and angular component map. We prove an Equivalence Theorem among whose consequences are a relative quantifier reduction and an NIP…

We look at equivalence relations on the set of models of a theory -- MERs, for short -- such that the class of equivalent pairs is itself an elementary class, in a language appropriate for pairs of models. We provide many examples of…

In this paper we address the decision problem for a fragment of set theory with restricted quantification which extends the language studied in [4] with pair related quantifiers and constructs, in view of possible applications in the field…

Within Bishop Set Theory, a reconstruction of Bishop's theory of sets, we study the so-called completely separated sets, that is sets equipped with a positive notion of an inequality, induced by a given set of real-valued functions. We…

Many problems can be specified by patterns of propositional formulae depending on a parameter, e.g. the specification of a circuit usually depends on the number of bits of its input. We define a logic whose formulae, called "iterated…

Transfinite set theory including the axiom of choice supplies the following basic theorems: (1) Mappings between infinite sets can always be completed, such that at least one of the sets is exhausted. (2) The real numbers can be well…

The compactness theorem for a logic states, roughly, that the satisfiability of a set of well-formed formulas can be determined from the satisfiability of its finite subsets, and vice versa. Usually, proofs of this theorem depend on the…

We describe one.

Theory of stable models is the mathematical basis of answer set programming. Several results in that theory refer to the concept of the positive dependency graph of a logic program. We describe a modification of that concept and show that…