We calculate the universal Euler characteristic and universal dimension function on semilinear and semibounded sets and obtain some criteria for definable equivalence of semilinear and semibounded sets in terms of these invariants.

Let L be a countable first-order language and M = (M, . . .) be an L-structure. “Definable set” means a subset of M which is L-definable in M with parameters. A set X ⊆ M is said to be immune if it is infinite and does not contain any infinite definable subset. X is said to be partially immune if for some definable A, A∩X is immune. X is said to be totally non-immune if for every definable A, A...

We study sets and groups definable in tame expansions of ominimal structures. Let M̃ = ⟨M, P ⟩ be an expansion of an o-minimal Lstructure M by a dense set P . We impose three tameness conditions on M̃ and prove a cone decomposition theorem for definable sets and functions in the realm of the o-minimal semi-bounded structures. The proof involves induction on the notion of ‘large dimension’ for def...

For G a group definable in some structure M , we define notions of “definable” compactification of G and “definable” action of G on a compact space X (definable G-flow), where the latter is under a definability of types assumption on M . We describe the universal definable compactification of G as G∗/(G∗)00 M and the universal definable G-ambit as the type space SG(M). We also point out the exi...

The set E of strings is said to be definable (strongly definable) if there is a context-free grammar G such that E is the set of all terminal strings generated from the initial symbol (from all non­ terminal symbols) of G. The classification of definable and strongly definable sets in dependence on minimal number of nonterminal symbols needed for their generation is given.

Let T be simple, work in Ceq over a boundedly closed set. Let p ∈ S(∅) be internal in a quasi-stably-embedded type-definable set Q (e.g. Q is definable or stably-embedded) and suppose (p,Q) is ACLembedded in Q (see definitions below). Then Aut(p/Q) = {σ|pC | σ ∈ Aut(C/Q)} with its action on pC is type-definable in Ceq over ∅. In particular, if p ∈ S(∅) is internal in a stably-embedded type-defi...

The only cardinality restriction is that the size of the collection of definable concepts is no greater than the size of the language (at most countably many in the case of a countable language). This restriction follows not from the diagonalization argument but from what it is to be a definable concept. Definable Basic Law V, which forces ^ to be injective only on the definable concepts, vivid...

An expansion of a definably complete field either defines a discrete subring, or the image of every definable discrete set under every definable map is nowhere dense. As an application we show a definable version of Lebesgue’s differentiation theorem.

We introduce bisimulations for the logic ITL with ◯ (‘next’), U (‘until’) and R (‘release’), an intuitionistic temporal logic based on structures (W,≼, S), where ≼ is used to interpret intuitionistic implication and S is a ≼-monotone function used to interpret the temporal modalities. Our main results are that ◇ (‘eventually’), which is definable in terms of U , cannot be defined in terms of ◯ ...

Given a subset of X ⊆ ℝn we can associate with every point x ∈ vector space V maximal dimension the property that for some ball centered at x, coincides inside union lines parallel to V. A is singular if has 0. In an earlier paper proved 〈ℝ, +, <, ℤ〉-definable relation 1〉-definable and only number points finite rational section 〈R, 1〉-definable, where set obtained from by fixing component va...